# Properties

 Label 476850.gj2 Conductor $476850$ Discriminant $-7.913\times 10^{17}$ j-invariant $$\frac{2053225511}{2098140}$$ CM no Rank $0$ Torsion structure $$\Z/{2}\Z$$

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, 1, 1, 191312, 28264781]) # or

sage: E = EllipticCurve("476850.gj2")

gp: E = ellinit([1, 1, 1, 191312, 28264781]) \\ or

gp: E = ellinit("476850.gj2")

magma: E := EllipticCurve([1, 1, 1, 191312, 28264781]); // or

magma: E := EllipticCurve("476850.gj2");

$$y^2+xy+y=x^3+x^2+191312x+28264781$$

## Mordell-Weil group structure

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

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(-135, 67\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-135, 67\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$476850$$ = $$2 \cdot 3 \cdot 5^{2} \cdot 11 \cdot 17^{2}$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-791312484713437500$$ = $$-1 \cdot 2^{2} \cdot 3 \cdot 5^{7} \cdot 11^{2} \cdot 17^{8}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{2053225511}{2098140}$$ = $$2^{-2} \cdot 3^{-1} \cdot 5^{-1} \cdot 11^{-2} \cdot 17^{-2} \cdot 31^{3} \cdot 41^{3}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $$0$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$1$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.18692960394532006507470980938$$ 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: $$32$$  = $$2\cdot1\cdot2\cdot2\cdot2^{2}$$ 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)

## Modular invariants

Modular form 476850.2.a.gj

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);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 7077888 $$\Gamma_0(N)$$-optimal: unknown* (one of 2 curves in this isogeny class which might be optimal) Manin constant: 1 (conditional*)
* The optimal curve in each isogeny class has not been determined in all cases for conductors over 400000. The Manin constant is correct provided that this curve is optimal.

#### 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);

$$L(E,1)$$ ≈ $$1.4954368315625605205976784750521804971$$

## 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$$(j)_{-}$$
$$2$$ $$2$$ $$I_{2}$$ Split multiplicative -1 1 2 2
$$3$$ $$1$$ $$I_{1}$$ Non-split multiplicative 1 1 1 1
$$5$$ $$2$$ $$I_1^{*}$$ Additive 1 2 7 1
$$11$$ $$2$$ $$I_{2}$$ Non-split multiplicative 1 1 2 2
$$17$$ $$4$$ $$I_2^{*}$$ Additive 1 2 8 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 X6.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^1\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right)$ and has index 3.

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
$$2$$ B

## $p$-adic data

### $p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(3,20) if E.conductor().valuation(p)<2]

All $$p$$-adic regulators are identically $$1$$ since the rank is $$0$$.

No Iwasawa invariant data is available for this curve.

## Isogenies

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2.
Its isogeny class 476850.gj consists of 2 curves linked by isogenies of degree 2.