# Properties

 Label 56a4 Conductor $56$ Discriminant $14336$ j-invariant $$\frac{1443468546}{7}$$ CM no Rank $0$ Torsion structure $$\Z/{2}\Z$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Simplified equation

 $$y^2=x^3-299x+1990$$ y^2=x^3-299x+1990 (homogenize, simplify) $$y^2z=x^3-299xz^2+1990z^3$$ y^2z=x^3-299xz^2+1990z^3 (dehomogenize, simplify) $$y^2=x^3-299x+1990$$ y^2=x^3-299x+1990 (homogenize, minimize)

sage: E = EllipticCurve([0, 0, 0, -299, 1990])

gp: E = ellinit([0, 0, 0, -299, 1990])

magma: E := EllipticCurve([0, 0, 0, -299, 1990]);

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

## Mordell-Weil group structure

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

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(10, 0\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(10, 0\right)$$

## Invariants

 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: $14336$ = $2^{11} \cdot 7$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{1443468546}{7}$$ = $2 \cdot 3^{3} \cdot 7^{-1} \cdot 13^{3} \cdot 23^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $-0.0020328267985874883465435369936\dots$ Stable Faltings height: $-0.63741774231187068864567298166\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $0$ sage: E.regulator()  magma: Regulator(E); Regulator: $1$ sage: E.period_lattice().omega()  gp: if(E.disc>0,2,1)*E.omega[1]  magma: (Discriminant(E) gt 0 select 2 else 1) * 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: $1$ 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: $1$ (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: $L(E,1)$ ≈ $0.87454831418833636530086034973$

## Modular invariants

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);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 8 $\Gamma_0(N)$-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$(j)_{-}$
$2$ $1$ $II^{*}$ Additive -1 3 11 0
$7$ $1$ $I_{1}$ Non-split multiplicative 1 1 1 1

## 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 for all primes $\ell$ except those listed in the table below.

prime $\ell$ mod-$\ell$ image $\ell$-adic image
$2$ 2B 8.24.0.103
sage: gens = [[49, 8, 48, 9], [24, 3, 29, 2], [1, 0, 8, 1], [8, 43, 11, 40], [1, 4, 4, 17], [7, 6, 50, 51], [24, 1, 27, 14], [1, 8, 0, 1]]

sage: GL(2,Integers(56)).subgroup(gens)

magma: Gens := [[49, 8, 48, 9], [24, 3, 29, 2], [1, 0, 8, 1], [8, 43, 11, 40], [1, 4, 4, 17], [7, 6, 50, 51], [24, 1, 27, 14], [1, 8, 0, 1]];

magma: sub<GL(2,Integers(56))|Gens>;

The image of the adelic Galois representation has level $56$, index $48$, genus $0$, and generators

$\left(\begin{array}{rr} 49 & 8 \\ 48 & 9 \end{array}\right),\left(\begin{array}{rr} 24 & 3 \\ 29 & 2 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 8 & 43 \\ 11 & 40 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 50 & 51 \end{array}\right),\left(\begin{array}{rr} 24 & 1 \\ 27 & 14 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right)$

## $p$-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

$p$ Reduction type 2 7 add nonsplit - 0 - 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.

## Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2 and 4.
Its isogeny class 56a consists of 4 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{14})$$ $$\Z/2\Z \oplus \Z/2\Z$$ 2.2.56.1-56.1-b5 $2$ $$\Q(\sqrt{2})$$ $$\Z/4\Z$$ 2.2.8.1-392.1-b6 $2$ $$\Q(\sqrt{7})$$ $$\Z/4\Z$$ 2.2.28.1-56.1-a4 $4$ $$\Q(\sqrt{2}, \sqrt{7})$$ $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $4$ 4.4.14336.1 $$\Z/8\Z$$ Not in database $8$ 8.0.493455671296.14 $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $8$ 8.0.481890304.4 $$\Z/8\Z$$ Not in database $8$ 8.8.40282095616.2 $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $8$ 8.2.5377010688.2 $$\Z/6\Z$$ Not in database $16$ deg 16 $$\Z/4\Z \oplus \Z/4\Z$$ Not in database $16$ 16.0.59447875862838378496.3 $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $16$ deg 16 $$\Z/16\Z$$ Not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/6\Z$$ Not in database $16$ deg 16 $$\Z/12\Z$$ Not in database $16$ deg 16 $$\Z/12\Z$$ 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.