curve #745

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y² = x³ + x² − 16489339774237010364548523362386016208x + 25613929920741954473166448583590674302293472150393717988

a-invariants

[0, 1, 0, -16489339774237010364548523362386016208, 25613929920741954473166448583590674302293472150393717988]

rank (lower bound)

≥ 23

torsion subgroup

trivial

conductor (N)

9252467452476159857081397435096950329303577724136752038772152521917883645245369769040427481200

naive height

268.7009

Faltings height

20.2362

discriminant (Δ)

3515417228832982492329928296324410223992511163996413555036738271741064785181095671456611579715879770046833280000

primes of bad reduction

2, 3, 5, 7, 23, 29, 47, 71, 101, 665384696019408514388819, 747023040411454510195699, 9857528942884130812331333025182357

regulator

5519807073021843845879008.30169833212536154084933988776187368473375637471529351446117

submitted by

Valery Asiryan

submitted at

2026-09-16 18:17:19 UTC

later contributions

primes of bad reduction recorded · Daniel Błażewicz · 2026-09-16 18:31:23 UTC

Witness: 23 independent points log in to add more points →

(-1634362246264492722, -6942468870763540819471040880)
(996254020109771418, -3189853129309052080503977820)
(1765557752902092288, -1415851189864345822793680470)
(1843142222932715754, -1217869047783228832757035356)
(2176754143087178022, -186320390093194151335226112)
(2189408618032776558, -83544178339163105110942080)
(10133805900376268661273/4624, -15590820477211366767485604483579/314432)
(2500581670112044152, -129992235860218052402005722)
(3451625416216673297751/1369, -13119052919174815319713169283582/50653)
(2711974193085279882, -917226635122052366791791132)
(178661910817532517642/49, -1281783419819844597729683135940/343)
(4067271018630868038, -5082414204084045526330296720)
(4132832456951993162, -5296818529871385234965636508)
(39396369459739067998/9, -164922414729999438824062806176/27)
(5152324515803640992, -8799512188662219173110026198)
(6985441099347841578, -15852220912926771286494790620)
(8153019903270669114, -20811568444776558483111944604)
(305379508227563800073568893358/15507471841, -165418385031077625253704425273005551933862080/1931129960887889)
(32337159207680902512, -182502328383054701820873350442)
(2519632362730008163017/64, -125827432672560226517064348539685/512)
(218226015730735421898, -3223184162543500841886601573020)
(1508929761406622728917/4, -58610911728423938773486766403465/8)
(66805243169880876255469/36, -17266916340536774471424801198618571/216)

Commentary

Specialization T=57/85 of the family identified by wgxli in the commentary to curve #302. The points were obtained by specializing reconstructed generic sections and running minimized quartic searches. The 23 listed points are certified independent by exact quadratic-character certificates.

last edited by Valery Asiryan at 2026-09-16 18:17:19 UTC · history