# Properties

 Label 468270.bf2 Conductor $468270$ Discriminant $5.373\times 10^{16}$ j-invariant $$\frac{76711450249}{41602500}$$ CM no Rank $1$ Torsion structure $$\Z/{2}\Z \times \Z/{2}\Z$$

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, -1, 0, -96399, -2864295])

gp: E = ellinit([1, -1, 0, -96399, -2864295])

magma: E := EllipticCurve([1, -1, 0, -96399, -2864295]);

$$y^2+xy=x^3-x^2-96399x-2864295$$

## 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);

 $$P$$ = $$\left(-159, 2982\right)$$ $$\hat{h}(P)$$ ≈ $1.1384058885660341890258037788$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(-294, 147\right)$$, $$\left(-30, 15\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-294, 147\right)$$, $$\left(-159, 2982\right)$$, $$\left(-159, -2823\right)$$, $$\left(-129, 2787\right)$$, $$\left(-129, -2658\right)$$, $$\left(-30, 15\right)$$, $$\left(331, 1022\right)$$, $$\left(331, -1353\right)$$, $$\left(696, 15987\right)$$, $$\left(696, -16683\right)$$, $$\left(916, 25557\right)$$, $$\left(916, -26473\right)$$, $$\left(35445, 6655125\right)$$, $$\left(35445, -6690570\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$468270$$ = $$2 \cdot 3^{2} \cdot 5 \cdot 11^{2} \cdot 43$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$53728296180322500$$ = $$2^{2} \cdot 3^{8} \cdot 5^{4} \cdot 11^{6} \cdot 43^{2}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{76711450249}{41602500}$$ = $$2^{-2} \cdot 3^{-2} \cdot 5^{-4} \cdot 7^{3} \cdot 43^{-2} \cdot 607^{3}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $$1.9011698513779527540067172252\dots$$ Stable Faltings height: $$0.15291607064471263627812281776\dots$$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $$1$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$1.1384058885660341890258037788\dots$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.28888484015132798951419476365\dots$$ 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: $$256$$  = $$2\cdot2^{2}\cdot2^{2}\cdot2^{2}\cdot2$$ 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 468270.2.a.bf

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);

$$q - q^{2} + q^{4} + q^{5} - 4q^{7} - q^{8} - q^{10} + 2q^{13} + 4q^{14} + q^{16} + 2q^{17} - 4q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 4423680 $$\Gamma_0(N)$$-optimal: not computed* (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 468270.bf3 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);

$$L'(E,1)$$ ≈ $$5.2618912503316686479850795976225445037$$

## Local data

This elliptic curve is not 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$$(j)_{-}$$
$$2$$ $$2$$ $$I_{2}$$ Non-split multiplicative 1 1 2 2
$$3$$ $$4$$ $$I_2^{*}$$ Additive -1 2 8 2
$$5$$ $$4$$ $$I_{4}$$ Split multiplicative -1 1 4 4
$$11$$ $$4$$ $$I_0^{*}$$ Additive -1 2 6 0
$$43$$ $$2$$ $$I_{2}$$ 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(5,20) 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 468270.bf consists of 2 curves linked by isogenies of degrees dividing 4.