curve #508

y² + xy + y = x³ − x² − 53792886242560750174351881255074187083988864182x + 4797642165509852394570540630340076783744661179810401137268571754467525

a-invariants: [1, -1, 1, -53792886242560750174351881255074187083988864182, 4797642165509852394570540630340076783744661179810401137268571754467525]
rank (lower bound): ≥ 17
torsion subgroup: ℤ/2ℤ
conductor (N): 2824480633135096954485482008313880173502754314875888014401478071245887272764416254
discriminant (Δ): 18678994192039697456639190949581220336849768420906118062509930672458958527217540760329299428836334039534028340248795738486520823306590093312
Faltings height: 25.6393
naive height: 334.4180
primes of bad reduction: 2, 3, 7, 17, 19, 31, 43, 53, 61, 71, 79, 131, 179, 263, 1009, 2729, 2767, 3593, 29527, 135301, 762877, 1249603, 2633311, 21043613, 80581197209
regulator: 3645826786026580420239587.07685750327310926337262105310703419073633372035
submitted by: Matthias Breddin
submitted at: 2026-09-02 13:41:34 UTC
last updated: 2026-09-02 17:33:23 UTC

Witness: 17 independent points log in to add more points →

(214654283443990116384063, 56047109830495339631998543804198837)
(-935816174180979197550056965741166585031/26250819670976209, 347357971384075637614457310293749485385697932164930332273741/4253185287699760894270823)
(2011139059169064525855030359930823035773617/24103353608979789769, 3530601412092410758989455975820106661320404441100107312449308247/118335815550780270467667886853)
(741037380176440388212870502711654761/5701020557761, 17318895712939965606469553358533851457867541381661067/13612218466375342241)
(8658888625985292707426253923450073/40130906929, 457319295407546475809820757156982748061271120664221/8039304192365783)
(17173934981470959373051777879017/132733441, 2905710959616661362673536128052902177924197675/1529221973761)
(-11954133171633191227816246777676655/2653448781481, 306850471568829589573794813695615866887155339437100935/4322311511554441621)
(6985868630425762971677974847016782121/48209345457025, 2256365193808282702621187063792754353876148321239765909/334731707265034397375)
(37201949704061014023415812481917017433/261149252172769, 21300474946917688757104575063486566251251223591635976555/4220201424977442562897)
(8980038470842565894980915906227415557201/69479571593268841, 37155314071950776952556763579323143423766160590469754678809/18314104921284411686776139)
(5549376664773102652715645176017169456239201/42525900613303810921, 101609782675025274479902470453598769642102051468861292778338743/277319417816424913368238780469)
(33145021988186079265571608508910962425/258614031153361, 12163419260233643993793502984257534213154283056806974259/4158896628326183007641)
(1076713607012983322491488093744201/7456495201, 4116815484452726414003389818220235032981403021861/643875817101551)
(-1899964530443937372841299608453183241/10424277682225, 3111603990428640999184461865347348257513415795177005093/33656500502880979625)
(-493444943963856306114291618755542835/2799594893601, 439508421030790023937434770669286883690614378412808305/4684279376378299599)
(844180456472167124814463676999254662297/3372257649326449, 16406317109771456253864592796905717490934384735432365223293/195830856186880073561207)
(31120677685172612816823710643230600208288161514601/238827059197155333453030001, 2989659702837021538250608604148943262345565217828335698184600084083328739/3690840660938844921315372966147314986249)

Commentary

Exact rank 17, unconditional: rational 2-torsion point, 17 independent points (Neron-Tate height matrix of rank 17; 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 17 (PARI/GP 2.15.4 ellrank, effort 1). Fiber t=431/1490 on branch u=1/17 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 13:41:34 UTC · history