# Properties

 Label 256a2 Conductor $256$ Discriminant $32768$ j-invariant $$8000$$ CM yes ($$D=-8$$) Rank $1$ Torsion structure $$\Z/{2}\Z$$

# Learn more

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([0, 1, 0, -13, -21])

gp: E = ellinit([0, 1, 0, -13, -21])

magma: E := EllipticCurve([0, 1, 0, -13, -21]);

$$y^2=x^3+x^2-13x-21$$

## Mordell-Weil group structure

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

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(5, 8\right)$$ $\hat{h}(P)$ ≈ $0.96025159501659529387304053183$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(-3, 0\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-3, 0\right)$$, $$(-2,\pm 1)$$, $$(5,\pm 8)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$256$$ = $2^{8}$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $32768$ = $2^{15}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$8000$$ = $2^{6} \cdot 5^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z[\sqrt{-2}]$$ (potential complex multiplication) Sato-Tate group: $N(\mathrm{U}(1))$ Faltings height: $-0.40397265035989411975627760561\dots$ Stable Faltings height: $-1.2704066260598257565278177574\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $0.96025159501659529387304053183\dots$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $2.5189270468096534385807611190\dots$ 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: $2$  = $2$ 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: $1$ (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: $L'(E,1)$ ≈ $1.2094018572147058551904833617186879585$

## 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);

$$q - 2q^{3} + q^{9} - 6q^{11} - 6q^{17} - 2q^{19} + O(q^{20})$$

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 16 $\Gamma_0(N)$-optimal: no Manin constant: 1

## Local data

This elliptic curve is not semistable. There is only one prime 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 8 15 0

## 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 16.192.5.624

## $p$-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

 $p$ Reduction type $\lambda$-invariant(s) $\mu$-invariant(s) 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 add ordinary ss ss ordinary ss ordinary ordinary ss ss ss ss ordinary ordinary ss - 3 1,1 3,1 1 1,1 1 1 1,1 1,1 1,1 1,1 1 1 3,1 - 0 0,0 0,0 0 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.
Its isogeny class 256a consists of 2 curves linked by isogenies of degree 2.

## 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 $K$ $2$ $$\Q(\sqrt{2})$$ $$\Z/2\Z \times \Z/2\Z$$ 2.2.8.1-1024.1-d3 $4$ 4.0.2048.2 $$\Z/2\Z \times \Z/4\Z$$ Not in database $4$ 4.2.18432.2 $$\Z/6\Z$$ Not in database $4$ 4.0.6144.1 $$\Z/6\Z$$ Not in database $8$ 8.4.67108864.1 $$\Z/2\Z \times \Z/4\Z$$ Not in database $8$ 8.0.339738624.10 $$\Z/3\Z \times \Z/6\Z$$ Not in database $8$ 8.4.1358954496.3 $$\Z/2\Z \times \Z/6\Z$$ Not in database $8$ 8.0.150994944.2 $$\Z/2\Z \times \Z/6\Z$$ Not in database $12$ 12.0.169075682574336.4 $$\Z/18\Z$$ Not in database $16$ 16.0.18014398509481984.1 $$\Z/4\Z \times \Z/8\Z$$ Not in database $16$ 16.4.4611686018427387904.1 $$\Z/2\Z \times \Z/8\Z$$ Not in database $16$ 16.0.1846757322198614016.7 $$\Z/6\Z \times \Z/6\Z$$ Not in database $16$ 16.0.29548117155177824256.5 $$\Z/2\Z \times \Z/12\Z$$ Not in database $16$ 16.0.364791569817010176.1 $$\Z/2\Z \times \Z/12\Z$$ Not in database $20$ 20.0.84954018740373771557797888.2 $$\Z/22\Z$$ Not in database

We only show fields where the torsion growth is primitive.