Properties

Label 256.d1
Conductor $256$
Discriminant $32768$
j-invariant \( 8000 \)
CM yes (\(D=-8\))
Rank $0$
Torsion structure \(\Z/{2}\Z\)

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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]);
 

\(y^2=x^3-x^2-13x+21\)  Toggle raw display

Mordell-Weil group structure

$\Z/{2}\Z$

Torsion generators

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

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

Integral points

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

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

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: $0$
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: $1$
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: $3.5623007922266199348181877138\dots$
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) $ ≈ $ 1.7811503961133099674090938569022932435 $

Modular invariants

Modular form   256.2.a.d

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);
 

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

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 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 16.192.5.602

$p$-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
Reduction type add
$\lambda$-invariant(s) -
$\mu$-invariant(s) -

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.
Its isogeny class 256.d 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:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$2$ \(\Q(\sqrt{2}) \) \(\Z/2\Z \times \Z/2\Z\) 2.2.8.1-1024.1-d4
$4$ \(\Q(\zeta_{16})^+\) \(\Z/2\Z \times \Z/4\Z\) 4.4.2048.1-64.1-a6
$4$ 4.0.6144.2 \(\Z/6\Z\) Not in database
$4$ 4.2.18432.1 \(\Z/6\Z\) Not in database
$8$ 8.0.67108864.1 \(\Z/2\Z \times \Z/4\Z\) Not in database
$8$ 8.4.67108864.1 \(\Z/2\Z \times \Z/8\Z\) Not in database
$8$ 8.0.339738624.9 \(\Z/3\Z \times \Z/6\Z\) Not in database
$8$ 8.0.150994944.2 \(\Z/2\Z \times \Z/6\Z\) Not in database
$8$ 8.4.1358954496.3 \(\Z/2\Z \times \Z/6\Z\) Not in database
$12$ 12.0.169075682574336.3 \(\Z/18\Z\) Not in database
$16$ 16.0.18014398509481984.1 \(\Z/4\Z \times \Z/8\Z\) Not in database
$16$ 16.0.1846757322198614016.7 \(\Z/6\Z \times \Z/6\Z\) Not in database
$16$ 16.0.364791569817010176.1 \(\Z/2\Z \times \Z/12\Z\) Not in database
$16$ 16.8.29548117155177824256.1 \(\Z/2\Z \times \Z/12\Z\) Not in database
$20$ 20.0.84954018740373771557797888.1 \(\Z/22\Z\) Not in database

We only show fields where the torsion growth is primitive.