# Properties

 Label 463680.jr2 Conductor $463680$ Discriminant $-3.116\times 10^{17}$ j-invariant $$\frac{4716275733}{44023437500}$$ CM no Rank $2$ Torsion structure $$\Z/{2}\Z$$

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([0, 0, 0, 6708, 26855824])

gp: E = ellinit([0, 0, 0, 6708, 26855824])

magma: E := EllipticCurve([0, 0, 0, 6708, 26855824]);

$$y^2=x^3+6708x+26855824$$

## Mordell-Weil group structure

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

### Infinite order Mordell-Weil generators and heights

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(-142, 4800\right)$$ $$\left(-192, 4300\right)$$ $$\hat{h}(P)$$ ≈ $0.63473550322778924223387583551$ $2.9328076628640162744947609811$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(-292, 0\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-292, 0\right)$$, $$(-192,\pm 4300)$$, $$(-142,\pm 4800)$$, $$(128,\pm 5460)$$, $$(333,\pm 8125)$$, $$(434,\pm 10560)$$, $$(1458,\pm 56000)$$, $$(2333,\pm 112875)$$, $$(33458,\pm 6120000)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$463680$$ = $$2^{6} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 23$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-311592960000000000$$ = $$-1 \cdot 2^{20} \cdot 3^{3} \cdot 5^{10} \cdot 7^{2} \cdot 23$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{4716275733}{44023437500}$$ = $$2^{-2} \cdot 3^{3} \cdot 5^{-10} \cdot 7^{-2} \cdot 13^{3} \cdot 23^{-1} \cdot 43^{3}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $$2.0356174953654966717545882786\dots$$ Stable Faltings height: $$0.72124365235855128477992878718\dots$$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $$2$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$1.8614754463467886312785560552\dots$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.24121705345941256880587644926\dots$$ 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: $$160$$  = $$2^{2}\cdot2\cdot( 2 \cdot 5 )\cdot2\cdot1$$ 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: $$1$$ (rounded)

## Modular invariants

Modular form 463680.2.a.jr

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);

$$q + q^{5} - q^{7} + 2q^{11} - 2q^{13} + 2q^{17} - 4q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 3932160 $$\Gamma_0(N)$$-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);

$$L^{(2)}(E,1)/2!$$ ≈ $$17.960784890192687449046420818306008887$$

## 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$$(j)_{-}$$
$$2$$ $$4$$ $$I_{10}^{*}$$ Additive -1 6 20 2
$$3$$ $$2$$ $$III$$ Additive 1 2 3 0
$$5$$ $$10$$ $$I_{10}$$ Split multiplicative -1 1 10 10
$$7$$ $$2$$ $$I_{2}$$ Non-split multiplicative 1 1 2 2
$$23$$ $$1$$ $$I_{1}$$ Split multiplicative -1 1 1 1

## 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(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 463680.jr consists of 2 curves linked by isogenies of degree 2.