Properties

Label 13680bh2
Conductor $13680$
Discriminant $970146201600$
j-invariant \( \frac{758800078561}{324900} \)
CM no
Rank $2$
Torsion structure \(\Z/{2}\Z \times \Z/{2}\Z\)

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Minimal Weierstrass equation

sage: E = EllipticCurve([0, 0, 0, -27363, 1741538]) # or
 
sage: E = EllipticCurve("13680bh2")
 
gp: E = ellinit([0, 0, 0, -27363, 1741538]) \\ or
 
gp: E = ellinit("13680bh2")
 
magma: E := EllipticCurve([0, 0, 0, -27363, 1741538]); // or
 
magma: E := EllipticCurve("13680bh2");
 

\( y^2 = x^{3} - 27363 x + 1741538 \)

Mordell-Weil group structure

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

Infinite order Mordell-Weil generators and heights

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

\(P\) =  \( \left(-31, 1600\right) \)\( \left(-1, 1330\right) \)
\(\hat{h}(P)\) ≈  $2.557603475845549$$1.953808032170655$

Torsion generators

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

\( \left(94, 0\right) \), \( \left(97, 0\right) \)

Integral points

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

\( \left(-191, 0\right) \), \((-146,\pm 1620)\), \((-31,\pm 1600)\), \((-1,\pm 1330)\), \((49,\pm 720)\), \((79,\pm 270)\), \( \left(94, 0\right) \), \( \left(97, 0\right) \), \((98,\pm 34)\), \((103,\pm 126)\), \((113,\pm 304)\), \((151,\pm 1026)\), \((241,\pm 3024)\), \((322,\pm 5130)\), \((823,\pm 23166)\), \((961,\pm 29376)\), \((1633,\pm 65664)\)

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 13680 \)  =  \(2^{4} \cdot 3^{2} \cdot 5 \cdot 19\)
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: \(970146201600 \)  =  \(2^{14} \cdot 3^{8} \cdot 5^{2} \cdot 19^{2} \)
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{758800078561}{324900} \)  =  \(2^{-2} \cdot 3^{-2} \cdot 5^{-2} \cdot 7^{3} \cdot 19^{-2} \cdot 1303^{3}\)
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: \(1.67495223327\)
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: \(0.86631329634\)
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: \( 64 \)  = \( 2^{2}\cdot2^{2}\cdot2\cdot2 \)
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
Torsion order: \(4\)
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 
Analytic order of Ш: \(1\) (rounded)

Modular invariants

Modular form 13680.2.a.a

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);
 

\( q - q^{5} - 4q^{7} - 4q^{11} - 2q^{13} + 2q^{17} + q^{19} + O(q^{20}) \)

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 36864
\( \Gamma_0(N) \)-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);
 

\( L^{(2)}(E,1)/2! \) ≈ \( 5.80413356166 \)

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\((j)_{-}\)
\(2\) \(4\) \( I_6^{*} \) Additive -1 4 14 2
\(3\) \(4\) \( I_2^{*} \) Additive -1 2 8 2
\(5\) \(2\) \( I_{2} \) Non-split multiplicative 1 1 2 2
\(19\) \(2\) \( I_{2} \) 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.

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
\(2\) 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 add add nonsplit ordinary ordinary ordinary ordinary split ordinary ordinary ordinary ordinary ordinary ordinary ss
$\lambda$-invariant(s) - - 2 2 6 2 2 3 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

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.
Its isogeny class 13680bh 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
$4$ \(\Q(\sqrt{2}, \sqrt{3})\) \(\Z/2\Z \times \Z/4\Z\) Not in database
$4$ \(\Q(\sqrt{-3}, \sqrt{-95})\) \(\Z/2\Z \times \Z/4\Z\) Not in database
$4$ \(\Q(\sqrt{-2}, \sqrt{-285})\) \(\Z/2\Z \times \Z/4\Z\) 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.