curve #4184

y² + xy = x³ − x² − 1019402439308929x + 12645004805360636001409

a-invariants: [1, -1, 0, -1019402439308929, 12645004805360636001409]
c-invariants: [48931317086828601, -10925283931640662614488685]
rank (lower bound): ≥ 16
torsion subgroup: trivial
conductor (N): 638558705203558998344002920820447810861615354
discriminant (Δ): -1277117410407117996688005841640895621723230708 ★ record for rank ≥ 16
Faltings height: 7.4707 ★ record for rank ≥ 16
naive height: 115.3062 ★ record for rank ≥ 16
primes of bad reduction: 2, 15233, 11817766043, 1773576820123072327762918950983
regulator: 50264258944901.41876483176115798707785283179065239534857355604567163576
submitted by: Andrii Prokhorenkov
submitted at: 2026-10-11 23:04:01 UTC
last updated: 2026-10-11 23:04:01 UTC

Witness: 16 independent points log in to add more points →

(18375848, 10836353017)
(-2495214974/81, 88743218573861/729)
(1371565106, 50781104070165)
(73246295/4, -87072560481/8)
(-34805268, -77198171075)
(38265764, 172225317943)
(475680384/25, -1464977395133/125)
(-19562603288342797/1533583921, -9220415050164788363962635/60056679930281)
(2927203630953/157609, -681741570823967701/62570773)
(218839048255/484, -102127293785736751/10648)
(834265445429/44944, -103747406774419153/9528128)
(15561697996/289, 1657816522933069/4913)
(587882384356/30625, 65594728908191979/5359375)
(69288795, 524048347088)
(4009496739529269/215854864, -34558856836081578607915/3171339661888)
(3192650291/169, -24985825959024/2197)

Commentary

Independently constructed eight-point plane cubic pencil 1005a6f72cab52f0 at t=-114/1709. Rank exactly 16: exact local 2-descent characters certify 16 independent rational points, and unconditional PARI algebraic 2-descent gives upper bound 16. The displayed quadratic base change embeds the curve in the independently derived K3 family recorded in companion metadata.

last edited by Andrii Prokhorenkov at 2026-10-11 23:04:01 UTC · history