Properties

Label 32490ba5
Conductor $32490$
Discriminant $1.158\times 10^{15}$
j-invariant \( \frac{2656166199049}{33750} \)
CM no
Rank $2$
Torsion structure \(\Z/{2}\Z\)

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Minimal Weierstrass equation

sage: E = EllipticCurve([1, -1, 0, -937404, 349562578])
 
gp: E = ellinit([1, -1, 0, -937404, 349562578])
 
magma: E := EllipticCurve([1, -1, 0, -937404, 349562578]);
 

\(y^2+xy=x^3-x^2-937404x+349562578\)

Mordell-Weil group structure

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

Infinite order Mordell-Weil generators and heights

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

\(P\) =  \( \left(537, 634\right) \)\( \left(557, -211\right) \)
\(\hat{h}(P)\) ≈  $0.73846880124009612570128153239$$1.1845002601503894810985557896$

Torsion generators

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

\( \left(\frac{2243}{4}, -\frac{2243}{8}\right) \)

Integral points

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

\( \left(-793, 24764\right) \), \( \left(-793, -23971\right) \), \( \left(-523, 26654\right) \), \( \left(-523, -26131\right) \), \( \left(347, 7949\right) \), \( \left(347, -8296\right) \), \( \left(537, 634\right) \), \( \left(537, -1171\right) \), \( \left(557, -211\right) \), \( \left(557, -346\right) \), \( \left(567, 29\right) \), \( \left(567, -596\right) \), \( \left(1259, 33485\right) \), \( \left(1259, -34744\right) \), \( \left(1373, 39926\right) \), \( \left(1373, -41299\right) \), \( \left(1967, 77204\right) \), \( \left(1967, -79171\right) \), \( \left(427049, 278857706\right) \), \( \left(427049, -279284755\right) \)

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 32490 \)  =  \(2 \cdot 3^{2} \cdot 5 \cdot 19^{2}\)
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: \(1157505094653750 \)  =  \(2 \cdot 3^{9} \cdot 5^{4} \cdot 19^{6} \)
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{2656166199049}{33750} \)  =  \(2^{-1} \cdot 3^{-3} \cdot 5^{-4} \cdot 11^{3} \cdot 1259^{3}\)
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);
 
Analytic rank: \(2\)
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: \(0.87274397810343758994403076587\)
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: \(0.44397639285732097984766907641\)
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: \( 64 \)  = \( 1\cdot2^{2}\cdot2^{2}\cdot2^{2} \)
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 32490.2.a.n

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);
 

\( q - q^{2} + q^{4} + q^{5} - 4q^{7} - q^{8} - q^{10} - 2q^{13} + 4q^{14} + q^{16} - 6q^{17} + O(q^{20}) \)

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 442368
\( \Gamma_0(N) \)-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);
 

\( L^{(2)}(E,1)/2! \) ≈ \( 6.1996435725810071431090883377542709939 \)

Local data

This elliptic curve is not semistable. There are 4 primes of bad reduction:

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
\(3\) \(4\) \(I_3^{*}\) Additive -1 2 9 3
\(5\) \(4\) \(I_{4}\) Split multiplicative -1 1 4 4
\(19\) \(4\) \(I_0^{*}\) 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 X13.

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

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
\(3\) B

$p$-adic data

$p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(5,20) 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 add split ordinary ss ordinary ordinary add ss ordinary ordinary ordinary ordinary ordinary ss
$\lambda$-invariant(s) 3 - 3 2 2,2 4 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, 3, 4, 6 and 12.
Its isogeny class 32490ba consists of 6 curves linked by isogenies of degrees dividing 12.

Growth of torsion in number fields

The number fields $K$ of degree less than 24 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
$2$ \(\Q(\sqrt{6}) \) \(\Z/2\Z \times \Z/2\Z\) Not in database
$2$ \(\Q(\sqrt{114}) \) \(\Z/4\Z\) Not in database
$2$ \(\Q(\sqrt{19}) \) \(\Z/4\Z\) Not in database
$2$ \(\Q(\sqrt{57}) \) \(\Z/6\Z\) Not in database
$4$ \(\Q(\sqrt{6}, \sqrt{19})\) \(\Z/2\Z \times \Z/4\Z\) Not in database
$4$ \(\Q(\sqrt{6}, \sqrt{38})\) \(\Z/2\Z \times \Z/6\Z\) Not in database
$4$ \(\Q(\sqrt{2}, \sqrt{57})\) \(\Z/12\Z\) Not in database
$4$ \(\Q(\sqrt{3}, \sqrt{19})\) \(\Z/12\Z\) Not in database
$6$ 6.0.617310000.2 \(\Z/6\Z\) Not in database
$8$ 8.0.398475694964736.89 \(\Z/2\Z \times \Z/4\Z\) Not in database
$8$ Deg 8 \(\Z/8\Z\) Not in database
$8$ 8.0.243210263040000.101 \(\Z/8\Z\) Not in database
$8$ 8.8.691798081536.1 \(\Z/2\Z \times \Z/12\Z\) Not in database
$12$ Deg 12 \(\Z/3\Z \times \Z/6\Z\) Not in database
$12$ Deg 12 \(\Z/2\Z \times \Z/6\Z\) Not in database
$12$ Deg 12 \(\Z/12\Z\) Not in database
$12$ Deg 12 \(\Z/12\Z\) Not in database
$16$ Deg 16 \(\Z/4\Z \times \Z/4\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \times \Z/8\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \times \Z/8\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \times \Z/12\Z\) Not in database
$16$ Deg 16 \(\Z/24\Z\) Not in database
$16$ Deg 16 \(\Z/24\Z\) Not in database
$18$ 18.6.179384045603582512084728341700000000.1 \(\Z/18\Z\) Not in database

We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.