curve #3518

y² + xy + y = x³ + x² − 7671715934167815930781994061299x + 8178285354508392679962576458844594775376567537

a-invariants: [1, 1, 1, -7671715934167815930781994061299, 8178285354508392679962576458844594775376567537]
rank (lower bound): ≥ 11
torsion subgroup: ℤ/4ℤ
conductor (N): 27145429606258964067875985943433158218
discriminant (Δ): 3276721386201583895266720381163929043277538584375571439373626135094247611847844117389889472 ☆ record for ℤ/4ℤ torsion, rank ≥ 11
Faltings height: 16.4248 ☆ record for ℤ/4ℤ torsion, rank ≥ 11
naive height: 224.9589 ☆ record for ℤ/4ℤ torsion, rank ≥ 11
primes of bad reduction: 2, 3, 7, 13, 17, 19, 23, 103, 1063, 6673, 43801, 76441, 1625287, 1683223
regulator: 879949790672.251315970506107851043949129922966222783444142151
submitted by: dujella
submitted at: 2026-09-27 15:16:36 UTC
last updated: 2026-09-27 15:16:36 UTC

Witness: 11 independent points log in to add more points →

(1538865762039849, 4091995471313073867694)
(1619928061801999, 1272426754397506409594)
(39699269930220669/25, 45592653178469853882212/125)
(1032795906892023, 36832298467066294689538)
(-200168577914205, -98518534079058408848954)
(1582877295233673, 894589483767100750318)
(2317104754481921, 53315963594447076854398)
(813599464909173, 49750765741963773993958)
(43925280402228791/25, 1386561391703091367406674/125)
(987829224002351285, 981795641366021819261814118)
(-57742117929904287476139/19333609, -5671160357763549657125856551188498/85009878773)

Commentary

It is the fiber at t = −94073/34016 of Elkies' K3 surface. This value of t lies on a rank-5 subfamily over Q(u), found by Dujella-Peral. The subfamily is defined by the condition 2(49t+22)(872t+791) = □, which gives an extra quadratic section. It is parametrized by t = (791 − 8624u²)/(19208u² − 872), and the curve is the fiber at u = 25/131. Found with the assistance of Claude, based on our old PARI/GP code.

last edited by dujella at 2026-09-27 16:27:04 UTC · history