Properties

Label 46410s2
Conductor 46410
Discriminant 82743765249600
j-invariant \( \frac{23304472877725373881}{82743765249600} \)
CM no
Rank 1
Torsion Structure \(\Z/{2}\Z \times \Z/{2}\Z\)

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Show commands for: Magma / SageMath / Pari/GP

Minimal Weierstrass equation

magma: E := EllipticCurve([1, 1, 0, -59507, 5545389]); // or
 
magma: E := EllipticCurve("46410s2");
 
sage: E = EllipticCurve([1, 1, 0, -59507, 5545389]) # or
 
sage: E = EllipticCurve("46410s2")
 
gp: E = ellinit([1, 1, 0, -59507, 5545389]) \\ or
 
gp: E = ellinit("46410s2")
 

\( y^2 + x y = x^{3} + x^{2} - 59507 x + 5545389 \)

Mordell-Weil group structure

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

Infinite order Mordell-Weil generator and height

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

\(P\) =  \( \left(110, -643\right) \)
\(\hat{h}(P)\) ≈  0.489267509593

Torsion generators

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

\( \left(134, -67\right) \), \( \left(\frac{587}{4}, -\frac{587}{8}\right) \)

Integral points

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

\( \left(-282, 141\right) \), \( \left(-250, 2333\right) \), \( \left(-2, 2381\right) \), \( \left(110, 533\right) \), \( \left(134, -67\right) \), \( \left(159, 288\right) \), \( \left(173, 596\right) \), \( \left(355, 5237\right) \), \( \left(593, 13091\right) \)

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: \(82743765249600 \)  =  \(2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 7^{6} \cdot 13^{2} \cdot 17^{2} \)
magma: jInvariant(E);
 
sage: E.j_invariant().factor()
 
gp: E.j
 
j-invariant: \( \frac{23304472877725373881}{82743765249600} \)  =  \(2^{-6} \cdot 3^{-2} \cdot 5^{-2} \cdot 7^{-6} \cdot 13^{-2} \cdot 17^{-2} \cdot 43^{3} \cdot 181^{3} \cdot 367^{3}\)
Endomorphism ring: \(\Z\)   (no Complex Multiplication)
Sato-Tate Group: $\mathrm{SU}(2)$

BSD invariants

magma: Rank(E);
 
sage: E.rank()
 
Rank: \(1\)
magma: Regulator(E);
 
sage: E.regulator()
 
Regulator: \(0.489267509593\)
magma: RealPeriod(E);
 
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
Real period: \(0.610472930288\)
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: \( 192 \)  = \( 2\cdot2\cdot2\cdot( 2 \cdot 3 )\cdot2\cdot2 \)
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: \(1\) (exact)

Modular invariants

Modular form 46410.2.a.n

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)
 

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

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

magma: ModularDegree(E);
 
sage: E.modular_degree()
 
Modular degree: 221184
\( \Gamma_0(N) \)-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])
 

\( L'(E,1) \) ≈ \( 3.58421484331 \)

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
\(3\) \(2\) \( I_{2} \) Non-split multiplicative 1 1 2 2
\(5\) \(2\) \( I_{2} \) Split multiplicative -1 1 2 2
\(7\) \(6\) \( I_{6} \) Split multiplicative -1 1 6 6
\(13\) \(2\) \( I_{2} \) 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 X8.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^1\Z_2)$ generated by $$ and has index 6.

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 split split ordinary split split ordinary ss ordinary ss ordinary ordinary ordinary ss
$\lambda$-invariant(s) 7 1 2 2 1 2 2 1 1,1 1 1,3 1 1 1 1,1
$\mu$-invariant(s) 0 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 46410s 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 \(\Q(\sqrt{-26}, \sqrt{-35})\) \(\Z/2\Z \times \Z/4\Z\) Not in database
\(\Q(\sqrt{26}, \sqrt{-51})\) \(\Z/2\Z \times \Z/4\Z\) Not in database
\(\Q(\sqrt{35}, \sqrt{51})\) \(\Z/2\Z \times \Z/4\Z\) 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.