# Properties

 Label 10626.o2 Conductor $10626$ Discriminant $-4.097\times 10^{12}$ j-invariant $$\frac{6360314548472639}{4097346156288}$$ CM no Rank $1$ Torsion structure $$\Z/{2}\Z$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, 0, 0, 3860, 31376])

gp: E = ellinit([1, 0, 0, 3860, 31376])

magma: E := EllipticCurve([1, 0, 0, 3860, 31376]);

$$y^2+xy=x^3+3860x+31376$$

## Mordell-Weil group structure

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

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(38, 464\right)$$ $\hat{h}(P)$ ≈ $0.093353212310205089306450475121$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(-8, 4\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-8, 4\right)$$, $$\left(-4, 128\right)$$, $$\left(-4, -124\right)$$, $$\left(14, 290\right)$$, $$\left(14, -304\right)$$, $$\left(28, 388\right)$$, $$\left(28, -416\right)$$, $$\left(38, 464\right)$$, $$\left(38, -502\right)$$, $$\left(80, 884\right)$$, $$\left(80, -964\right)$$, $$\left(176, 2396\right)$$, $$\left(176, -2572\right)$$, $$\left(248, 3908\right)$$, $$\left(248, -4156\right)$$, $$\left(542, 12434\right)$$, $$\left(542, -12976\right)$$, $$\left(1004, 31376\right)$$, $$\left(1004, -32380\right)$$, $$\left(7766, 680528\right)$$, $$\left(7766, -688294\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$10626$$ = $2 \cdot 3 \cdot 7 \cdot 11 \cdot 23$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $-4097346156288$ = $-1 \cdot 2^{8} \cdot 3^{6} \cdot 7^{3} \cdot 11^{2} \cdot 23^{2}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{6360314548472639}{4097346156288}$$ = $2^{-8} \cdot 3^{-6} \cdot 7^{-3} \cdot 11^{-2} \cdot 23^{-2} \cdot 41^{3} \cdot 4519^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $1.1090359097517722410312349719\dots$ Stable Faltings height: $1.1090359097517722410312349719\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $0.093353212310205089306450475121\dots$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $0.48703747738266073707669504138\dots$ 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: $576$  = $2^{3}\cdot( 2 \cdot 3 )\cdot3\cdot2\cdot2$ 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: $L'(E,1)$ ≈ $6.5471778761947541394441983718946273345$

## Modular invariants

Modular form 10626.2.a.o

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);

$$q + q^{2} + q^{3} + q^{4} - 4q^{5} + q^{6} + q^{7} + q^{8} + q^{9} - 4q^{10} + q^{11} + q^{12} + q^{14} - 4q^{15} + q^{16} - 2q^{17} + q^{18} + O(q^{20})$$

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 27648 $\Gamma_0(N)$-optimal: yes Manin constant: 1

## Local data

This elliptic curve is semistable. There are 5 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$(j)_{-}$
$2$ $8$ $I_{8}$ Split multiplicative -1 1 8 8
$3$ $6$ $I_{6}$ Split multiplicative -1 1 6 6
$7$ $3$ $I_{3}$ Split multiplicative -1 1 3 3
$11$ $2$ $I_{2}$ Split multiplicative -1 1 2 2
$23$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2

## 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]

Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.

## 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 split split ordinary split split ss ordinary ss nonsplit ordinary ordinary ordinary ss ordinary ordinary 4 2 1 2 2 1,1 1 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 0 0

## Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2.
Its isogeny class 10626.o 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 $K$ $2$ $$\Q(\sqrt{-7})$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database $4$ 4.2.448063.2 $$\Z/4\Z$$ Not in database $8$ Deg 8 $$\Z/2\Z \times \Z/4\Z$$ Not in database $8$ 8.0.9837262146481.2 $$\Z/2\Z \times \Z/4\Z$$ Not in database $8$ 8.2.143367545618352.4 $$\Z/6\Z$$ 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.