# Properties

 Label 6400.x1 Conductor $6400$ Discriminant $512000000$ j-invariant $$8000$$ CM yes ($$D=-8$$) Rank $1$ Torsion structure $$\Z/{2}\Z$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([0, -1, 0, -333, -1963])

gp: E = ellinit([0, -1, 0, -333, -1963])

magma: E := EllipticCurve([0, -1, 0, -333, -1963]);

$$y^2=x^3-x^2-333x-1963$$

## Mordell-Weil group structure

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

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(-\frac{68}{9}, \frac{217}{27}\right)$$ $\hat{h}(P)$ ≈ $4.7593679832490537374420804510$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(-13, 0\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-13, 0\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$6400$$ = $2^{8} \cdot 5^{2}$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $512000000$ = $2^{15} \cdot 5^{6}$ 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: $0.40074630585715606754410206100\dots$ Stable Faltings height: $-1.2704066260598257565278177574\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $4.7593679832490537374420804510\dots$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $1.1264984214058359704960458699\dots$ 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: $2$ 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: $L'(E,1)$ ≈ $5.3614205200195362094023405134712007004$

## 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);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 2304 $\Gamma_0(N)$-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 8 15 0
$5$ $2$ $I_0^{*}$ Additive 1 2 6 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 $\ell$.

## $p$-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 add ss ordinary ss ordinary ordinary ss ss ss ss ordinary ordinary ss - 1 - 3,1 1 1,1 1 3 1,1 3,1 1,1 1,3 1 1 1,1 - 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 6400.x 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 $K$ $2$ $$\Q(\sqrt{2})$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database $4$ 4.0.51200.2 $$\Z/2\Z \times \Z/4\Z$$ Not in database $4$ 4.2.460800.2 $$\Z/6\Z$$ Not in database $4$ 4.0.153600.3 $$\Z/6\Z$$ Not in database $8$ 8.4.41943040000.4 $$\Z/2\Z \times \Z/4\Z$$ Not in database $8$ 8.0.212336640000.40 $$\Z/3\Z \times \Z/6\Z$$ Not in database $8$ 8.4.849346560000.35 $$\Z/2\Z \times \Z/6\Z$$ Not in database $8$ 8.0.94371840000.12 $$\Z/2\Z \times \Z/6\Z$$ Not in database $12$ Deg 12 $$\Z/18\Z$$ Not in database $16$ Deg 16 $$\Z/4\Z \times \Z/4\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/8\Z$$ Not in database $16$ Deg 16 $$\Z/6\Z \times \Z/6\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/12\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/12\Z$$ Not in database $20$ 20.0.829629089261462612869120000000000.1 $$\Z/22\Z$$ 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.