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Properties

Label 422142.bs1
Conductor $422142$
Discriminant $-9.753\times 10^{23}$
j-invariant \( -\frac{60622636760828930041195168773529}{23543783424} \)
CM no
Rank $1$
Torsion structure trivial

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Show commands: Magma / Oscar / Pari/GP / SageMath

Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

\(y^2+xy+y=x^3-28318037769143x-58001986279690357606\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z+xyz+yz^2=x^3-28318037769143xz^2-58001986279690357606z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3-36700176948808707x-2706140561764702478027778\) Copy content Toggle raw display (homogenize, minimize)

Copy content comment:Define the curve
 
Copy content sage:E = EllipticCurve([1, 0, 1, -28318037769143, -58001986279690357606])
 
Copy content gp:E = ellinit([1, 0, 1, -28318037769143, -58001986279690357606])
 
Copy content magma:E := EllipticCurve([1, 0, 1, -28318037769143, -58001986279690357606]);
 
Copy content oscar:E = elliptic_curve([1, 0, 1, -28318037769143, -58001986279690357606])
 
Copy content comment:Simplified equation
 
Copy content sage:E.short_weierstrass_model()
 
Copy content magma:WeierstrassModel(E);
 
Copy content oscar:short_weierstrass_model(E)
 

Mordell-Weil group structure

\(\Z\)

Copy content comment:Mordell-Weil group
 
Copy content magma:MordellWeilGroup(E);
 

Mordell-Weil generators

$P$$\hat{h}(P)$Order
$(2457942252145050723247174883296770735688178255564660939421299941811532904726544036103293998924245779742810686570896270473632877708645969357829159550400164275150395769454121434756743822067403381397968800246940643093577140173664064374473548354257429462474364195949961959955624857748635637966739225776816021791220105649590011620904701072442295047536369973283135483768181700412026275543298090408848920792191497101669407189577244302332163960735639646024393776929398637247605395685464654270948184065408804656062644827769926707334405271017007387138182033876061276505055547270796983166339248164116714240030447065104363817057366152683724778892995247019671034667548244001770456547743947629443979762455282586816578303417396497719843650049207555878288339081490177962194458754042169916924716868865646269514412657013389690966322101720548076990565390146962021795793421451011757921022340911133801150371633775687696397140257051512065937476796959803898024034170926202890510419128339570893268425843437007489473311430712947568924136638328121991016041916291073304061814481565504155791287777446181744791235303470678465003375809889000874914753512376741254806409940859951268476534766191427799884763073167784659335325298848680647331342029845414829515295100962860925963348192003826493810324221457936127864030454475790771767512649290529682260160709600261473677256634183191580086872447889243276538073726575904460506515889210803301275076422887821375185257325314244463878535467877820732149083413922591705453763709941928951606510930404167264817174292696072538169815804700191710906601142204273931463159988279699851822065995483180899907883411473427038413315456954419970624174394286669939797096226602025705462854590262001547300496613466704162516472068884005265163203491090756593760120173637811078649400097353795830823333451938721437753817833981806158196875867790521577652901578796791730736271276262/292835919043609320673408924817231823041852180910797630606440447995959895595592604863837428520287419686349888649948657940596301056150045067906009610920153649579707562967242289441017517891453585629885386694024713923546699312165952210836672381202367733348440095394503600737927965219986677776981688636744718401095154653705991832833457112909975703970971321953623900394099027822503404821500396923897541356306768208992252145778284114581328041532630222525771698215630689725527605924248937033461612977762630865236649235935216070148104356411264678843206471814562778571160162937535380254992684635025074359395353357420879722098148678645490323137763452195922692219229754620597554812405653830514114298703885525773993371950749496567392945433058717363235774209726530163227780728134198231437936939915643778509635905305738555156335992266669156679566975741363312070406902984357755449969492434602249493932303571348294333629091835502342728819925370737687650776939477863415454611771779928881196870831055219906449501434549501496363305098335441143040002755049676814788195276283072052010763365399631268071849805983964471114189125232979641083927152947991381125136787748928980085292596266517038223334241545805436072149085560837550329480729944090401885449532469940564775960627198216798939638388498406138838402741875503730948646417456229262300199010235496379944513405923110619418149302234027255143579101695103368935200801955559667226991274717736067647620190983179255224965479409035625486057185771881455471720320588300069800233229595180905931490813468834865764136948144384812063405683624971607903823379035650576735934479601836141949402579192338051584002163897844378374903148789271891377918062939426042370392908144318535880963328793354790832003714850849061559393827057941934470530137727680598793164214680701249124776077745640241672778089996189363753782927974553672123486412088865302147409, 86143577281657743420577486887142805534093066863926685585555150278208224401437057448455184989906776608802141037565537801406750842823568970003486727706579176709910281950650204018021321908119397599420339912054693270845224088096389338467636209372405075850440975778946433580529259599299730819077703473260090223428535872624675076155065340945948595544082280880101287365378363333330750577761797028448269767169628527645968734011036666597981643583088717575900677559340134529756687625290691812814979668498562577299169152816632044410545168608674237937879023912920871498776113895478382876713166835077162380258296731403457352829040482195562198123755308491402302215128064184897628530227770876299566281471672157915263300598147332269402729455545759074356110801312232922700872116440389014777708830384715728033565292521656758805365700429166435066687108805959025591807391785863052658007792389573239075793977055633491422496951760785547290849632912802454427922571297484778320217090627622878171136777102921901196227314509219099556710211077215585260781591208727825738536867183834946847004708510179781887541848269265283844401887171972857188810556958974197875850684745253792631834453129531725197603138080434018043511898012305164951711864123683167355290421332262647197948232299665025905650181301305456863890728161317965999664090842991847613777227105251247614262534887739562229808650483401269535067193250924156632533873968147157821197959962294814861184522320104105393154594413366372571402730436242246823151734210023230947247562903579007439817376128609642937243578094318282684402638937778906575150772411956237647541715550359831047245350936158516942788246858974737469599305421947876890652435727013555441410288770238238111965924727274965165720046136006709678084287942030830774066456932859801513788763504351236074259966058601102243455196003480956790058657430392822496257899437974496215933211942129338386082120326062217478032395461890798274123937046992926735789955489026626136991203970258318448604314220758539093322188560239204772796962717566558868928808646113912831021780558603390581008939321931061868027427563238235594605009703353436148071461396852749058968585509903680238689360406879110318365322497283651455999411774997027254922937092014263626195755414383959102797992302934884247544497956221487070305135897730164260874435888341766591434915073991584977292618188480966621789121888071569302843844953991221395736526924169045766845377607430024422122287133584442871443462721570610456423845961560848858775350083804340377080663343469203995335174358474809784190404916200705652421012520383205079311767008026806664524988650174338033614474415525836535539633152123108776414670105188194720050878262766692953383376031279293666401406649327498121911363168100508859543420022655129818811763096244812011068772595188778613534951423151525198144744/5011139800727717683804758506268592429347781162743766583281701635132599133639659241867681849772679687732777660436888990166290669452643236780421381184905301670627706669178220399923955388385721066062723511735990454476390404248609670974526634644597172946466922263883606115879094110022834843536464088271919396824994993294478468377282088088645190984964376755141881760416456121593959172569150103888320896070295242173248314262789349555810342571857706847258828313839324815666415333896181652784869313380597275606671054333929604036958510804056769790780635099857452661621016524012291327515680516435452941403031364452657082287159386063895083405975354179901628663640853139323422777945800029512505295218914973731820659334735489380513390302579345727443177931356318163176539070776910815913731049757669833912718718657243045718406312104294495824023658408599983604841904117655277858856891425632803653575766043960551690958514011474577112828011242758809011562976183673969683192912283561880832669537588895828065859078778110726834157233751660418320894456107984614493240589730200028432281302102195308888265779243779866469991683204843826755977308821685104206959656751527832230190369263257165934801246287470395653905136634423093480189917918776956689979831440557429829134268738163984144955163777066628657107900321774992397889229576858643494237927884990103953938870658883601675795263858764267581588053081144322361799207044798461678799673089582170212683282886553999712053201350379442545853638662597532349334630436685006944885325410414301238385763371600629345152461665763803274335348989736890970661838688763739254968788743072671285034698372570557043213159956030275517002991394079742524106608757170768120142647205193638645094067413310032132487124972310283662404287097165732717319495031569229410548537253715617562795365682939463862879363031309095795144419159715353964918300990597777022330035366260663515406996314550521026003231680845795840013789808295350823182194893567120708001232559143784451285051152326169870630378988293936595523630961611376664639010551859763236689592557118596230472416940584013252356870715205992067525527741850145882184275398007551816523408185841398543184838542548102065205701747421203983133017194074151096483179638629214244782316060216971264450811247809447987539435458015917311961979399236539612525507882410347998140996369127122392963884059271346000732412690930566902290444143166528639597227443285846218786756653255016361226689756987697560845880935258323061804399768609576584069543376176847852181129053410839812135384938435375268606839374320105563228823469960779052380378637531045811254344716891389007890359873881639000800769121816265556595626683475515600647464225706239711149955029953481246587603177796457231850533856092210569208345898209800965257087496759101240919789673544209640803735768508327)$$4290.3779596324903652622317609$$\infty$

Integral points

None

Copy content comment:Integral points
 
Copy content sage:E.integral_points()
 
Copy content magma:IntegralPoints(E);
 

Invariants

Conductor: $N$  =  \( 422142 \) = $2 \cdot 3 \cdot 7 \cdot 19 \cdot 23^{2}$
Copy content comment:Conductor
 
Copy content sage:E.conductor().factor()
 
Copy content gp:ellglobalred(E)[1]
 
Copy content magma:Conductor(E);
 
Copy content oscar:conductor(E)
 
Discriminant: $\Delta$  =  $-975336808026059448754176$ = $-1 \cdot 2^{13} \cdot 3^{2} \cdot 7^{5} \cdot 19 \cdot 23^{10} $
Copy content comment:Discriminant
 
Copy content sage:E.discriminant().factor()
 
Copy content gp:E.disc
 
Copy content magma:Discriminant(E);
 
Copy content oscar:discriminant(E)
 
j-invariant: $j$  =  \( -\frac{60622636760828930041195168773529}{23543783424} \) = $-1 \cdot 2^{-13} \cdot 3^{-2} \cdot 7^{-5} \cdot 19^{-1} \cdot 23^{2} \cdot 167^{3} \cdot 4019^{3} \cdot 7237^{3}$
Copy content comment:j-invariant
 
Copy content sage:E.j_invariant().factor()
 
Copy content gp:E.j
 
Copy content magma:jInvariant(E);
 
Copy content oscar:j_invariant(E)
 
Endomorphism ring: $\mathrm{End}(E)$ = $\Z$
Geometric endomorphism ring: $\mathrm{End}(E_{\overline{\Q}})$  =  \(\Z\)    (no potential complex multiplication)
Copy content comment:Potential complex multiplication
 
Copy content sage:E.has_cm()
 
Copy content magma:HasComplexMultiplication(E);
 
Sato-Tate group: $\mathrm{ST}(E)$ = $\mathrm{SU}(2)$
Faltings height: $h_{\mathrm{Faltings}}$ ≈ $5.8435287542771173280861000059$
Copy content comment:Faltings height
 
Copy content gp:ellheight(E)
 
Copy content magma:FaltingsHeight(E);
 
Copy content oscar:faltings_height(E)
 
Stable Faltings height: $h_{\mathrm{stable}}$ ≈ $3.2306169076694925857471393127$
Copy content comment:Stable Faltings height
 
Copy content magma:StableFaltingsHeight(E);
 
Copy content oscar:stable_faltings_height(E)
 
$abc$ quality: $Q$ ≈ $1.0650502642075206$
Szpiro ratio: $\sigma_{m}$ ≈ $8.070437755528024$

BSD invariants

Analytic rank: $r_{\mathrm{an}}$ = $ 1$
Copy content comment:Analytic rank
 
Copy content sage:E.analytic_rank()
 
Copy content gp:ellanalyticrank(E)
 
Copy content magma:AnalyticRank(E);
 
Mordell-Weil rank: $r$ = $ 1$
Copy content comment:Mordell-Weil rank
 
Copy content sage:E.rank()
 
Copy content gp:[lower,upper] = ellrank(E)
 
Copy content magma:Rank(E);
 
Regulator: $\mathrm{Reg}(E/\Q)$ ≈ $4290.3779596324903652622317609$
Copy content comment:Regulator
 
Copy content sage:E.regulator()
 
Copy content gp:G = E.gen \\ if available matdet(ellheightmatrix(E,G))
 
Copy content magma:Regulator(E);
 
Real period: $\Omega$ ≈ $0.0010347936853587869971309904393$
Copy content comment:Real Period
 
Copy content sage:E.period_lattice().omega()
 
Copy content gp:if(E.disc>0,2,1)*E.omega[1]
 
Copy content magma:(Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E);
 
Tamagawa product: $\prod_{p}c_p$ = $ 2 $  = $ 1\cdot2\cdot1\cdot1\cdot1 $
Copy content comment:Tamagawa numbers
 
Copy content sage:E.tamagawa_numbers()
 
Copy content gp:gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
 
Copy content magma:TamagawaNumbers(E);
 
Copy content oscar:tamagawa_numbers(E)
 
Torsion order: $\#E(\Q)_{\mathrm{tor}}$ = $1$
Copy content comment:Torsion order
 
Copy content sage:E.torsion_order()
 
Copy content gp:elltors(E)[1]
 
Copy content magma:Order(TorsionSubgroup(E));
 
Copy content oscar:prod(torsion_structure(E)[1])
 
Special value: $ L'(E,1)$ ≈ $8.8793120408604355509804554618 $
Copy content comment:Special L-value
 
Copy content sage:r = E.rank(); E.lseries().dokchitser().derivative(1,r)/r.factorial()
 
Copy content gp:[r,L1r] = ellanalyticrank(E); L1r/r!
 
Copy content magma:Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
 
Analytic order of Ш: Ш${}_{\mathrm{an}}$  ≈  $1$    (rounded)
Copy content comment:Order of Sha
 
Copy content sage:E.sha().an_numerical()
 
Copy content magma:MordellWeilShaInformation(E);
 

BSD formula

$$\begin{aligned} 8.879312041 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.001035 \cdot 4290.377960 \cdot 2}{1^2} \\ & \approx 8.879312041\end{aligned}$$

Copy content comment:BSD formula
 
Copy content sage:# self-contained SageMath code snippet for the BSD formula (checks rank, computes analytic sha) E = EllipticCurve([1, 0, 1, -28318037769143, -58001986279690357606]); r = E.rank(); ar = E.analytic_rank(); assert r == ar; Lr1 = E.lseries().dokchitser().derivative(1,r)/r.factorial(); sha = E.sha().an_numerical(); omega = E.period_lattice().omega(); reg = E.regulator(); tam = E.tamagawa_product(); tor = E.torsion_order(); assert r == ar; print("analytic sha: " + str(RR(Lr1) * tor^2 / (omega * reg * tam)))
 
Copy content magma:/* self-contained Magma code snippet for the BSD formula (checks rank, computes analytic sha) */ E := EllipticCurve([1, 0, 1, -28318037769143, -58001986279690357606]); r := Rank(E); ar,Lr1 := AnalyticRank(E: Precision := 12); assert r eq ar; sha := MordellWeilShaInformation(E); omega := RealPeriod(E) * (Discriminant(E) gt 0 select 2 else 1); reg := Regulator(E); tam := &*TamagawaNumbers(E); tor := #TorsionSubgroup(E); assert r eq ar; print "analytic sha:", Lr1 * tor^2 / (omega * reg * tam);
 

Modular invariants

Modular form 422142.2.a.bs

\( q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} - q^{7} - q^{8} + q^{9} - q^{10} + 3 q^{11} + q^{12} - 2 q^{13} + q^{14} + q^{15} + q^{16} + 4 q^{17} - q^{18} - q^{19} + O(q^{20}) \) Copy content Toggle raw display

Copy content comment:q-expansion of modular form
 
Copy content sage:E.q_eigenform(20)
 
Copy content gp:\\ actual modular form, use for small N [mf,F] = mffromell(E) Ser(mfcoefs(mf,20),q) \\ or just the series Ser(ellan(E,20),q)*q
 
Copy content magma:ModularForm(E);
 

For more coefficients, see the Downloads section to the right.

Modular degree: 9763952640
Copy content comment:Modular degree
 
Copy content sage:E.modular_degree()
 
Copy content gp:ellmoddegree(E)
 
Copy content magma:ModularDegree(E);
 
$ \Gamma_0(N) $-optimal: yes
Manin constant: 1
Copy content comment:Manin constant
 
Copy content magma:ManinConstant(E);
 

Local data at primes of bad reduction

This elliptic curve is not semistable. There are 5 primes $p$ of bad reduction:

$p$ Tamagawa number Kodaira symbol Reduction type Root number $\mathrm{ord}_p(N)$ $\mathrm{ord}_p(\Delta)$ $\mathrm{ord}_p(\mathrm{den}(j))$
$2$ $1$ $I_{13}$ nonsplit multiplicative 1 1 13 13
$3$ $2$ $I_{2}$ split multiplicative -1 1 2 2
$7$ $1$ $I_{5}$ nonsplit multiplicative 1 1 5 5
$19$ $1$ $I_{1}$ nonsplit multiplicative 1 1 1 1
$23$ $1$ $II^{*}$ additive -1 2 10 0

Copy content comment:Local data
 
Copy content sage:E.local_data()
 
Copy content gp:ellglobalred(E)[5]
 
Copy content magma:[LocalInformation(E,p) : p in BadPrimes(E)];
 
Copy content oscar:[(p,tamagawa_number(E,p), kodaira_symbol(E,p), reduction_type(E,p)) for p in bad_primes(E)]
 

Galois representations

The $\ell$-adic Galois representation has maximal image for all primes $\ell$.

Copy content comment:Mod p Galois image
 
Copy content sage:rho = E.galois_representation(); [rho.image_type(p) for p in rho.non_surjective()]
 
Copy content magma:[GaloisRepresentation(E,p): p in PrimesUpTo(20)];
 

Copy content comment:Adelic image of Galois representation
 
Copy content sage:gens = [[1, 1, 1063, 0], [1009, 2, 1009, 3], [1063, 2, 1062, 3], [533, 2, 533, 3], [1, 0, 2, 1], [1, 2, 0, 1], [799, 2, 799, 3], [913, 2, 913, 3]] GL(2,Integers(1064)).subgroup(gens)
 
Copy content magma:Gens := [[1, 1, 1063, 0], [1009, 2, 1009, 3], [1063, 2, 1062, 3], [533, 2, 533, 3], [1, 0, 2, 1], [1, 2, 0, 1], [799, 2, 799, 3], [913, 2, 913, 3]]; sub<GL(2,Integers(1064))|Gens>;
 

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1064 = 2^{3} \cdot 7 \cdot 19 \), index $2$, genus $0$, and generators

$\left(\begin{array}{rr} 1 & 1 \\ 1063 & 0 \end{array}\right),\left(\begin{array}{rr} 1009 & 2 \\ 1009 & 3 \end{array}\right),\left(\begin{array}{rr} 1063 & 2 \\ 1062 & 3 \end{array}\right),\left(\begin{array}{rr} 533 & 2 \\ 533 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 799 & 2 \\ 799 & 3 \end{array}\right),\left(\begin{array}{rr} 913 & 2 \\ 913 & 3 \end{array}\right)$.

Input positive integer $m$ to see the generators of the reduction of $H$ to $\mathrm{GL}_2(\Z/m\Z)$:

The torsion field $K:=\Q(E[1064])$ is a degree-$190625218560$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1064\Z)$.

The table below list all primes $\ell$ for which the Serre invariants associated to the mod-$\ell$ Galois representation are exceptional.

$\ell$ Reduction type Serre weight Serre conductor
$2$ nonsplit multiplicative $4$ \( 70357 = 7 \cdot 19 \cdot 23^{2} \)
$3$ split multiplicative $4$ \( 140714 = 2 \cdot 7 \cdot 19 \cdot 23^{2} \)
$5$ good $2$ \( 60306 = 2 \cdot 3 \cdot 19 \cdot 23^{2} \)
$7$ nonsplit multiplicative $8$ \( 60306 = 2 \cdot 3 \cdot 19 \cdot 23^{2} \)
$13$ good $2$ \( 211071 = 3 \cdot 7 \cdot 19 \cdot 23^{2} \)
$19$ nonsplit multiplicative $20$ \( 22218 = 2 \cdot 3 \cdot 7 \cdot 23^{2} \)
$23$ additive $112$ \( 798 = 2 \cdot 3 \cdot 7 \cdot 19 \)

Isogenies

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This curve has no rational isogenies. Its isogeny class 422142.bs consists of this curve only.

Twists

The minimal quadratic twist of this elliptic curve is 422142.bd1, its twist by $-23$.

Iwasawa invariants

No Iwasawa invariant data is available for this curve.

$p$-adic regulators

$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.