# Properties

 Label 53130cr2 Conductor 53130 Discriminant 351347432901696000000 j-invariant $$\frac{1007588745830352584072161}{351347432901696000000}$$ CM no Rank 1 Torsion Structure $$\Z/{2}\Z \times \Z/{6}\Z$$

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, 0, 0, -2088590, -732603900]) # or

sage: E = EllipticCurve("53130cr2")

gp: E = ellinit([1, 0, 0, -2088590, -732603900]) \\ or

gp: E = ellinit("53130cr2")

magma: E := EllipticCurve([1, 0, 0, -2088590, -732603900]); // or

magma: E := EllipticCurve("53130cr2");

$$y^2 + x y = x^{3} - 2088590 x - 732603900$$

## Mordell-Weil group structure

$$\Z\times \Z/{2}\Z \times \Z/{6}\Z$$

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(-1148, 12922\right)$$ $$\hat{h}(P)$$ ≈ 1.3837175901661312

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(-1220, 610\right)$$, $$\left(8680, 792610\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-1220, 610\right)$$, $$\left(-1148, 12922\right)$$, $$\left(-1148, -11774\right)$$, $$\left(-980, 19810\right)$$, $$\left(-980, -18830\right)$$, $$\left(-770, 20860\right)$$, $$\left(-770, -20090\right)$$, $$\left(-560, 16450\right)$$, $$\left(-560, -15890\right)$$, $$\left(-428, 9322\right)$$, $$\left(-428, -8894\right)$$, $$\left(-420, 8610\right)$$, $$\left(-420, -8190\right)$$, $$\left(1596, -798\right)$$, $$\left(1750, 30310\right)$$, $$\left(1750, -32060\right)$$, $$\left(1780, 33610\right)$$, $$\left(1780, -35390\right)$$, $$\left(2380, 87010\right)$$, $$\left(2380, -89390\right)$$, $$\left(3850, 217840\right)$$, $$\left(3850, -221690\right)$$, $$\left(4060, 238210\right)$$, $$\left(4060, -242270\right)$$, $$\left(7996, 698722\right)$$, $$\left(7996, -706718\right)$$, $$\left(8680, 792610\right)$$, $$\left(8680, -801290\right)$$, $$\left(31780, 5643610\right)$$, $$\left(31780, -5675390\right)$$, $$\left(37660, 7284130\right)$$, $$\left(37660, -7321790\right)$$, $$\left(4790380, 10482279010\right)$$, $$\left(4790380, -10487069390\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$53130$$ = $$2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 23$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$351347432901696000000$$ = $$2^{12} \cdot 3^{6} \cdot 5^{6} \cdot 7^{6} \cdot 11^{2} \cdot 23^{2}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{1007588745830352584072161}{351347432901696000000}$$ = $$2^{-12} \cdot 3^{-6} \cdot 5^{-6} \cdot 7^{-6} \cdot 11^{-2} \cdot 13^{6} \cdot 23^{-2} \cdot 593209^{3}$$ Endomorphism ring: $$\Z$$ (no Complex Multiplication) Sato-Tate Group: $\mathrm{SU}(2)$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Rank: $$1$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$1.38371759017$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.129172844773$$ 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: $$10368$$  = $$( 2^{2} \cdot 3 )\cdot( 2 \cdot 3 )\cdot( 2 \cdot 3 )\cdot( 2 \cdot 3 )\cdot2\cdot2$$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $$12$$ sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Analytic order of Ш: $$1$$ (exact)

## Modular invariants

#### Modular form 53130.2.a.cs

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);

$$q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} + q^{7} + q^{8} + q^{9} + q^{10} - q^{11} + q^{12} + 2q^{13} + q^{14} + q^{15} + q^{16} - 6q^{17} + q^{18} - 4q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 2654208 $$\Gamma_0(N)$$-optimal: no Manin constant: 1

#### Special L-value

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);

$$L'(E,1)$$ ≈ $$12.8691890989$$

## Local data

This elliptic curve is semistable.

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$$(j)_{-}$$
$$2$$ $$12$$ $$I_{12}$$ Split multiplicative -1 1 12 12
$$3$$ $$6$$ $$I_{6}$$ Split multiplicative -1 1 6 6
$$5$$ $$6$$ $$I_{6}$$ Split multiplicative -1 1 6 6
$$7$$ $$6$$ $$I_{6}$$ Split multiplicative -1 1 6 6
$$11$$ $$2$$ $$I_{2}$$ Non-split multiplicative 1 1 2 2
$$23$$ $$2$$ $$I_{2}$$ Non-split multiplicative 1 1 2 2

## Galois representations

The image of the 2-adic representation attached to this elliptic curve is the subgroup of $\GL(2,\Z_2)$ with Rouse label X8.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^1\Z_2)$ generated by  and has index 6.

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 mod $$p$$ Galois representation has maximal image $$\GL(2,\F_p)$$ for all primes $$p$$ except those listed.

prime Image of Galois representation
$$2$$ Cs
$$3$$ B.1.1

## $p$-adic data

### $p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(3,20) 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 split split split split nonsplit ordinary ordinary ordinary nonsplit ordinary ordinary ordinary ordinary ordinary ss 5 4 2 2 1 1 1 1 1 1 1 1 1 1 1,1 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, 3 and 6.
Its isogeny class 53130cr consists of 8 curves linked by isogenies of degrees dividing 12.

## Growth of torsion in number fields

The number fields $K$ of degree up to 7 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \times \Z/{6}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
4 $$\Q(\sqrt{-11}, \sqrt{-15})$$ $$\Z/2\Z \times \Z/12\Z$$ Not in database
$$\Q(\sqrt{15}, \sqrt{-161})$$ $$\Z/2\Z \times \Z/12\Z$$ Not in database
$$\Q(\sqrt{11}, \sqrt{161})$$ $$\Z/2\Z \times \Z/12\Z$$ Not in database
6 6.0.110623106187.1 $$\Z/6\Z \times \Z/6\Z$$ Not in database

We only show fields where the torsion growth is primitive. For each field $K$ we either show its label, or a defining polynomial when $K$ is not in the database.