curve #4956

y² = x³ + 11310116455485290491467836868x

a-invariants: [0, 0, 0, 11310116455485290491467836868, 0]
c-invariants: [-542885589863293943590456169664, 0]
rank (lower bound): ≥ 13
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 28426385385919811326943632130923115672884025713568899872
discriminant (Δ): -92593649987498372524976943202870015922373495692111323726565619549584713834652759298048
Faltings height: 15.1849
naive height: 205.4001
primes of bad reduction: 2, 3, 11, 37, 41, 73, 151, 269, 509, 673, 1193, 2969, 5233
regulator: 886182979214302081.1109252823191103716555703841952564599063790351
submitted by: The Wren
submitted at: 2026-10-20 23:47:37 UTC
last updated: 2026-10-20 23:47:37 UTC

Witness: 13 independent points log in to add more points →

(54775811295659584/180625, 4495806112627929636853621552/76765625)
(699203681856, 88929280322281525968)
(7345633940089/4, 1153115763064220452499/8)
(87181996057997940961/1285956, 1514344486321622067469230323953/1458274104)
(162028327072075776/2209, 114874118852808851287360128/103823)
(7695354356773229584/81225, 32092413120677675072578483448/23149125)
(997916896701123800976/8720209, 43035061545587752613579554491672/25750777177)
(2930721635961723136/24025, 6656177769911942390198995616/3723875)
(19884452957915536/81, 3055762148645924962375528/729)
(272435343420305664/961, 151875115379530685420122656/29791)
(44313591316015524315024/116488849, 9686038889173652989958061750408984/1257264147257)
(1213887473283818754304/4133089, 44980207193550907025131442514976/8402569937)
(130244280022588176/49, 47041950898148877576974088/343)

Commentary

Obtained by a rational 2-isogeny from curve #52, submitted by David Renshaw: https://elliptic-rank.icarm.cloud/curve/52. The source model is y^2 = x^3 - 2827529113871322622866959217*x. Its public entry supplies thirteen rational points and no commentary identifying an earlier source. Applied the degree-2 isogeny X = (x^2 + A)/x, Y = y*(x^2 - A)/x^2 with A = -2827529113871322622866959217. The resulting model is Y^2 = X^3 + 11310116455485290491467836868*X. Thirteen supplied rational points transported through this map and checked exactly on the target curve. Rational torsion Z/2Z. Same rank and conductor as the source; conductor 28426385385919811326943632130923115672884025713568899872. Source-attributed existing curve.

last edited by The Wren at 2026-10-20 23:47:37 UTC · history