Label 56.a3
Conductor $56$
Discriminant $50176$
j-invariant \( \frac{740772}{49} \)
CM no
Rank $0$
Torsion structure \(\Z/{2}\Z \times \Z/{2}\Z\)

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Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([0, 0, 0, -19, 30])
gp: E = ellinit([0, 0, 0, -19, 30])
magma: E := EllipticCurve([0, 0, 0, -19, 30]);

\(y^2=x^3-19x+30\)  Toggle raw display

Mordell-Weil group structure

$\Z/{2}\Z \times \Z/{2}\Z$

Torsion generators

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

\( \left(2, 0\right) \), \( \left(3, 0\right) \)  Toggle raw display

Integral points

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

\( \left(-5, 0\right) \), \( \left(2, 0\right) \), \( \left(3, 0\right) \)  Toggle raw display


sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
Conductor: \( 56 \)  =  $2^{3} \cdot 7$
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
Discriminant: $50176 $  =  $2^{10} \cdot 7^{2} $
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
j-invariant: \( \frac{740772}{49} \)  =  $2^{2} \cdot 3^{3} \cdot 7^{-2} \cdot 19^{3}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $-0.34860641707856014305515959772\dots$
Stable Faltings height: $-0.92622906754518123423618636560\dots$

BSD invariants

sage: E.rank()
magma: Rank(E);
Analytic rank: $0$
sage: E.regulator()
magma: Regulator(E);
Regulator: $1$
sage: E.period_lattice().omega()
magma: RealPeriod(E);
Real period: $3.4981932567533454612034413989\dots$
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: $ 4 $  = $ 2\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: $1$ (exact)
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);
Special value: $ L(E,1) $ ≈ $ 0.87454831418833636530086034972735623668 $

Modular invariants

Modular form   56.2.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);

\( q + 2q^{5} - q^{7} - 3q^{9} - 4q^{11} + 2q^{13} - 6q^{17} + 8q^{19} + O(q^{20}) \)  Toggle raw display

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

sage: E.modular_degree()
magma: ModularDegree(E);
Modular degree: 4
$ \Gamma_0(N) $-optimal: no
Manin constant: 1

Local data

This elliptic curve is not semistable. There are 2 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$(j)_{-}$
$2$ $2$ $III^{*}$ Additive -1 3 10 0
$7$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2

Galois representations

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 $\ell$-adic Galois representation has maximal image $\GL(2,\Z_\ell)$ for all primes $\ell$ except those listed in the table below.

prime $\ell$ mod-$\ell$ image $\ell$-adic image
$2$ 2Cs

$p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(5,20) if E.conductor().valuation(p)<2]

All $p$-adic regulators are identically $1$ since the rank is $0$.

Iwasawa invariants

$p$ 2 7
Reduction type add nonsplit
$\lambda$-invariant(s) - 0
$\mu$-invariant(s) - 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ of good reduction are zero.

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


This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2.
Its isogeny class 56.a consists of 2 curves linked by isogenies of degrees dividing 4.

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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \times \Z/{2}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$2$ \(\Q(\sqrt{2}) \) \(\Z/2\Z \times \Z/4\Z\)
$4$ \(\Q(i, \sqrt{7})\) \(\Z/2\Z \times \Z/4\Z\) Not in database
$4$ \(\Q(\sqrt{-2}, \sqrt{-7})\) \(\Z/2\Z \times \Z/4\Z\) Not in database
$8$ 8.0.157351936.1 \(\Z/4\Z \times \Z/4\Z\) Not in database
$8$ 8.4.822083584.1 \(\Z/2\Z \times \Z/8\Z\) Not in database
$8$ 8.2.5377010688.2 \(\Z/2\Z \times \Z/6\Z\) Not in database
$16$ 16.0.232218265089212416.1 \(\Z/2\Z \times \Z/8\Z\) 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

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.