curve #504

y² + xy = x³ − 12618522080408152055199453857711427352754247646247874600x + 17207833052907998450566888545722239644098659143943632690620976330023163085245920832

a-invariants: [1, 0, 0, -12618522080408152055199453857711427352754247646247874600, 17207833052907998450566888545722239644098659143943632690620976330023163085245920832]
rank (lower bound): ≥ 18
torsion subgroup: ℤ/2ℤ
conductor (N): 11031717583137458840012928216026292466241694901282318502893118943250561112352498132666738170010
discriminant (Δ): 670170989959873166859151934009159079059583626016331570725624939615465483546860353317834952168048922001399288429873171623904695137117452347881416324448262553600000000
Faltings height: 30.4962
naive height: 392.2379
primes of bad reduction: 2, 3, 5, 11, 31, 37, 41, 59, 73, 79, 251, 271, 397, 449, 2719, 4349, 5861, 14537, 15647, 23869, 389911, 20532121, 149619199, 6479642180953553953521049
regulator: 186263818106675153257036228248.26303453534996070685184774288420110562924984
submitted by: Matthias Breddin
submitted at: 2026-09-02 13:41:17 UTC
last updated: 2026-09-02 17:33:21 UTC

Witness: 18 independent points log in to add more points →

(1549524844244852693202219288218420304/1004826601, 1198640794663799023899302918692448797477146212050903752/31851998425099)
(-445965264814589414915949782620603084806576/709184009599081, 2979804253975496992596668401462373701016071110336412931826948792/18885931150284412962229)
(31009780066114482853617951185323738914768/21332583675841, 4336094290817578552538296661238489660299563672236687881806296/98529252207864019361)
(668356674473902998175842554780789949730177456176/638532829332611736769, 1157549464958012562067806255856418162329878738824687246196773028469309944/16135218328180435445613180120353)
(-481961424377322891994940278658360930431670141184/147408362932782668929, 274433867935186381712913597295727738240431691086226296184177014712620584/1789712142560319427693671908033)
(194208469774266631765132753146868935401276011887696/64223939740566419244649, 1332393212527004644550211775397224045066480190470817978647039910507738174136/16275915070184460107823203386226357)
(-4379606039050463657926541845624004221157857644/5857091254676695489, 2295526045645765182701615379614475381364542924741999251989242414561924/14174997328519322244999652513)
(6616789721602019859635100245337186092794804783056/3376047747376193782489, 445216271363892749840899049642290066398343251734135920307381669113695624/196161091920616423958794519489763)
(40652492836387064485065111401890305132906671905790736/22285140062268436624536409, 1694802684146046275228743572160120236644135499111379257348130677637123325208696/105201771249929572944774677996981690077)
(800954095523321826934341486938866419640034416378968256/422679819958226728808723289, 87260079755263465838813601145252692501275134931277064772085284593936725886778824/8689949860324694208636410013512623816413)
(34993980567642900760321807490752782715227410226/17969961917224060969, 339204739402268986366401083179448939872269011479096021574734121885244/76176450953725478150526332203)
(31348679488646773438137889202272036179866805411264/16138507345295053150441, 10460243954342539750459823110027890088678989709026827506496614490991466872/2050194416150960763218966138265739)
(841701373806334824729821125964248501141805806801591569889104/43553277337163076157237472941201, 759991779903511251593550420026976183404047912962124058199626412154751943794455243054450152/287429448205832531489666599385738604162102048199)
(-4510001278947725231910366531422296803480108336/7072216996189288441, 2973469861993042643731589024656927571829322663271867439954630303585528/18807599428370487785586581261)
(3148739075775956949978709969285223945239305015549924/35840406776433107662784881, 27225209204530878719274763641080585592878306416849973086829505262034548295030892/214565254040839312115940059886050524279)
(88911089604560736055826899858005261378914734861918842641014181554416/78109757585968123376461319624497712948329, 1434700793062117306060116496550979894240697796434728295388906115383122705014746153501780284134314949816/21830211078127176076425934310414034259549841713321317735022517)
(-27597360743615939604725028753216975837882201834225304752/82786125288497470802442622129, 3482670619090043848935022593200539499798029016902558835337968292522921399993004624536/23819702588611368758818321480766415231008617)
(3629625851321827937372667998482677632870257238775893004251261376/1552786511968463011542062594373407569, 42563761245544696858886552076922733494369215299187052330129806783627586059752001603484861413144/1934940544891910545556611493801162846323421087623889753)

Commentary

Exact rank 18, unconditional: rational 2-torsion point, 18 independent points (Neron-Tate height matrix of rank 18; the archived certificate additionally saturates the lattice at all primes up to 97 — the points listed here are the ellrank output, not the saturated basis) with matching 2-descent upper bound 18 (PARI/GP 2.15.4 ellrank, effort 3). Fiber t=56199/13346 on branch u=29/13 of the Elkies-Klagsbrun Z/2Z rank-9 family (arXiv:2003.00077), found by a Mestre-Nagao residue-table sieve followed by an exact 2-descent-type upper bound and ellrank (Matthias Breddin, 2026-09-02; method: doi:10.5281/zenodo.22242107).

last edited by Matthias Breddin at 2026-09-02 13:41:18 UTC · history