Properties

Label 32490.n8
Conductor 32490
Discriminant -74080326057840
j-invariant \( \frac{357911}{2160} \)
CM no
Rank 2
Torsion Structure \(\Z/{2}\Z\)

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Minimal Weierstrass equation

sage: E = EllipticCurve([1, -1, 0, 4806, 392548]) # or
 
sage: E = EllipticCurve("32490ba1")
 
gp: E = ellinit([1, -1, 0, 4806, 392548]) \\ or
 
gp: E = ellinit("32490ba1")
 
magma: E := EllipticCurve([1, -1, 0, 4806, 392548]); // or
 
magma: E := EllipticCurve("32490ba1");
 

\( y^2 + x y = x^{3} - x^{2} + 4806 x + 392548 \)

Mordell-Weil group structure

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

Infinite order Mordell-Weil generators and heights

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

\(P\) =  \( \left(24, -734\right) \)\( \left(-16, 566\right) \)
\(\hat{h}(P)\) ≈  1.18450026015038962.9538752049603847

Torsion generators

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

\( \left(-52, 26\right) \)

Integral points

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

\( \left(-52, 26\right) \), \( \left(-27, 506\right) \), \( \left(-27, -479\right) \), \( \left(-16, 566\right) \), \( \left(-16, -550\right) \), \( \left(24, 710\right) \), \( \left(24, -734\right) \), \( \left(119, 1565\right) \), \( \left(119, -1684\right) \), \( \left(309, 5441\right) \), \( \left(309, -5750\right) \), \( \left(9164, 872666\right) \), \( \left(9164, -881830\right) \)

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: \(-74080326057840 \)  =  \(-1 \cdot 2^{4} \cdot 3^{9} \cdot 5 \cdot 19^{6} \)
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{357911}{2160} \)  =  \(2^{-4} \cdot 3^{-3} \cdot 5^{-1} \cdot 71^{3}\)
Endomorphism ring: \(\Z\)   (no 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: \(3.49097591241\)
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: \(0.443976392857\)
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: \( 16 \)  = \( 2\cdot2\cdot1\cdot2^{2} \)
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: 110592
\( \Gamma_0(N) \)-optimal: yes
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);
 

\( L^{(2)}(E,1)/2! \) ≈ \( 6.19964357258 \)

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
\(3\) \(2\) \( I_3^{*} \) Additive -1 2 9 3
\(5\) \(1\) \( I_{1} \) Split multiplicative -1 1 1 1
\(19\) \(4\) \( I_0^{*} \) 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(3,20) if E.conductor().valuation(p)<2]
 

Note: \(p\)-adic regulator data only exists for primes \(p\ge5\) of good ordinary reduction.

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 32490.n consists of 8 curves linked by isogenies of degrees dividing 12.

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 \(\Q(\sqrt{-15}) \) \(\Z/2\Z \times \Z/2\Z\) Not in database
\(\Q(\sqrt{-19}) \) \(\Z/4\Z\) Not in database
\(\Q(\sqrt{57}) \) \(\Z/6\Z\) Not in database
\(\Q(\sqrt{285}) \) \(\Z/4\Z\) Not in database
4 \(\Q(\sqrt{-3}, \sqrt{-19})\) \(\Z/12\Z\) Not in database
\(\Q(\sqrt{-15}, \sqrt{-19})\) \(\Z/2\Z \times \Z/4\Z\) Not in database
\(\Q(\sqrt{-15}, \sqrt{57})\) \(\Z/2\Z \times \Z/6\Z\) Not in database
\(\Q(\sqrt{5}, \sqrt{57})\) \(\Z/12\Z\) Not in database
6 6.0.617310000.2 \(\Z/12\Z\) 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.