# Properties

 Label 12696.p1 Conductor $12696$ Discriminant $3027538944$ j-invariant $$\frac{15043017316604}{243}$$ CM no Rank $2$ Torsion structure $$\Z/{2}\Z$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([0, 1, 0, -119224, 15805376])

gp: E = ellinit([0, 1, 0, -119224, 15805376])

magma: E := EllipticCurve([0, 1, 0, -119224, 15805376]);

$$y^2=x^3+x^2-119224x+15805376$$

## Mordell-Weil group structure

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

### Infinite order Mordell-Weil generators and heights

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(200, 24\right)$$ $$\left(215, 396\right)$$ $\hat{h}(P)$ ≈ $0.51047435559892144628407288162$ $2.4679349026126002518021409787$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(199, 0\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$(-376,\pm 2760)$$, $$(107,\pm 2070)$$, $$(176,\pm 552)$$, $$\left(199, 0\right)$$, $$(200,\pm 24)$$, $$(215,\pm 396)$$, $$(235,\pm 906)$$, $$(368,\pm 4680)$$, $$(1160,\pm 37944)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$12696$$ = $2^{3} \cdot 3 \cdot 23^{2}$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $3027538944$ = $2^{10} \cdot 3^{5} \cdot 23^{3}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{15043017316604}{243}$$ = $2^{2} \cdot 3^{-5} \cdot 15551^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $1.3646460450840540515759335759\dots$ Stable Faltings height: $0.0031498406351455376932186000658\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $2$ sage: E.regulator()  magma: Regulator(E); Regulator: $1.2571135648524588511231738037\dots$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $1.0160969965156070809275590632\dots$ 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: $20$  = $2\cdot5\cdot2$ 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: $1$ (rounded) 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: $L^{(2)}(E,1)/2!$ ≈ $6.3867465876280563870394107373782985934$

## Modular invariants

Modular form 12696.2.a.p

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);

$$q + q^{3} - 2q^{5} - 2q^{7} + q^{9} - 4q^{11} - 6q^{13} - 2q^{15} - 6q^{17} - 6q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 46080 $\Gamma_0(N)$-optimal: no Manin constant: 1

## Local data

This elliptic curve is not 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$(j)_{-}$
$2$ $2$ $III^{*}$ Additive -1 3 10 0
$3$ $5$ $I_{5}$ Split multiplicative -1 1 5 5
$23$ $2$ $III$ Additive -1 2 3 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 2.3.0.1

## $p$-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 split ordinary ordinary ordinary ordinary ordinary ordinary add ordinary ordinary ordinary ordinary ordinary ordinary - 3 4 4 2 2 2 2 - 2 2 2 2 2 2 - 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 12696.p 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 $K$ $2$ $$\Q(\sqrt{69})$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database $4$ 4.0.584016.1 $$\Z/4\Z$$ Not in database $8$ 8.4.442032795979776.7 $$\Z/2\Z \times \Z/4\Z$$ Not in database $8$ 8.0.3069672194304.25 $$\Z/2\Z \times \Z/4\Z$$ Not in database $8$ Deg 8 $$\Z/6\Z$$ Not in database $16$ Deg 16 $$\Z/8\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/6\Z$$ 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.