# Properties

 Label 474320.bj1 Conductor 474320 Discriminant 54372396828097964000000000000 j-invariant $$\frac{1969902499564819009}{63690429687500}$$ CM no Rank 1 Torsion Structure $$\Z/{2}\Z$$

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## Minimal Weierstrass equation

magma: E := EllipticCurve([0, 1, 0, -2477470200, 46117850491348]); // or

magma: E := EllipticCurve("474320bj4");

sage: E = EllipticCurve([0, 1, 0, -2477470200, 46117850491348]) # or

sage: E = EllipticCurve("474320bj4")

gp: E = ellinit([0, 1, 0, -2477470200, 46117850491348]) \\ or

gp: E = ellinit("474320bj4")

$$y^2 = x^{3} + x^{2} - 2477470200 x + 46117850491348$$

## Mordell-Weil group structure

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

### Infinite order Mordell-Weil generator and height

magma: Generators(E);

sage: E.gens()

 $$P$$ = $$\left(-9684, -8318750\right)$$ $$\hat{h}(P)$$ ≈ 1.32513107042

## Torsion generators

magma: TorsionSubgroup(E);

sage: E.torsion_subgroup().gens()

gp: elltors(E)

$$\left(24691, 0\right)$$

## Integral points

magma: IntegralPoints(E);

sage: E.integral_points()

$$\left(-9684, 8318750\right)$$, $$\left(24691, 0\right)$$, $$\left(36791, 2184050\right)$$, $$\left(43556, 4565330\right)$$

Note: only one of each pair $\pm P$ is listed.

## Invariants

 magma: Conductor(E);  sage: E.conductor().factor()  gp: ellglobalred(E)[1] Conductor: $$474320$$ = $$2^{4} \cdot 5 \cdot 7^{2} \cdot 11^{2}$$ magma: Discriminant(E);  sage: E.discriminant().factor()  gp: E.disc Discriminant: $$54372396828097964000000000000$$ = $$2^{14} \cdot 5^{12} \cdot 7^{8} \cdot 11^{9}$$ magma: jInvariant(E);  sage: E.j_invariant().factor()  gp: E.j j-invariant: $$\frac{1969902499564819009}{63690429687500}$$ = $$2^{-2} \cdot 5^{-12} \cdot 7^{-2} \cdot 11^{-3} \cdot 23^{3} \cdot 54503^{3}$$ Endomorphism ring: $$\Z$$ (no Complex Multiplication) Sato-Tate Group: $\mathrm{SU}(2)$

## BSD invariants

 magma: Rank(E);  sage: E.rank() Rank: $$1$$ magma: Regulator(E);  sage: E.regulator() Regulator: $$1.32513107042422$$ magma: RealPeriod(E);  sage: E.period_lattice().omega()  gp: E.omega[1] Real period: $$0.0351999988836624$$ magma: TamagawaNumbers(E);  sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]] Tamagawa product: $$384$$  = $$2\cdot( 2^{2} \cdot 3 )\cdot2^{2}\cdot2^{2}$$ magma: Order(TorsionSubgroup(E));  sage: E.torsion_order()  gp: elltors(E)[1] Torsion order: $$2$$ magma: MordellWeilShaInformation(E);  sage: E.sha().an_numerical() Analytic order of Ш: $$1$$ (exact)

## Modular invariants

#### Modular form 474320.2.a.bj

magma: ModularForm(E);

sage: E.q_eigenform(20)

gp: xy = elltaniyama(E);

gp: x*deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+E.a3)

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

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

 magma: ModularDegree(E);  sage: E.modular_degree() Modular degree: 477757440 $$\Gamma_0(N)$$-optimal: unknown* (one of 4 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 curve 474320.bj3 is optimal.

#### Special L-value

magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);

sage: r = E.rank();

sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()

gp: ar = ellanalyticrank(E);

gp: ar[2]/factorial(ar[1])

$$L'(E,1)$$ ≈ $$4.477882771165344$$

## Local data

magma: [LocalInformation(E,p) : p in BadPrimes(E)];

sage: E.local_data()

gp: ellglobalred(E)[5]

prime Tamagawa number Kodaira symbol Reduction type Root number ord($$N$$) ord($$\Delta$$) ord$$(j)_{-}$$
$$2$$ $$2$$ $$I_6^{*}$$ Additive -1 4 14 2
$$5$$ $$12$$ $$I_{12}$$ Split multiplicative -1 1 12 12
$$7$$ $$4$$ $$I_2^{*}$$ Additive -1 2 8 2
$$11$$ $$4$$ $$I_3^{*}$$ Additive -1 2 9 3

## 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.

magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];

sage: rho = E.galois_representation();

sage: [rho.image_type(p) for p in rho.non_surjective()]

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
$$3$$ B

## $p$-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.

No Iwasawa invariant data is available for this curve.

## Isogenies

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 3 and 6.
Its isogeny class 474320.bj consists of 4 curves linked by isogenies of degrees dividing 6.