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Properties

Label 413270.ci2
Conductor $413270$
Discriminant $1.970\times 10^{25}$
j-invariant \( \frac{87501897507774086005761}{815991377947460000} \)
CM no
Rank $2$
Torsion structure \(\Z/{2}\Z\)

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Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

\(y^2+xy+y=x^3-x^2-267298033x+1668520212481\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z+xyz+yz^2=x^3-x^2z-267298033xz^2+1668520212481z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3-4276768523x+106781016830278\) Copy content Toggle raw display (homogenize, minimize)

Copy content comment:Define the curve
 
Copy content sage:E = EllipticCurve([1, -1, 1, -267298033, 1668520212481])
 
Copy content gp:E = ellinit([1, -1, 1, -267298033, 1668520212481])
 
Copy content magma:E := EllipticCurve([1, -1, 1, -267298033, 1668520212481]);
 
Copy content oscar:E = elliptic_curve([1, -1, 1, -267298033, 1668520212481])
 
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

\(\Z \oplus \Z \oplus \Z/{2}\Z\)

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

Mordell-Weil generators

$P$$\hat{h}(P)$Order
$(-6005, 1751452)$$0.60427038020531291048730495081$$\infty$
$(33455/4, 1037391/8)$$4.5920991103264560677362309077$$\infty$
$(-75445/4, 75441/8)$$0$$2$

Integral points

\( \left(-6005, 1751452\right) \), \( \left(-6005, -1745448\right) \), \( \left(10417, 114726\right) \), \( \left(10417, -125144\right) \), \( \left(13001, 618606\right) \), \( \left(13001, -631608\right) \), \( \left(14081, 827496\right) \), \( \left(14081, -841578\right) \), \( \left(14803, 969996\right) \), \( \left(14803, -984800\right) \), \( \left(43363, 8440816\right) \), \( \left(43363, -8484180\right) \) Copy content Toggle raw display

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

Invariants

Conductor: $N$  =  \( 413270 \) = $2 \cdot 5 \cdot 11 \cdot 13 \cdot 17^{2}$
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: $\Delta$  =  $19696048188611894124740000$ = $2^{5} \cdot 5^{4} \cdot 11^{12} \cdot 13 \cdot 17^{6} $
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: $j$  =  \( \frac{87501897507774086005761}{815991377947460000} \) = $2^{-5} \cdot 3^{3} \cdot 5^{-4} \cdot 11^{-12} \cdot 13^{-1} \cdot 3229^{3} \cdot 4583^{3}$
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: $h_{\mathrm{Faltings}}$ ≈ $3.6748097479472025147225331674$
Copy content comment:Faltings height
 
Copy content gp:ellheight(E)
 
Copy content magma:FaltingsHeight(E);
 
Copy content oscar:faltings_height(E)
 
Stable Faltings height: $h_{\mathrm{stable}}$ ≈ $2.2582030759190944745977658585$
Copy content comment:Stable Faltings height
 
Copy content magma:StableFaltingsHeight(E);
 
Copy content oscar:stable_faltings_height(E)
 
$abc$ quality: $Q$ ≈ $1.0322408901773137$
Szpiro ratio: $\sigma_{m}$ ≈ $5.399474389057654$

BSD invariants

Analytic rank: $r_{\mathrm{an}}$ = $ 2$
Copy content comment:Analytic rank
 
Copy content sage:E.analytic_rank()
 
Copy content gp:ellanalyticrank(E)
 
Copy content magma:AnalyticRank(E);
 
Mordell-Weil rank: $r$ = $ 2$
Copy content comment:Mordell-Weil rank
 
Copy content sage:E.rank()
 
Copy content gp:[lower,upper] = ellrank(E)
 
Copy content magma:Rank(E);
 
Regulator: $\mathrm{Reg}(E/\Q)$ ≈ $2.6919786342632700698197607371$
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: $\Omega$ ≈ $0.068828093541502785373486838189$
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: $\prod_{p}c_p$ = $ 480 $  = $ 5\cdot2\cdot( 2^{2} \cdot 3 )\cdot1\cdot2^{2} $
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: $\#E(\Q)_{\mathrm{tor}}$ = $2$
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^{(2)}(E,1)/2!$ ≈ $22.234050870095912096297608782 $
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

$$\begin{aligned} 22.234050870 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.068828 \cdot 2.691979 \cdot 480}{2^2} \\ & \approx 22.234050870\end{aligned}$$

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, -1, 1, -267298033, 1668520212481]); 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, -1, 1, -267298033, 1668520212481]); 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 413270.2.a.ci

\( q + q^{2} + q^{4} - q^{5} + q^{8} - 3 q^{9} - q^{10} + q^{11} + q^{13} + q^{16} - 3 q^{18} + 4 q^{19} + O(q^{20}) \) 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: 137625600
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: no
Manin constant: 1 (conditional*)
Copy content comment:Manin constant
 
Copy content magma:ManinConstant(E);
 
* The Manin constant is correct provided that curve 413270.ci4 is optimal.

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 $\mathrm{ord}_p(N)$ $\mathrm{ord}_p(\Delta)$ $\mathrm{ord}_p(\mathrm{den}(j))$
$2$ $5$ $I_{5}$ split multiplicative -1 1 5 5
$5$ $2$ $I_{4}$ nonsplit multiplicative 1 1 4 4
$11$ $12$ $I_{12}$ split multiplicative -1 1 12 12
$13$ $1$ $I_{1}$ split multiplicative -1 1 1 1
$17$ $4$ $I_0^{*}$ additive 1 2 6 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 $\ell$ except those listed in the table below.

prime $\ell$ mod-$\ell$ image $\ell$-adic image
$2$ 2B 4.6.0.1

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 = [[60776, 6443, 33609, 62238], [38897, 91528, 6868, 74393], [1, 0, 8, 1], [1, 8, 0, 1], [1, 4, 4, 17], [22879, 0, 0, 97239], [70721, 91528, 36924, 74393], [95099, 72216, 80818, 20741], [7, 6, 97234, 97235], [97233, 8, 97232, 9], [72608, 34323, 64685, 11442]] GL(2,Integers(97240)).subgroup(gens)
 
Copy content magma:Gens := [[60776, 6443, 33609, 62238], [38897, 91528, 6868, 74393], [1, 0, 8, 1], [1, 8, 0, 1], [1, 4, 4, 17], [22879, 0, 0, 97239], [70721, 91528, 36924, 74393], [95099, 72216, 80818, 20741], [7, 6, 97234, 97235], [97233, 8, 97232, 9], [72608, 34323, 64685, 11442]]; sub<GL(2,Integers(97240))|Gens>;
 

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 97240 = 2^{3} \cdot 5 \cdot 11 \cdot 13 \cdot 17 \), index $48$, genus $0$, and generators

$\left(\begin{array}{rr} 60776 & 6443 \\ 33609 & 62238 \end{array}\right),\left(\begin{array}{rr} 38897 & 91528 \\ 6868 & 74393 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 22879 & 0 \\ 0 & 97239 \end{array}\right),\left(\begin{array}{rr} 70721 & 91528 \\ 36924 & 74393 \end{array}\right),\left(\begin{array}{rr} 95099 & 72216 \\ 80818 & 20741 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 97234 & 97235 \end{array}\right),\left(\begin{array}{rr} 97233 & 8 \\ 97232 & 9 \end{array}\right),\left(\begin{array}{rr} 72608 & 34323 \\ 64685 & 11442 \end{array}\right)$.

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

The torsion field $K:=\Q(E[97240])$ is a degree-$416255915851776000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/97240\Z)$.

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
$2$ split multiplicative $4$ \( 3757 = 13 \cdot 17^{2} \)
$3$ good $2$ \( 37570 = 2 \cdot 5 \cdot 13 \cdot 17^{2} \)
$5$ nonsplit multiplicative $6$ \( 41327 = 11 \cdot 13 \cdot 17^{2} \)
$11$ split multiplicative $12$ \( 37570 = 2 \cdot 5 \cdot 13 \cdot 17^{2} \)
$13$ split multiplicative $14$ \( 31790 = 2 \cdot 5 \cdot 11 \cdot 17^{2} \)
$17$ additive $146$ \( 1430 = 2 \cdot 5 \cdot 11 \cdot 13 \)

Isogenies

Copy content comment:Isogenies
 
Copy content gp:ellisomat(E)
 

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2 and 4.
Its isogeny class 413270.ci consists of 4 curves linked by isogenies of degrees dividing 4.

Twists

The minimal quadratic twist of this elliptic curve is 1430.h2, its twist by $17$.

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.