Properties

Label 46410b2
Conductor 46410
Discriminant 15197834433600
j-invariant \( \frac{127787213284071769}{15197834433600} \)
CM no
Rank 2
Torsion Structure \(\Z/{2}\Z \times \Z/{2}\Z\)

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Minimal Weierstrass equation

magma: E := EllipticCurve([1, 1, 0, -10493, 364413]); // or
 
magma: E := EllipticCurve("46410b2");
 
sage: E = EllipticCurve([1, 1, 0, -10493, 364413]) # or
 
sage: E = EllipticCurve("46410b2")
 
gp: E = ellinit([1, 1, 0, -10493, 364413]) \\ or
 
gp: E = ellinit("46410b2")
 

\( y^2 + x y = x^{3} + x^{2} - 10493 x + 364413 \)

Mordell-Weil group structure

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

Infinite order Mordell-Weil generators and heights

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

\(P\) =  \( \left(91, -462\right) \)\( \left(-79, 881\right) \)
\(\hat{h}(P)\) ≈  1.22299464412.51419662016

Torsion generators

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

\( \left(42, -21\right) \), \( \left(74, -37\right) \)

Integral points

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

\( \left(-86, 843\right) \), \( \left(-79, 881\right) \), \( \left(-62, 915\right) \), \( \left(-7, 665\right) \), \( \left(29, 278\right) \), \( \left(34, 203\right) \), \( \left(42, -21\right) \), \( \left(74, -37\right) \), \( \left(91, 371\right) \), \( \left(123, 915\right) \), \( \left(146, 1331\right) \), \( \left(159, 1578\right) \), \( \left(434, 8603\right) \), \( \left(679, 17178\right) \), \( \left(10442, 1061819\right) \), \( \left(55154, 12925403\right) \)

Note: only one of each pair $\pm P$ is listed.

Invariants

magma: Conductor(E);
 
sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
Conductor: \( 46410 \)  =  \(2 \cdot 3 \cdot 5 \cdot 7 \cdot 13 \cdot 17\)
magma: Discriminant(E);
 
sage: E.discriminant().factor()
 
gp: E.disc
 
Discriminant: \(15197834433600 \)  =  \(2^{6} \cdot 3^{4} \cdot 5^{2} \cdot 7^{4} \cdot 13^{2} \cdot 17^{2} \)
magma: jInvariant(E);
 
sage: E.j_invariant().factor()
 
gp: E.j
 
j-invariant: \( \frac{127787213284071769}{15197834433600} \)  =  \(2^{-6} \cdot 3^{-4} \cdot 5^{-2} \cdot 7^{-4} \cdot 13^{-2} \cdot 17^{-2} \cdot 109^{3} \cdot 4621^{3}\)
Endomorphism ring: \(\Z\)   (no Complex Multiplication)
Sato-Tate Group: $\mathrm{SU}(2)$

BSD invariants

magma: Rank(E);
 
sage: E.rank()
 
Rank: \(2\)
magma: Regulator(E);
 
sage: E.regulator()
 
Regulator: \(1.97492848297\)
magma: RealPeriod(E);
 
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
Real period: \(0.676447010267\)
magma: TamagawaNumbers(E);
 
sage: E.tamagawa_numbers()
 
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
 
Tamagawa product: \( 64 \)  = \( 2\cdot2\cdot2\cdot2\cdot2\cdot2 \)
magma: Order(TorsionSubgroup(E));
 
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
Torsion order: \(4\)
magma: MordellWeilShaInformation(E);
 
sage: E.sha().an_numerical()
 
Analytic order of Ш: \(1\) (rounded)

Modular invariants

Modular form 46410.2.a.a

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

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

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

magma: ModularDegree(E);
 
sage: E.modular_degree()
 
Modular degree: 147456
\( \Gamma_0(N) \)-optimal: no
Manin constant: 1

Special L-value

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

\( L^{(2)}(E,1)/2! \) ≈ \( 5.34373787119 \)

Local data

magma: [LocalInformation(E,p) : p in BadPrimes(E)];
 
sage: E.local_data()
 
gp: ellglobalred(E)[5]
 
prime Tamagawa number Kodaira symbol Reduction type Root number ord(\(N\)) ord(\(\Delta\)) ord\((j)_{-}\)
\(2\) \(2\) \( I_{6} \) Non-split multiplicative 1 1 6 6
\(3\) \(2\) \( I_{4} \) Non-split multiplicative 1 1 4 4
\(5\) \(2\) \( I_{2} \) Non-split multiplicative 1 1 2 2
\(7\) \(2\) \( I_{4} \) Non-split multiplicative 1 1 4 4
\(13\) \(2\) \( I_{2} \) Non-split multiplicative 1 1 2 2
\(17\) \(2\) \( I_{2} \) Split multiplicative -1 1 2 2

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 X8c.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 0 & 7 \end{array}\right),\left(\begin{array}{rr} 3 & 6 \\ 0 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 5 \end{array}\right)$ and has index 12.

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

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.

Iwasawa invariants

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Reduction type nonsplit nonsplit nonsplit nonsplit ordinary nonsplit split ss ordinary ordinary ss ordinary ordinary ordinary ss
$\lambda$-invariant(s) 5 2 4 2 2 2 3 2,2 2 2 2,2 2 2 2 2,2
$\mu$-invariant(s) 1 0 0 0 0 0 0 0,0 0 0 0,0 0 0 0 0,0

Isogenies

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2.
Its isogeny class 46410b consists of 4 curves linked by isogenies of degrees dividing 4.

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
4 \(\Q(\sqrt{-13}, \sqrt{-85})\) \(\Z/2\Z \times \Z/4\Z\) Not in database
\(\Q(\sqrt{2}, \sqrt{85})\) \(\Z/2\Z \times \Z/4\Z\) Not in database
\(\Q(\sqrt{-2}, \sqrt{13})\) \(\Z/2\Z \times \Z/4\Z\) 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.