Properties

Label 287490bc4
Conductor $287490$
Discriminant $3.119\times 10^{21}$
j-invariant \( \frac{47595748626367201}{1215506250000} \)
CM no
Rank $2$
Torsion structure \(\Z/{2}\Z \times \Z/{2}\Z\)

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Show commands for: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([1, 0, 1, -10335979, -12505609594]) # or
 
sage: E = EllipticCurve("287490bc4")
 
gp: E = ellinit([1, 0, 1, -10335979, -12505609594]) \\ or
 
gp: E = ellinit("287490bc4")
 
magma: E := EllipticCurve([1, 0, 1, -10335979, -12505609594]); // or
 
magma: E := EllipticCurve("287490bc4");
 

\( y^2 + x y + y = x^{3} - 10335979 x - 12505609594 \)

Mordell-Weil group structure

\(\Z^2 \times \Z/{2}\Z \times \Z/{2}\Z\)

Infinite order Mordell-Weil generators and heights

sage: E.gens()
 
magma: Generators(E);
 

\(P\) =  \( \left(5578, -324352\right) \)\( \left(\frac{4431148}{441}, \frac{8730041753}{9261}\right) \)
\(\hat{h}(P)\) ≈  $2.923941408312019$$8.643224427902947$

Torsion generators

sage: E.torsion_subgroup().gens()
 
gp: elltors(E)
 
magma: TorsionSubgroup(E);
 

\( \left(3703, -1852\right) \), \( \left(-1625, 812\right) \)

Integral points

sage: E.integral_points()
 
magma: IntegralPoints(E);
 

\( \left(-1736, 15242\right) \), \( \left(-1736, -13507\right) \), \( \left(-1625, 812\right) \), \( \left(3703, -1852\right) \), \( \left(5578, 318773\right) \), \( \left(5578, -324352\right) \), \( \left(20131, 2807336\right) \), \( \left(20131, -2827468\right) \)

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 287490 \)  =  \(2 \cdot 3 \cdot 5 \cdot 7 \cdot 37^{2}\)
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: \(3118656485929556250000 \)  =  \(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4} \cdot 37^{6} \)
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{47595748626367201}{1215506250000} \)  =  \(2^{-4} \cdot 3^{-4} \cdot 5^{-8} \cdot 7^{-4} \cdot 13^{3} \cdot 61^{3} \cdot 457^{3}\)
Endomorphism ring: \(\Z\)
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Rank: \(2\)
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: \(17.8427753728\)
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: \(0.0843311739452\)
sage: E.tamagawa_numbers()
 
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
 
magma: TamagawaNumbers(E);
 
Tamagawa product: \( 128 \)  = \( 2\cdot2^{2}\cdot2\cdot2\cdot2^{2} \)
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
Torsion order: \(4\)
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 
Analytic order of Ш: \(1\) (rounded)

Modular invariants

Modular form 287490.2.a.bc

sage: E.q_eigenform(20)
 
gp: xy = elltaniyama(E);
 
gp: x*deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+E.a3)
 
magma: ModularForm(E);
 

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

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 25952256
\( \Gamma_0(N) \)-optimal: no
Manin constant: 1

Special L-value

sage: r = E.rank();
 
sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()
 
gp: ar = ellanalyticrank(E);
 
gp: ar[2]/factorial(ar[1])
 
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
 

\( L^{(2)}(E,1)/2! \) ≈ \( 12.037617549 \)

Local data

This elliptic curve is not semistable.

sage: E.local_data()
 
gp: ellglobalred(E)[5]
 
magma: [LocalInformation(E,p) : p in BadPrimes(E)];
 
prime Tamagawa number Kodaira symbol Reduction type Root number ord(\(N\)) ord(\(\Delta\)) ord\((j)_{-}\)
\(2\) \(2\) \( I_{4} \) Non-split multiplicative 1 1 4 4
\(3\) \(4\) \( I_{4} \) Split multiplicative -1 1 4 4
\(5\) \(2\) \( I_{8} \) Non-split multiplicative 1 1 8 8
\(7\) \(2\) \( I_{4} \) Non-split multiplicative 1 1 4 4
\(37\) \(4\) \( I_0^{*} \) Additive 1 2 6 0

Galois representations

The image of the 2-adic representation attached to this elliptic curve is the subgroup of $\GL(2,\Z_2)$ with Rouse label X189.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 0 \\ 6 & 7 \end{array}\right),\left(\begin{array}{rr} 7 & 0 \\ 0 & 7 \end{array}\right),\left(\begin{array}{rr} 3 & 0 \\ 4 & 7 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 4 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right)$ and has index 48.

sage: rho = E.galois_representation();
 
sage: [rho.image_type(p) for p in rho.non_surjective()]
 
magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];
 

The mod \( p \) Galois representation has maximal image \(\GL(2,\F_p)\) for all primes \( p \) except those listed.

prime Image of Galois representation
\(2\) Cs

$p$-adic data

$p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(3,20) if E.conductor().valuation(p)<2]
 

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

No Iwasawa invariant data is available for this curve.

Isogenies

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2 and 4.
Its isogeny class 287490bc consists of 8 curves linked by isogenies of degrees dividing 16.

Growth of torsion in number fields

The number fields $K$ of degree up to 7 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \times \Z/{2}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$2$ \(\Q(\sqrt{37}) \) \(\Z/2\Z \times \Z/4\Z\) Not in database
$2$ \(\Q(\sqrt{-37}) \) \(\Z/2\Z \times \Z/4\Z\) Not in database
$4$ \(\Q(i, \sqrt{37})\) \(\Z/4\Z \times \Z/4\Z\) Not in database
$4$ \(\Q(\sqrt{-6}, \sqrt{37})\) \(\Z/2\Z \times \Z/8\Z\) Not in database
$4$ \(\Q(\sqrt{6}, \sqrt{37})\) \(\Z/2\Z \times \Z/8\Z\) Not in database

We only show fields where the torsion growth is primitive. For each field $K$ we either show its label, or a defining polynomial when $K$ is not in the database.