# Properties

 Label 494190.bn3 Conductor $494190$ Discriminant $5.717\times 10^{15}$ j-invariant $$\frac{758800078561}{324900}$$ CM no Rank $1$ Torsion structure $$\Z/{2}\Z \times \Z/{2}\Z$$

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Show commands for: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, -1, 0, -494244, -133566692]) # or

sage: E = EllipticCurve("494190bn2")

gp: E = ellinit([1, -1, 0, -494244, -133566692]) \\ or

gp: E = ellinit("494190bn2")

magma: E := EllipticCurve([1, -1, 0, -494244, -133566692]); // or

magma: E := EllipticCurve("494190bn2");

$$y^2 + x y = x^{3} - x^{2} - 494244 x - 133566692$$

## Mordell-Weil group structure

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

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(-\frac{3674}{9}, \frac{20}{27}\right)$$ $$\hat{h}(P)$$ ≈ $2.3713764986387216$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(812, -406\right)$$, $$\left(-412, 206\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-412, 206\right)$$, $$\left(-403, 404\right)$$, $$\left(-403, -1\right)$$, $$\left(812, -406\right)$$, $$\left(1322, 38354\right)$$, $$\left(1322, -39676\right)$$, $$\left(3719, 220526\right)$$, $$\left(3719, -224245\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$494190$$ = $$2 \cdot 3^{2} \cdot 5 \cdot 17^{2} \cdot 19$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$5717033906544900$$ = $$2^{2} \cdot 3^{8} \cdot 5^{2} \cdot 17^{6} \cdot 19^{2}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{758800078561}{324900}$$ = $$2^{-2} \cdot 3^{-2} \cdot 5^{-2} \cdot 7^{3} \cdot 19^{-2} \cdot 1303^{3}$$ 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); Rank: $$1$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$2.37137649863872$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.180063348587361$$ 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: $$128$$  = $$2\cdot2^{2}\cdot2\cdot2^{2}\cdot2$$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $$4$$ sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Analytic order of Ш: $$1$$ (exact)

## Modular invariants

Modular form 494190.2.a.bn

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);

$$q - q^{2} + q^{4} + q^{5} - 4q^{7} - q^{8} - q^{10} - 4q^{11} - 2q^{13} + 4q^{14} + q^{16} - q^{19} + O(q^{20})$$

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 7864320 $$\Gamma_0(N)$$-optimal: unknown* (one of 3 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 494190.bn4 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);

$$L'(E,1)$$ ≈ $$3.4159839448500833$$

## Local data

This elliptic curve is not 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$$(j)_{-}$$
$$2$$ $$2$$ $$I_{2}$$ Non-split multiplicative 1 1 2 2
$$3$$ $$4$$ $$I_2^{*}$$ Additive -1 2 8 2
$$5$$ $$2$$ $$I_{2}$$ Split multiplicative -1 1 2 2
$$17$$ $$4$$ $$I_0^{*}$$ Additive 1 2 6 0
$$19$$ $$2$$ $$I_{2}$$ 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

## $p$-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.
Its isogeny class 494190.bn consists of 4 curves linked by isogenies of degrees dividing 4.