Properties

Label 59290br3
Conductor $59290$
Discriminant $1.140\times 10^{18}$
j-invariant \( \frac{74565301329}{5468750} \)
CM no
Rank $2$
Torsion structure \(\Z/{2}\Z\)

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Minimal Weierstrass equation

sage: E = EllipticCurve([1, -1, 0, -519899, 134964143]) # or
 
sage: E = EllipticCurve("59290br3")
 
gp: E = ellinit([1, -1, 0, -519899, 134964143]) \\ or
 
gp: E = ellinit("59290br3")
 
magma: E := EllipticCurve([1, -1, 0, -519899, 134964143]); // or
 
magma: E := EllipticCurve("59290br3");
 

\( y^2 + x y = x^{3} - x^{2} - 519899 x + 134964143 \)

Mordell-Weil group structure

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

Infinite order Mordell-Weil generators and heights

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

\(P\) =  \( \left(-593, 15609\right) \)\( \left(-173, 14909\right) \)
\(\hat{h}(P)\) ≈  $1.351088065370785$$0.9743155089393529$

Torsion generators

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

\( \left(\frac{2003}{4}, -\frac{2003}{8}\right) \)

Integral points

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

\( \left(-593, 15609\right) \), \( \left(-593, -15016\right) \), \( \left(-173, 14909\right) \), \( \left(-173, -14736\right) \), \( \left(157, 7484\right) \), \( \left(157, -7641\right) \), \( \left(289, 2820\right) \), \( \left(289, -3109\right) \), \( \left(557, 3959\right) \), \( \left(557, -4516\right) \), \( \left(1367, 43784\right) \), \( \left(1367, -45151\right) \), \( \left(1493, 51071\right) \), \( \left(1493, -52564\right) \), \( \left(7307, 617984\right) \), \( \left(7307, -625291\right) \)

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 59290 \)  =  \(2 \cdot 5 \cdot 7^{2} \cdot 11^{2}\)
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: \(1139809891111718750 \)  =  \(2 \cdot 5^{8} \cdot 7^{7} \cdot 11^{6} \)
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{74565301329}{5468750} \)  =  \(2^{-1} \cdot 3^{3} \cdot 5^{-8} \cdot 7^{-1} \cdot 23^{3} \cdot 61^{3}\)
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: \(0.893328971989\)
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: \(0.269016327463\)
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: \( 128 \)  = \( 1\cdot2^{3}\cdot2^{2}\cdot2^{2} \)
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
Torsion order: \(2\)
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 
Analytic order of Ш: \(1\) (rounded)

Modular invariants

Modular form 59290.2.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);
 

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

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 983040
\( \Gamma_0(N) \)-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);
 

\( L^{(2)}(E,1)/2! \) ≈ \( 7.69024253635 \)

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\((j)_{-}\)
\(2\) \(1\) \( I_{1} \) Non-split multiplicative 1 1 1 1
\(5\) \(8\) \( I_{8} \) Split multiplicative -1 1 8 8
\(7\) \(4\) \( I_1^{*} \) Additive -1 2 7 1
\(11\) \(4\) \( I_0^{*} \) 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 X34.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 5 & 0 \\ 4 & 3 \end{array}\right),\left(\begin{array}{rr} 7 & 7 \\ 4 & 3 \end{array}\right),\left(\begin{array}{rr} 3 & 0 \\ 0 & 3 \end{array}\right),\left(\begin{array}{rr} 7 & 0 \\ 4 & 3 \end{array}\right)$ and has index 12.

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
\(2\) B

$p$-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 ss split add add ordinary ordinary ss ss ordinary ordinary ordinary ordinary ordinary ordinary
$\lambda$-invariant(s) 5 2,2 3 - - 2 2 2,2 2,2 2 2 2 2 2 2
$\mu$-invariant(s) 0 0,0 0 - - 0 0 0,0 0,0 0 0 0 0 0 0

An entry - indicates that the invariants are not computed because the reduction is additive.

Isogenies

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2 and 4.
Its isogeny class 59290br 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$2$ \(\Q(\sqrt{11}) \) \(\Z/4\Z\) Not in database
$2$ \(\Q(\sqrt{14}) \) \(\Z/2\Z \times \Z/2\Z\) Not in database
$2$ \(\Q(\sqrt{154}) \) \(\Z/4\Z\) Not in database
$4$ \(\Q(\sqrt{11}, \sqrt{14})\) \(\Z/2\Z \times \Z/4\Z\) 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.