curve #4643

y² = x³ + x² − 99435257165280622013699609620964018752760x + 11995883739748268985378191386564723783114025998377130895653908

a-invariants: [0, 1, 0, -99435257165280622013699609620964018752760, 11995883739748268985378191386564723783114025998377130895653908]
c-invariants: [4772892343933469856657581261806272900132496, -10364443551142504403395394712055522167750463950168678731245771456]
rank (lower bound): ≥ 12
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 12437980303867946022779738389472071607816438224008357360
discriminant (Δ): 756475838384846306411404854972231958880719207616893514885793182318648285224885027809180961142297533074833089591708224000000
Faltings height: 22.4115
naive height: 294.8146
primes of bad reduction: 2, 3, 5, 7, 13, 17, 19, 23, 43, 47, 53, 59, 61, 71, 73, 103, 151, 181, 571, 607, 853, 1627, 1657, 4241, 23929, 24071
regulator: 547265061691.82923750425841607816903340261791761337443652866217
submitted by: The Wren
submitted at: 2026-10-19 16:21:22 UTC
last updated: 2026-10-19 16:21:22 UTC

Witness: 12 independent points log in to add more points →

(167385317948221952686, 204101546508351336343067805600)
(161337559769180883511, 390931872175009353829503545250)
(-309812336698634859164, 3614584858107150170418441616350)
(202005801306169330486, 390516463003494440227652692200)
(224866462883959490071, 1003287227001190069500029967030)
(-363217493650920574748, 440832554837922752423801025570)
(40125135492800341636, 2840885971773919997777362217250)
(166730363407080419566, 227918619974573869049649394560)
(299680196131537818691, 3018418682015030155631802534060)
(-137468744640802205132, 4802841523750242568921386068322)
(-1357785397315014314, 3482943148648559003802414906600)
(99778532263451016502, 1751499470735363195920708006680)

Commentary

From Andrej Dujella and Juan Carlos Peral’s prescribed-torsion rank table, z2z2old1011121314.html: https://dujella.github.io/tors/z2z2old1011121314.html Twelve supplied rational points checked exactly on the displayed model; the source lists them as independent. Rational torsion Z/2Z × Z/2Z.

last edited by The Wren at 2026-10-19 16:21:23 UTC · history