# Properties

 Label 663b6 Conductor $663$ Discriminant $-7.345\times 10^{12}$ j-invariant $$\frac{6439735268725823}{7345472585373}$$ CM no Rank $1$ Torsion structure $$\Z/{2}\Z$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, 1, 1, 3876, -89910])

gp: E = ellinit([1, 1, 1, 3876, -89910])

magma: E := EllipticCurve([1, 1, 1, 3876, -89910]);

$$y^2+xy+y=x^3+x^2+3876x-89910$$

## Mordell-Weil group structure

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

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(25, 140\right)$$ $\hat{h}(P)$ ≈ $0.74395301777770823327716607750$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(\frac{83}{4}, -\frac{87}{8}\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(25, 140\right)$$, $$\left(25, -166\right)$$, $$\left(1249, 43592\right)$$, $$\left(1249, -44842\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$663$$ = $3 \cdot 13 \cdot 17$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $-7345472585373$ = $-1 \cdot 3^{4} \cdot 13 \cdot 17^{8}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{6439735268725823}{7345472585373}$$ = $3^{-4} \cdot 13^{-1} \cdot 17^{-8} \cdot 23^{3} \cdot 8089^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $1.1551559452628778738618944154\dots$ Stable Faltings height: $1.1551559452628778738618944154\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $0.74395301777770823327716607750\dots$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $0.40059551251326720618099269335\dots$ 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: $16$  = $2\cdot1\cdot2^{3}$ 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: $L'(E,1)$ ≈ $1.1920969617698112754438579084628276344$

## 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);

$$q - q^{2} - q^{3} - q^{4} - 2q^{5} + q^{6} + 3q^{8} + q^{9} + 2q^{10} + 4q^{11} + q^{12} + q^{13} + 2q^{15} - q^{16} + q^{17} - q^{18} - 4q^{19} + O(q^{20})$$

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 1024 $\Gamma_0(N)$-optimal: no Manin constant: 1

## Local data

This elliptic curve is semistable. There are 3 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$(j)_{-}$
$3$ $2$ $I_{4}$ Non-split multiplicative 1 1 4 4
$13$ $1$ $I_{1}$ Split multiplicative -1 1 1 1
$17$ $8$ $I_{8}$ Split multiplicative -1 1 8 8

## 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
$2$ 2B 8.48.0.178

## $p$-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 ordinary nonsplit ordinary ss ordinary split split ordinary ss ordinary ordinary ordinary ordinary ordinary ordinary 2 1 1 1,1 1 2 2 1 1,1 1 1 1 1 1 1 3 0 0 0,0 0 0 0 0 0,0 0 0 0 0 0 0

## Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2, 4 and 8.
Its isogeny class 663b consists of 4 curves linked by isogenies of degrees dividing 8.

## 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 $K$ $2$ $$\Q(\sqrt{-13})$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database $2$ $$\Q(\sqrt{13})$$ $$\Z/4\Z$$ Not in database $2$ $$\Q(\sqrt{-1})$$ $$\Z/4\Z$$ Not in database $4$ $$\Q(i, \sqrt{13})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database $4$ 4.0.562432.1 $$\Z/2\Z \times \Z/4\Z$$ Not in database $4$ 4.2.35152.1 $$\Z/8\Z$$ Not in database $4$ $$\Q(\zeta_{8})$$ $$\Z/8\Z$$ Not in database $4$ $$\Q(i, \sqrt{26})$$ $$\Z/8\Z$$ Not in database $8$ 8.0.316329754624.3 $$\Z/4\Z \times \Z/4\Z$$ Not in database $8$ 8.0.19770609664.3 $$\Z/2\Z \times \Z/8\Z$$ Not in database $8$ 8.0.1871773696.1 $$\Z/2\Z \times \Z/8\Z$$ Not in database $8$ 8.2.422574120899307.3 $$\Z/6\Z$$ Not in database $16$ Deg 16 $$\Z/4\Z \times \Z/8\Z$$ Not in database $16$ Deg 16 $$\Z/4\Z \times \Z/8\Z$$ Not in database $16$ Deg 16 $$\Z/16\Z$$ Not in database $16$ Deg 16 $$\Z/16\Z$$ Not in database $16$ Deg 16 $$\Z/16\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/6\Z$$ 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.