curve #414

y² + xy = x³ − 417912674922888853976693575239883106296909250x + 2674272079811740358679570958520093732597884037078754632469690562500

a-invariants: [1, 0, 0, -417912674922888853976693575239883106296909250, 2674272079811740358679570958520093732597884037078754632469690562500]
rank (lower bound): ≥ 25
torsion subgroup: trivial
conductor (N): 16760725130923377828936183343653472405052507400204269535633169891162525316371894847775702540918638174274090
discriminant (Δ): 1581739705138847952897657581979669295980470647074499285199050633127943304037864743566952295543770172880782758840137269478960433920000000
Faltings height: 24.6590
naive height: 319.8451
primes of bad reduction: 2, 3, 5, 11, 13, 17, 29, 31, 107, 2389148369575121535136373159566497236058454879555115678910972748319347864775402115477928908335141
regulator: 3932032436300986209338186096.240109828705554754518125658728216810084174523195871294191491
submitted by: 7fff-zip
submitted at: 2026-08-29 05:03:40 UTC
last updated: 2026-08-29 05:03:40 UTC

Witness: 25 independent points log in to add more points →

(15822380316923547828580, 151643060249117670730726936276270)
(-22816408878790378304900, 575828230148625383476004261500750)
(3502597587318607010039500/9, 6546290214313260910388604210152505250/27)
(-12847639393307369174300, -2433682077246885894706777800506450)
(37368154869795085727692, 6264004564904152704618026454846502)
(-8147032401957697531268, -2353352489348946565146929622436298)
(7314863982403396321060, -93251557207261203580730036421650)
(15878183809166079843600, -204287664445226886637292709486750)
(6276682928817426954072556, -15725084384127481121029416422467664498)
(17105485255858886411800, -728489777122742608029142376831750)
(3466564719651636987652, -1125703646276725605809873386784498)
(191318624420912972948980, 83219710469615739584503837268461870)
(1088549103290770089465850, -1135522877485039954086011907487005500)
(748748481806784583411555/4, 644151500019090092797555295593415075/8)
(15765778007347823814550, 65509983713733457724183865693250)
(7027681705713219135600, -290517027792982258106875094586750)
(54104839457060278926340, -11766311708087028814291592566257650)
(7795942312580552683974204100/1129969, -415032435232650929609295747255897247543550/1201157047)
(1855795325004069684186400/49, 2200868019361747716785742237904337950/343)
(3558348719572905341759424148/214369, 55340098718074725998937110659283186702194/99252847)
(1124939783631397150741848478974625/330790620736, 216284368675656747861486004558093648562126796669125/190252240772585984)
(357805967957512958460505193065/48888064, -29920995894753812033971276258593973961533075/341825343488)
(10348737837123444886619656146347740/7375889689, -1052652882191382866990397839697257143810371738739870/633463534160387)
(46702079872170180690148303697500/2900576449, 54003469752825789415439788310295350829323449750/156216345813793)
(5654320077321690108631172995/37636, -421392451679890918317802354675473245564005/7301384)

Commentary

Found in this search by specializing Noam D. Elkies's rank-17 elliptic K3 family (arXiv:2608.25406) at t=-216/329. These 25 rational points are certified independent exactly by Cremona-Brumer quadratic characters; the claimed lower bound is rank >= 25.

last edited by 7fff-zip at 2026-08-29 05:03:40 UTC · history