Table of modular threefolds which are twisted fiber products of elliptic curve families
| Minimal twist form | Family A | Family B | Representative (a,b,c,d) |
|---|---|---|---|
| 5.4.a.a | B2_Gamma4cap2 | B4_Gamma1_6 | (1, -8, -1, 0) |
| 5.4.a.a | B3_Gamma1_5 | B3_Gamma1_5 | (1, 0, 0, 1) |
| 5.4.a.a | W_5_III_III_1 | W_5_III_III_1 | (1, 0, 0, 1) |
| 6.4.a.a | B2_Gamma4cap2 | B2_Gamma4cap2 | (4, 0, -1, -4) |
| 6.4.a.a | B2_Gamma4cap2 | W_2_2_IV_IV | (0, 8, -1, 0) |
| 6.4.a.a | B4_Gamma1_6 | B4_Gamma1_6 | (1, 0, 0, 1) |
| 6.4.a.a | B4_Gamma1_6 | HV_k_9 | (1, 1, 0, 1) |
| 6.4.a.a | HV_k_4 | Legendre_plane | (0, 16, 1, 0) |
| 6.4.a.a | J_Family | Legendre_plane | (0, 1, 1, 0) |
| 6.4.a.a | W_3_3_2_IV | W_3_3_2_IV | (1, 0, 0, 1) |
| 7.4.a.a | W_7_II_II | W_7_II_II | (1, 0, 0, 1) |
| 8.4.a.a | B2_Gamma4cap2 | B2_Gamma4cap2 | (1, 0, 0, 1) |
| 8.4.a.a | B4_Gamma1_6 | Legendre_plane | (0, 1, -1, 0) |
| 8.4.a.a | Legendre_plane | Legendre_plane | (1, 0, 0, 1) |
| 8.4.a.a | Legendre_plane | MP_JLR_X321 | (1, 0, 0, 1) |
| 8.4.a.a | W_4_2_III_III | W_4_2_III_III | (1, 0, 0, 1) |
| 8.4.a.a | W_4_4_II_II | W_4_4_II_II | (1, 0, 0, 1) |
| 8.4.a.a | W_4_III_III_II | W_4_III_III_II | (1, 0, 0, 1) |
| 8.4.a.a | W_4_II_III_III | W_4_II_III_III | (1, 0, 0, 1) |
| 9.4.a.a | B1_Gamma3_Hesse | B1_Gamma3_Hesse | (1, 0, 0, 1) |
| 9.4.a.a | B2_Gamma4cap2 | MP_JLR_X431 | (1, 0, 1, -8) |
| 9.4.a.a | MP_JLR_X321 | MP_JLR_X431 | (1, 0, 0, 1) |
| 9.4.a.a | W_3_3_III_III | W_3_3_III_III | (1, 0, 0, 1) |
| 9.4.a.a | W_3_III_III_III | W_3_III_III_III | (1, 0, 0, 1) |
| 10.4.a.a | B4_Gamma1_6 | B4_Gamma1_6 | (1, 0, -1, -1) |
| 10.4.a.a | HV_1_9_9 | HV_k_9 | (1, 0, 0, 1) |
| 10.4.a.a | W_5_2_1_IV | W_5_2_1_IV | (1, 0, 0, 1) |
| 10.4.a.a | W_5_3_2_II | W_5_3_2_II | (1, 0, 0, 1) |
| 12.4.a.a | B2_Gamma4cap2 | B2_Gamma4cap2 | (1, -8, 0, -1) |
| 12.4.a.a | B2_Gamma4cap2 | W_3_3_2_IV | (0, 8, -1, 0) |
| 12.4.a.a | B4_Gamma1_6 | HV_1_4_4 | (1, 1, 0, 1) |
| 12.4.a.a | HV_k_4 | HV_k_4 | (1, 0, 0, 1) |
| 12.4.a.a | HV_k_4 | Legendre_plane | (0, 4, 1, 0) |
| 12.4.a.a | J_Family | MP_JLR_X431 | (0, 1, 1, 0) |
| 12.4.a.a | Legendre_plane | MP_JLR_X431 | (1, 0, 0, 1) |
| 13.4.a.a | W_7_II_II | W_7_II_II | (1, 0, 26, -1) |
| 14.4.a.b | B4_Gamma1_6 | B4_Gamma1_6 | (1, 0, 0, -1) |
| 14.4.a.b | W_7_1_1_III | W_7_1_1_III | (1, 0, 0, 1) |
| 15.4.a.a | W_5_1_II_IV | W_5_1_II_IV | (1, 0, 0, 1) |
| 17.4.a.a | B4_Gamma1_6 | B4_Gamma1_6 | (1, 1, 0, -1) |
| 20.4.a.a | Legendre_plane | MP_JLR_X431 | (0, 27, 5, 27) |
| 21.4.a.a | B4_Gamma1_6 | B4_Gamma1_6 | (0, 1, 1, 0) |
| 21.4.a.a | C3_u_2_norm | C3_u_2_norm | (1, 2, 0, -1) |
| 21.4.a.b | W_4_2_II_IV | W_4_2_II_IV | (1, 0, 9, -8) |
| 21.4.a.b | W_6_1_1_II | W_6_1_1_II | (1, 0, 0, -1) |
| 22.4.a.a | B3_Gamma1_5 | B3_Gamma1_5 | (0, 1, 1, 0) |
| 22.4.a.a | B3_Gamma1_5 | B3_Gamma1_5 | (2, 11, 11, -2) |
| 24.4.a.a | B4_Gamma1_6 | Legendre_plane | (0, 1, -1, 0) |
| 24.4.a.a | HV_k_4 | Legendre_plane | (1, -16, -3, 0) |
| 24.4.a.a | HV_k_9 | Legendre_plane | (1, 0, 0, 1) |
| 24.4.a.a | Legendre_plane | Legendre_plane | (0, 1, -3, 0) |
| 24.4.a.a | Legendre_plane | MP_JLR_X211 | (1, 0, 0, 1) |
| 24.4.a.a | W_2_2_IV_IV | W_2_2_IV_IV | (1, 0, 0, 1) |
| 24.4.a.a | W_4_2_II_IV | W_4_2_II_IV | (1, 0, 0, 1) |
| 24.4.a.a | W_6_1_1_II | W_6_1_1_II | (1, 0, 0, 1) |
| 26.4.a.b | W_7_1_1_III | W_7_1_1_III | (1, 0, 0, -1) |
| 27.4.a.a | B1_Gamma3_Hesse | B1_Gamma3_Hesse | (1, 3, 0, -1) |
| 27.4.a.a | B4_Gamma1_6 | MP_JLR_X431 | (1, 0, 0, -1) |
| 27.4.a.a | MP_JLR_X431 | MP_JLR_X431 | (0, 1, 1, 0) |
| 27.4.a.a | W_3_3_2_IV | W_3_3_2_IV | (0, 1, -1, 0) |
| 28.4.a.a | B2_Gamma4cap2 | B4_Gamma1_6 | (1, 0, 0, 1) |
| 28.4.a.b | W_4_3_II_III | W_4_3_II_III | (1, 0, 0, 1) |
| 30.4.a.a | B4_Gamma1_6 | HV_1_1_25 | (1, 1, 0, 1) |
| 30.4.a.b | B4_Gamma1_6 | HV_1_1_16 | (4, 4, 0, 1) |
| 30.4.a.b | HV_1_4_16 | HV_1_4_4 | (1, 0, 0, 1) |
| 30.4.a.b | HV_k_4 | HV_k_4 | (1, 0, 0, 4) |
| 30.4.a.b | Legendre_plane | W_5_3_2_II | (3, 0, 1, -1) |
| 30.4.a.b | W_5_3_2_II | W_7_2_1_II | (1, -3, 0, 4) |
| 30.4.a.b | W_5_3_2_II | W_7_2_1_II | (3, -9, 0, 4) |
| 32.4.a.a | B2_Gamma4cap2 | Legendre_plane | (0, 8, -1, 0) |
| 32.4.a.a | B4_Gamma1_6 | MP_JLR_X321 | (0, 1, -1, 0) |
| 32.4.a.a | HV_k_-8 | HV_k_-8 | (1, 8, 0, -1) |
| 32.4.a.a | Legendre_plane | Legendre_plane | (0, 1, -1, 0) |
| 32.4.a.a | Legendre_plane | W_5_2_1_IV | (0, 1, -2, 0) |
| 32.4.a.a | MP_JLR_X321 | MP_JLR_X321 | (0, 1, 1, 0) |
| 32.4.a.b | B2_Gamma4cap2 | Legendre_plane | (0, 8, -1, 0) |
| 32.4.a.b | Legendre_plane | W_3_3_2_IV | (1, -1, 1, 0) |
| 40.4.a.a | HV_k_-4 | HV_k_-4 | (1, 0, 0, -1) |
| 40.4.a.a | HV_k_4 | HV_k_4 | (1, 0, 0, -1) |
| 40.4.a.a | J_Family | W_5_3_2_II | (0, 3, -1, 1) |
| 40.4.a.a | MP_JLR_X321 | W_5_3_2_II | (32, -27, 0, -9) |
| 40.4.a.b | W_4_4_II_II | W_4_4_II_II | (2, -5, 0, -2) |
| 42.4.a.b | W_7_2_1_II | W_7_2_1_II | (1, 0, 0, 1) |
| 46.4.a.a | W_5_2_1_IV | W_5_2_1_IV | (1, 0, -2, -1) |
| 46.4.a.b | W_3_3_2_IV | W_5_2_1_IV | (1, -1, -2, 0) |
| 54.4.a.a | B1_Gamma3_Hesse | B1_Gamma3_Hesse | (1, -6, 0, -1) |
| 54.4.a.a | B1_Gamma3_Hesse | B1_Gamma3_Hesse | (3, 0, -1, -3) |
| 54.4.a.a | B4_Gamma1_6 | MP_JLR_X431 | (0, 1, -1, 0) |
| 54.4.a.a | W_2_2_IV_IV | W_3_3_2_IV | (1, 0, 0, 1) |
| 54.4.a.a | W_3_3_III_III | W_3_3_III_III | (3, -2, 0, -3) |
| 54.4.a.a | W_4_IV_II_II | W_4_IV_II_II | (0, 1, -1, 0) |
| 54.4.a.a | W_6_II_II_II | W_6_II_II_II | (0, 1, 1, 0) |
| 54.4.a.b | B1_Gamma3_Hesse | B1_Gamma3_Hesse | (0, 18, -1, 0) |
| 54.4.a.b | HV_k_-8 | MP_JLR_X431 | (1, 0, 0, -8) |
| 54.4.a.b | MP_JLR_X431 | MP_JLR_X431 | (0, 1, -8, 0) |
| 54.4.a.b | W_2_2_IV_IV | W_2_2_IV_IV | (1, -1, -1, -1) |
| 54.4.a.b | W_3_III_III_III | W_3_III_III_III | (0, 1, 1, 0) |
| 55.4.a.a | B3_Gamma1_5 | B3_Gamma1_5 | (0, 1, 1, 11) |
| 58.4.a.b | W_5_2_1_IV | W_5_2_1_IV | (1, 0, 0, -1) |
| 60.4.a.a | B4_Gamma1_6 | B4_Gamma1_6 | (1, 2, 0, -1) |
| 60.4.a.a | HV_1_4_4 | HV_k_9 | (1, 0, 0, 1) |
| 60.4.a.a | HV_1_4_9 | HV_k_4 | (1, 0, 0, 1) |
| 60.4.a.a | HV_k_-1 | HV_k_-1 | (1, 0, 0, -1) |
| 60.4.a.a | MP_JLR_X431 | W_5_3_2_II | (3, 0, 1, -1) |
| 68.4.a.a | B2_Gamma4cap2 | B4_Gamma1_6 | (1, -8, 0, -1) |
| 72.4.a.a | MP_JLR_X211 | MP_JLR_X321 | (1, 0, 0, 1) |
| 72.4.a.a | W_4_IV_II_II | W_4_IV_II_II | (1, 0, 0, 1) |
| 73.4.a.a | B4_Gamma1_6 | B4_Gamma1_6 | (0, 1, -1, -1) |
| 78.4.a.b | HV_k_4 | HV_k_4 | (4, 0, 1, -3) |
| 78.4.a.c | W_5_2_1_IV | W_5_2_1_IV | (27, 0, -4, 25) |
| 78.4.a.d | W_3_3_2_IV | W_5_2_1_IV | (1, -1, -4, 0) |
| 88.4.a.a | B2_Gamma4cap2 | B3_Gamma1_5 | (0, 8, -1, 0) |
| 96.4.a.a | HV_k_4 | HV_k_4 | (4, 0, 1, -4) |
| 96.4.a.b | B2_Gamma4cap2 | Legendre_plane | (1, -8, -1, 0) |
| 96.4.a.b | B4_Gamma1_6 | MP_JLR_X321 | (1, 1, 0, 1) |
| 96.4.a.b | J_Family | J_Family | (1, 0, 1, -1) |
| 96.4.a.b | J_Family | MP_JLR_X321 | (1, 0, 0, 1) |
| 96.4.a.b | Legendre_plane | MP_JLR_X211 | (0, 1, -1, 0) |
| 96.4.a.b | Legendre_plane | W_3_3_2_IV | (1, -2, 1, 0) |
| 96.4.a.c | HV_1_1_16 | MP_JLR_X321 | (1, 0, 0, 4) |
| 98.4.a.e | B4_Gamma1_6 | B4_Gamma1_6 | (1, -8, 1, 1) |
| 102.4.a.b | W_4_2_II_IV | W_5_1_II_IV | (3, 0, 0, 1) |
| 102.4.a.c | B4_Gamma1_6 | B4_Gamma1_6 | (1, 0, -2, -1) |
| 102.4.a.d | W_7_2_1_II | W_7_2_1_II | (8, 0, -9, -8) |
| 108.4.a.a | B2_Gamma4cap2 | MP_JLR_X431 | (0, 8, -1, 0) |
| 108.4.a.a | MP_JLR_X211 | MP_JLR_X431 | (1, 0, 0, 1) |
| 108.4.a.a | MP_JLR_X431 | MP_JLR_X431 | (1, 0, 2, -1) |
| 108.4.a.a | W_3_II_III_IV | W_3_II_III_IV | (1, 0, 0, 1) |
| 108.4.a.b | W_6_II_II_II | W_6_II_II_II | (1, 0, 0, 1) |
| 110.4.a.g | B3_Gamma1_5 | B3_Gamma1_5 | (1, -1, 1, 1) |
| 118.4.a.b | W_5_3_2_II | W_5_3_2_II | (1, -6, 0, -1) |
| 120.4.a.b | HV_k_4 | HV_k_4 | (2, 0, -1, 6) |
| 120.4.a.c | HV_k_-4 | HV_k_-4 | (0, 16, 1, 0) |
| 120.4.a.c | W_4_4_II_II | W_4_4_II_II | (10, 1, 8, -10) |
| 120.4.a.c | W_4_III_III_II | W_4_III_III_II | (54, -5, 0, -54) |
| 126.4.a.b | W_4_2_II_IV | W_4_2_II_IV | (1, 0, 2, -1) |
| 128.4.a.a | B2_Gamma4cap2 | MP_JLR_X321 | (0, 8, -1, 0) |
| 128.4.a.a | MP_JLR_X321 | MP_JLR_X321 | (1, -2, 0, -1) |
| 130.4.a.b | W_5_2_1_IV | W_5_2_1_IV | (1, 0, -4, -1) |
| 162.4.a.a | B4_Gamma1_6 | MP_JLR_X431 | (1, 1, 1, 4) |
| 162.4.a.a | HV_k_4 | MP_JLR_X431 | (0, 4, 1, 0) |
| 162.4.a.a | W_3_3_2_IV | W_3_3_2_IV | (0, 1, 2, -1) |
| 162.4.a.b | HV_k_4 | MP_JLR_X431 | (0, 16, 1, 0) |
| 162.4.a.b | W_3_3_2_IV | W_3_3_2_IV | (0, 1, -1, 2) |
| 168.4.a.b | HV_k_4 | HV_k_4 | (2, 0, 1, -2) |
| 168.4.a.e | HV_k_4 | HV_k_4 | (12, 0, -1, 16) |
| 168.4.a.f | W_6_1_1_II | W_6_1_1_II | (27, 0, 14, -27) |
| 174.4.a.d | W_3_3_2_IV | W_5_2_1_IV | (1, -1, -2, -2) |
| 180.4.a.b | MP_JLR_X321 | MP_JLR_X431 | (4, 0, 0, -1) |
| 182.4.a.d | W_7_1_1_III | W_7_1_1_III | (32, 0, -13, -32) |
| 210.4.a.d | HV_k_9 | HV_k_9 | (1, 0, 0, 9) |
| 216.4.a.a | B4_Gamma1_6 | MP_JLR_X211 | (0, 1, 1, 1) |
| 216.4.a.b | B4_Gamma1_6 | MP_JLR_X211 | (1, 0, 1, 1) |
| 216.4.a.b | MP_JLR_X211 | MP_JLR_X211 | (0, 1, -1, 1) |
| 243.4.a.a | B4_Gamma1_6 | MP_JLR_X431 | (0, 1, 1, 1) |
| 256.4.a.a | B2_Gamma4cap2 | MP_JLR_X321 | (1, -8, 1, 0) |
| 256.4.a.b | B4_Gamma1_6 | MP_JLR_X321 | (1, 0, 0, -1) |
| 288.4.a.d | B2_Gamma4cap2 | MP_JLR_X211 | (0, 8, -1, 0) |
| 324.4.a.a | B4_Gamma1_6 | MP_JLR_X431 | (1, 1, 1, -2) |
| 324.4.a.a | MP_JLR_X321 | MP_JLR_X431 | (0, 4, -1, 1) |
| 330.4.a.h | W_5_3_2_II | W_5_3_2_II | (0, 9, 1, 0) |
| 370.4.a.b | W_5_3_2_II | W_5_3_2_II | (32, 0, 9, 5) |
| 384.4.a.b | HV_k_4 | HV_k_4 | (8, 0, 1, -8) |
| 390.4.a.j | W_7_2_1_II | W_7_2_1_II | (9, 0, -4, -9) |
| 480.4.a.b | HV_k_4 | HV_k_4 | (4, 0, -1, -4) |
| 486.4.a.e | HV_k_4 | MP_JLR_X431 | (1, -16, -3, 0) |
| 678.4.a.a | W_7_2_1_II | W_7_2_1_II | (1, 0, 0, -1) |
| 864.4.a.a | B2_Gamma4cap2 | MP_JLR_X211 | (1, -8, 1, 0) |
| 864.4.a.a | J_Family | J_Family | (1, -1, 0, -1) |
| 864.4.a.a | MP_JLR_X211 | MP_JLR_X211 | (0, 1, -1, 0) |
| 870.4.a.h | W_5_2_1_IV | W_5_2_1_IV | (2, 0, 0, 27) |
Table of paramodular threefolds which are twisted fiber products of elliptic curve families
| Minimal twist form | Family A | Family B | Representative (a,b,c,d) |
|---|---|---|---|
| 82.a | B4_Gamma1_6 | HV_k_-1 | (1, 0, 0, 1) |
| 96.a | Legendre_plane | Legendre_plane | (2, 0, 1, -2) |
| 105.a | B2_Gamma4cap2 | B4_Gamma1_6 | (1, -6, 0, -2) |
| 133.b | B2_Gamma4cap2 | B1_Gamma3_Hesse | (1, -2, 0, 2) |
| 140.a | B4_Gamma1_6 | B4_Gamma1_6 | (1, -1, 0, 1) |
| 150.a | W_5_2_1_IV | W_5_3_2_II | (4, 27, 0, 9) |
| 160.a | B4_Gamma1_6 | HV_k_-1 | (1, 1, 0, -1) |
| 160.a | HV_k_-1 | HV_k_-1 | (1, 0, 0, 1) |
| 160.b | Legendre_plane | MP_JLR_X321 | (4, 0, 0, -1) |
| 162.a | B4_Gamma1_6 | W_5_2_1_IV | (1, 1, 0, -2) |
| 172.a | B4_Gamma1_6 | W_5_2_1_IV | (1, 0, 0, 2) |
| 182.a | B1_Gamma3_Hesse | B1_Gamma3_Hesse | (0, 1, 1, 0) |
| 192.a | HV_k_9 | Legendre_plane | (1, 0, 1, -9) |
| 192.b | Legendre_plane | MP_JLR_X321 | (4, 0, 0, 1) |
| 209.b | B3_Gamma1_5 | B4_Gamma1_6 | (1, -1, -1, 0) |
| 224.b | HV_k_8 | HV_k_8 | (1, 0, 0, 1) |
| 240.a | B2_Gamma4cap2 | Legendre_plane | (1, 0, 0, -2) |
| 240.a | MP_JLR_X321 | MP_JLR_X321 | (1, -1, -15, -1) |
| 240.d | HV_k_-1 | HV_k_-1 | (0, 1, 1, 0) |
| 255.b | B2_Gamma4cap2 | B2_Gamma4cap2 | (0, 4, -1, 0) |
| 270.b | HV_k_-3 | HV_k_-3 | (1, -2, 0, -1) |
| 275.a | B3_Gamma1_5 | B4_Gamma1_6 | (0, 1, 1, 0) |
| 315.b | B2_Gamma4cap2 | B2_Gamma4cap2 | (1, -2, 0, -1) |
| 320.d | Legendre_plane | MP_JLR_X321 | (1, 0, 0, -4) |
| 360.a | B2_Gamma4cap2 | B4_Gamma1_6 | (1, 0, 0, -2) |
| 360.g | B4_Gamma1_6 | B4_Gamma1_6 | (2, 0, -1, -2) |
| 360.g | W_3_3_2_IV | W_4_2_II_IV | (1, -1, -1, -1) |
| 368.i | B2_Gamma4cap2 | W_5_2_1_IV | (1, 0, -1, -8) |
| 378.a | HV_k_-3 | HV_k_-3 | (1, 0, 0, 1) |
| 378.b | B2_Gamma4cap2 | W_2_2_IV_IV | (1, -1, -1, -1) |
| 378.b | C3_u_2_norm | C3_u_2_norm | (1, 0, 0, 1) |
| 384.a | HV_k_-2 | HV_k_-2 | (1, 0, 0, -1) |
| 384.a | HV_k_2 | HV_k_2 | (1, 0, 0, -1) |
| 384.d | Legendre_plane | W_6_1_1_II | (0, 1, 1, 0) |
| 384.e | B2_Gamma4cap2 | B2_Gamma4cap2 | (1, 0, 0, -2) |
| 384.e | HV_k_4 | Legendre_plane | (1, 0, 0, 4) |
| 384.e | Legendre_plane | Legendre_plane | (1, 3, 1, -1) |
| 384.g | B2_Gamma4cap2 | Legendre_plane | (1, -4, 0, 4) |
| 384.g | Legendre_plane | W_2_2_IV_IV | (1, -1, -1, -1) |
| 396.e | B4_Gamma1_6 | HV_k_9 | (3, 1, 0, 1) |
| 417.a | B1_Gamma3_Hesse | Legendre_plane | (0, 16, -1, 3) |
| 420.a | B2_Gamma4cap2 | B2_Gamma4cap2 | (4, 0, -3, -4) |
| 420.a | B4_Gamma1_6 | HV_k_9 | (9, 9, 0, 1) |
| 420.a | HV_1_9_9 | HV_1_9_9 | (1, 0, 0, 1) |
| 436.a | B4_Gamma1_6 | W_5_2_1_IV | (1, 1, -2, 16) |
| 448.a | HV_k_2 | HV_k_2 | (1, -4, 0, -1) |
| 448.g | B4_Gamma1_6 | Legendre_plane | (2, -7, 2, -16) |
| 448.l | B4_Gamma1_6 | Legendre_plane | (1, 1, 0, 8) |
| 475.a | B3_Gamma1_5 | B4_Gamma1_6 | (1, 8, 0, 1) |
| 476.d | B4_Gamma1_6 | B4_Gamma1_6 | (1, 0, -1, 1) |
| 480.b | B2_Gamma4cap2 | B2_Gamma4cap2 | (1, 0, 0, -4) |
| 480.i | B2_Gamma4cap2 | B2_Gamma4cap2 | (1, -24, 0, -2) |
| 482.a | B4_Gamma1_6 | W_5_2_1_IV | (1, 0, -2, 16) |
| 486.a | B1_Gamma3_Hesse | B1_Gamma3_Hesse | (0, 9, -1, -3) |
| 486.a | MP_JLR_X431 | W_3_3_2_IV | (1, -1, -1, -1) |
| 510.b | B2_Gamma4cap2 | HV_k_9 | (9, 0, 0, -8) |
| 513.a | B1_Gamma3_Hesse | B4_Gamma1_6 | (1, -2, 0, -1) |
| 513.c | B1_Gamma3_Hesse | B4_Gamma1_6 | (0, 3, -1, 0) |
| 520.f | B4_Gamma1_6 | HV_k_-1 | (1, 1, 0, 1) |
| 532.d | B1_Gamma3_Hesse | B1_Gamma3_Hesse | (1, 2, 0, -1) |
| 540.d | B2_Gamma4cap2 | B2_Gamma4cap2 | (3, -16, 0, -3) |
| 540.d | HV_k_9 | HV_k_9 | (1, 0, 0, 1) |
| 544.c | B4_Gamma1_6 | Legendre_plane | (2, 0, 0, -1) |
| 544.h | Legendre_plane | Legendre_plane | (1, -17, 1, -1) |
| 553.a | B2_Gamma4cap2 | B1_Gamma3_Hesse | (1, -6, 0, -2) |
| 560.n | B4_Gamma1_6 | B4_Gamma1_6 | (2, 2, -1, -2) |
| 580.a | W_5_2_1_IV | W_5_2_1_IV | (0, 1, 4, 0) |
| 585.b | B2_Gamma4cap2 | B2_Gamma4cap2 | (1, -10, 0, -1) |
| 585.e | B2_Gamma4cap2 | HV_k_9 | (1, 0, 0, -8) |
| 594.a | MP_JLR_X431 | W_5_3_2_II | (3, 3, 1, -1) |
| 600.q | Legendre_plane | Legendre_plane | (5, 3, 5, 4) |
| 603.h | B1_Gamma3_Hesse | Legendre_plane | (3, 7, 3, -9) |
| 612.b | B4_Gamma1_6 | HV_k_-8 | (1, 0, 0, -1) |
| 612.h | B4_Gamma1_6 | HV_k_-8 | (1, -8, 0, 1) |
| 640.c | HV_k_2 | HV_k_2 | (2, -4, 5, -2) |
| 640.d | HV_k_-4 | HV_k_-4 | (1, 0, 0, 1) |
| 640.j | HV_k_-4 | Legendre_plane | (1, 0, 0, -4) |
| 640.j | Legendre_plane | Legendre_plane | (1, -5, 1, -1) |
| 640.l | HV_k_2 | HV_k_2 | (10, -4, 1, -2) |
| 640.n | HV_k_-4 | HV_k_4 | (1, 0, 0, 1) |
| 640.t | B2_Gamma4cap2 | HV_k_-1 | (1, 0, 0, -8) |
| 650.a | W_5_2_1_IV | W_5_2_1_IV | (2, 1, 0, -2) |
| 656.i | HV_k_-1 | HV_k_-1 | (1, 2, 0, -1) |
| 672.a | MP_JLR_X321 | MP_JLR_X321 | (1, -9, -7, -1) |
| 672.h | B2_Gamma4cap2 | B2_Gamma4cap2 | (0, 8, -1, 0) |
| 672.j | HV_k_-1 | HV_k_-1 | (1, -6, 0, -1) |
| 703.b | B1_Gamma3_Hesse | B4_Gamma1_6 | (1, -3, 0, -1) |
| 704.v | B3_Gamma1_5 | Legendre_plane | (0, 1, 1, 0) |
| 720.b | B4_Gamma1_6 | B4_Gamma1_6 | (1, 0, 0, -2) |
| 720.l | B2_Gamma4cap2 | B2_Gamma4cap2 | (2, 0, -1, -2) |
| 720.l | B4_Gamma1_6 | HV_k_4 | (4, 4, 0, 1) |
| 720.l | HV_1_4_4 | HV_1_4_4 | (1, 0, 0, 1) |
| 725.a | B2_Gamma4cap2 | B3_Gamma1_5 | (0, 2, -1, 8) |
| 730.d | B1_Gamma3_Hesse | B1_Gamma3_Hesse | (0, 1, -1, 0) |
| 750.d | B3_Gamma1_5 | B3_Gamma1_5 | (1, 3, 8, -1) |
| 765.c | B4_Gamma1_6 | Legendre_plane | (1, 17, 1, 1) |
| 768.a | B2_Gamma4cap2 | HV_k_4 | (1, -8, 0, -2) |
| 768.h | B2_Gamma4cap2 | Legendre_plane | (3, -8, -1, 8) |
| 768.i | B2_Gamma4cap2 | HV_k_4 | (1, -8, 0, -1) |
| 768.t | Legendre_plane | Legendre_plane | (1, 0, -8, 8) |
| 768.t | Legendre_plane | MP_JLR_X321 | (1, 0, 0, 4) |
| 814.a | W_7_1_1_III | W_7_1_1_III | (1, 0, 0, 2) |
| 814.b | W_5_3_2_II | W_5_3_2_II | (1, 3, 0, 2) |
| 832.k | B2_Gamma4cap2 | W_5_2_1_IV | (1, 0, -2, -16) |
| 840.h | B4_Gamma1_6 | B4_Gamma1_6 | (1, -6, 0, -1) |
| 868.h | B1_Gamma3_Hesse | B1_Gamma3_Hesse | (0, 8, 1, 0) |
| 896.c | HV_k_2 | HV_k_2 | (1, 0, 1, -1) |
| 896.g | B4_Gamma1_6 | Legendre_plane | (1, -7, 0, 1) |
| 908.a | B4_Gamma1_6 | W_5_2_1_IV | (1, 0, -2, -2) |
| 910.b | B4_Gamma1_6 | B4_Gamma1_6 | (0, 1, -1, 0) |
| 924.f | B4_Gamma1_6 | HV_k_9 | (1, 1, 1, -7) |
| 928.a | B2_Gamma4cap2 | W_5_2_1_IV | (1, -8, -2, -16) |
| 944.g | B3_Gamma1_5 | MP_JLR_X321 | (1, 0, 0, 4) |
| 945.f | B4_Gamma1_6 | Legendre_plane | (16, 0, 1, -8) |
| 945.k | B2_Gamma4cap2 | B2_Gamma4cap2 | (9, 0, 0, -16) |
| 952.p | B1_Gamma3_Hesse | B1_Gamma3_Hesse | (0, 1, 2, 0) |
| 960.g | HV_k_2 | HV_k_2 | (1, -12, 0, -1) |
| 960.h | B4_Gamma1_6 | Legendre_plane | (1, 1, 0, 4) |
| 960.h | HV_k_4 | Legendre_plane | (1, 0, 0, 1) |
| 960.h | Legendre_plane | Legendre_plane | (8, -3, 8, 1) |
| 960.l | B2_Gamma4cap2 | B2_Gamma4cap2 | (8, 0, -3, -8) |
| 960.p | B2_Gamma4cap2 | HV_k_4 | (1, 0, 0, -2) |
| 960.x | B2_Gamma4cap2 | Legendre_plane | (1, -8, 0, 2) |
| 975.c | B2_Gamma4cap2 | B4_Gamma1_6 | (1, -2, 0, 2) |
| 990.e | W_3_3_2_IV | W_5_3_2_II | (0, 3, 1, 0) |
Family equations used in the tables
| Family | Equation | Allowed (a,b,c,d) |
|---|---|---|
| B1_Gamma3_Hesse | x^3+y^3+z^3 = t xyz | (1, 0, 0, 1); (3, 18, 1, -3) |
| B2_Gamma4cap2 | (x+y)(x^2+y^2+2xy+4z^2+4yz-4xz)=txyz | (1, 0, 0, 1); (1, 0, 0, -1); (0, 64, -1, 0); (0, 64, 1, 0); (8, -64, -1, -8); (8, -64, 1, 8); (8, 64, -1, 8); (8, 64, 1, -8) |
| B3_Gamma1_5 | (x+y)(x+y-z)(y-z)=txyz | (1, 0, 0, 1); (0, 1, -1, 0) |
| B4_Gamma1_6 | (x+y)(y+z)(z+x)=txyz | (1, 0, 0, 1); (0, 8, -1, 0); (1, -8, -1, -1); (8, 8, 1, -8) |
| C3_u_2_norm | 2(A+2B-9xyz)=27t xyz, A=x^2y+y^2z+z^2x, B=xy^2+yz^2+zx^2 | (1, 0, 0, 1) |
| HV_1_1_16 | (x+y+z)(xy+xz+16yz)=txyz | (1, 0, 0, 1) |
| HV_1_1_25 | (x+y+z)(xy+xz+25yz)=txyz | (1, 0, 0, 1) |
| HV_1_4_16 | (x+y+z)(xy+4xz+16yz)=txyz | (1, 0, 0, 1) |
| HV_1_4_4 | (x+y+z)(xy+4xz+4yz)=txyz | (1, 0, 0, 1) |
| HV_1_4_9 | (x+y+z)(xy+4xz+9yz)=txyz | (1, 0, 0, 1) |
| HV_1_9_9 | (x+y+z)(xy+9xz+9yz)=txyz | (1, 0, 0, 1) |
| HV_k_-1 | (x+y+z)(xy+xz-1 yz)=txyz | (1, 0, 0, 1) |
| HV_k_-2 | (x+y+z)(xy+xz-2 yz)=txyz | (1, 0, 0, 1) |
| HV_k_-3 | (x+y+z)(xy+xz-3 yz)=txyz | (1, 0, 0, 1) |
| HV_k_-4 | (x+y+z)(xy+xz-4 yz)=txyz | (1, 0, 0, 1) |
| HV_k_-8 | (x+y+z)(xy+xz-8 yz)=txyz | (1, 0, 0, 1) |
| HV_k_2 | (x+y+z)(xy+xz+2 yz)=txyz | (1, 0, 0, 1) |
| HV_k_4 | (x+y+z)(xy+xz+4 yz)=txyz | (1, 0, 0, 1); (0, 64, 1, 0); (4, -64, 1, -4); (16, -64, 1, -16) |
| HV_k_8 | (x+y+z)(xy+xz+8 yz)=txyz | (1, 0, 0, 1) |
| HV_k_9 | (x+y+z)(xy+xz+9 yz)=txyz | (1, 0, 0, 1) |
| J_Family | Y^2 Z = X^3 - 3t(t-1)^3 XZ^2 + 2t(t-1)^5 Z^3 | (1, 0, 0, 1); (0, 1, 1, 0); (0, 1, -1, 1); (1, -1, 0, -1); (1, -1, 1, 0); (1, 0, 1, -1) |
| Legendre_plane | y^2z = x(x-z)(x-tz) | (1, 0, 0, 1); (0, 1, 1, 0); (0, 1, -1, 1); (1, -1, 0, -1); (1, -1, 1, 0); (1, 0, 1, -1) |
| MP_JLR_X211 | Y^2 Z + 6(t+1)XYZ = X^3 + 216t(t^2+1)XZ^2 -108t(t+1)(t^4-6t^3+4t^2-6t-1)Z^3 | (1, 0, 0, 1); (0, 1, 1, 0); (0, 1, -1, 1); (1, -1, 0, -1); (1, -1, 1, 0); (1, 0, 1, -1) |
| MP_JLR_X321 | Y^2 Z + 4tXYZ = X^3 + 4t^3XZ^2 | (1, 0, 0, 1); (0, 1, 1, 0); (0, 1, -1, 1); (1, -1, 0, -1); (1, -1, 1, 0); (1, 0, 1, -1) |
| MP_JLR_X431 | Y^2 Z + 3XYZ = X^3 + 9tX^2Z + 27t^2XZ^2 + 27t(t^2-t-1)Z^3 | (1, 0, 0, 1); (0, 1, 1, 0); (0, 1, -1, 1); (1, -1, 0, -1); (1, -1, 1, 0); (1, 0, 1, -1) |
| W_2_2_IV_IV | Y^2=4X^3-G2X-G3, G2=3(t-1)^2(t+1)^2, G3=(t-1)^2(t+1)^2(t^2+1) | (1, 0, 0, 1); (0, 1, -1, 0); (0, 1, 1, 0); (1, 0, 0, -1); (1, -1, -1, -1); (1, -1, 1, 1); (1, 1, -1, 1); (1, 1, 1, -1) |
| W_3_3_2_IV | Y^2=4X^3-G2X-G3, G2=3(t-1)^2(9t^2+14t+9), G3=(t-1)^2(27t^4+36t^3+2t^2+36t+27) | (1, 0, 0, 1); (0, 1, -1, 0); (0, 1, 1, 0); (1, 0, 0, -1); (1, -1, -1, -1); (1, -1, 1, 1); (1, 1, -1, 1); (1, 1, 1, -1) |
| W_3_3_III_III | Y^2=4X^3-G2X-G3, G2=(81t^2-54t-3)(9t+1)(3t-1), G3=-(729t^4-972t^3+270t^2+36t+1)(27t^2+1) | (1, 0, 0, 1); (0, 1, -27, 0) |
| W_3_III_III_III | Y^2=4X^3-G2X-G3, G2=3t(t^3-1), G3=(t^3-1)^2 | (1, 0, 0, 1) |
| W_3_II_III_IV | Y^2=4X^3-G2X-G3, G2=3t^2(t-1)(t+3), G3=t^2(t-1)^2(t+3)(t+2) | (1, 0, 0, 1); (0, 3, -1, 0) |
| W_4_2_III_III | Y^2=4X^3-G2X-G3, G2=3(t-1)(t+1)(4t^2-1), G3=(t-1)^2(t+1)^2(8t^2+1) | (1, 0, 0, 1); (0, 1, -1, 0); (0, 1, 1, 0); (1, 0, 0, -1); (1, -1, -1, -1); (1, -1, 1, 1); (1, 1, -1, 1); (1, 1, 1, -1) |
| W_4_2_II_IV | Y^2=4X^3-G2X-G3, G2=36t^2(t-1)(3t-1), G3=4t^2(t-1)(54t^3-54t^2+9t-1) | (1, 0, 0, 1) |
| W_4_3_II_III | Y^2=4X^3-G2X-G3, G2=3(t-1)(t+27)(16t^2+80t-243), G3=(t-1)^2(t+27)(64t^3+1376t^2+486t+19683) | (1, 0, 0, 1) |
| W_4_4_II_II | Y^2=4X^3-G2X-G3, G2=3(64t^4-192t^3+64t^2+24t+1), G3=512t^6-2304t^5+2496t^4+312t^2+36t+1 | (1, 0, 0, 1); (0, 1, -8, 0) |
| W_4_III_III_II | Y^2=4X^3-G2X-G3, G2=-3(432t^2-40t+1)(16t-1), G3=(432t^2-40t+1)^2(4t-1) | (1, 0, 0, 1); (0, 1, 432, 0) |
| W_4_II_III_III | Y^2=4X^3-G2X-G3, G2=3(t^2+2)(t+5)(t+1), G3=(t^2+2)^2(t+5)(t+4) | (1, 0, 0, 1) |
| W_4_IV_II_II | Y^2=4X^3-G2X-G3, G2=12t^2(t-1)(t+1), G3=4t^2(t-1)(t+1)(2t^2-1) | (1, 0, 0, 1); (1, 0, 0, -1) |
| W_5_1_II_IV | Y^2=4X^3-G2X-G3, G2=3t^2(t-3)(t+5), G3=t^2(t-3)(t^3+6t^2-3t-32) | (1, 0, 0, 1) |
| W_5_2_1_IV | Y^2=4X^3-G2X-G3, G2=12t^2(t^2+8t+10), G3=4t^2(2t^4+24t^3+78t^2+56t+27) | (1, 0, 0, 1); (0, 27, 8, 0); (4, 27, -8, -4); (54, 27, -8, -54) |
| W_5_3_2_II | Y^2=4X^3-G2X-G3, G2=3(t-3)(81t^3-9t^2-53t-27), G3=(t-3)(729t^5-1215t^4-1350t^3-350t^2-675t-243) | (1, 0, 0, 1) |
| W_5_III_III_1 | Y^2=4X^3-G2X-G3, G2=3(500t^2+44t+1)(20t^2+20t+1), G3=-(500t^2+44t+1)^2(4t^2-8t-1) | (1, 0, 0, 1); (0, 1, 500, 0) |
| W_6_1_1_II | Y^2=4X^3-G2X-G3, G2=12t(t^3-6t^2+15t-12), G3=4t(2t^5-18t^4+72t^3-144t^2+135t-27) | (1, 0, 0, 1) |
| W_6_II_II_II | Y^2=4X^3-G2X-G3, G2=12t(t^3-1), G3=4(t^3-1)(2t^3-1) | (1, 0, 0, 1) |
| W_7_1_1_III | Y^2=4X^3-G2X-G3, G2=12t(t^3+4t^2+10t+6), G3=4t^2(2t^4+12t^3+42t^2+70t+63) | (1, 0, 0, 1) |
| W_7_2_1_II | Y^2=4X^3-G2X-G3, G2=12t(9t^3+36t^2+42t+14), G3=12t(18t^5+108t^4+234t^3+222t^2+87t+8) | (1, 0, 0, 1); (0, 2, 1, 0) |
| W_7_II_II | Y^2=4X^3-G2X-G3, G2=147(196t^2-26t+1)(9604t^2-490t+1), G3=343(196t^2-26t+1)(13176688t^4-1882384t^3+86436t^2-980t-1) | (1, 0, 0, 1); (0, 1, 196, 0) |