# Properties

 Label 468270.v1 Conductor $468270$ Discriminant $6.963\times 10^{16}$ j-invariant $$\frac{1481933914201}{53916840}$$ CM no Rank $2$ Torsion structure $$\Z/{2}\Z$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, -1, 0, -258660, -48951864])

gp: E = ellinit([1, -1, 0, -258660, -48951864])

magma: E := EllipticCurve([1, -1, 0, -258660, -48951864]);

$$y^2+xy=x^3-x^2-258660x-48951864$$

## Mordell-Weil group structure

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

### Infinite order Mordell-Weil generators and heights

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(-315, 1251\right)$$ $$\left(3087, 167463\right)$$ $\hat{h}(P)$ ≈ $2.8809757024776720330853434630$ $5.3787079393083428591569911272$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(-\frac{1341}{4}, \frac{1341}{8}\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-315, 1251\right)$$, $$\left(-315, -936\right)$$, $$\left(3087, 167463\right)$$, $$\left(3087, -170550\right)$$, $$\left(3567, 208938\right)$$, $$\left(3567, -212505\right)$$

## 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: $69631871849697960$ = $2^{3} \cdot 3^{12} \cdot 5 \cdot 11^{6} \cdot 43^{2}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{1481933914201}{53916840}$$ = $2^{-3} \cdot 3^{-6} \cdot 5^{-1} \cdot 13^{3} \cdot 43^{-2} \cdot 877^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $2.0015594764367132483586423929\dots$ Stable Faltings height: $0.25330569570347313063004798546\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $2$ sage: E.regulator()  magma: Regulator(E); Regulator: $15.092194851135392291226685676\dots$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $0.21217188251604169292303840660\dots$ 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: $16$  = $1\cdot2^{2}\cdot1\cdot2\cdot2$ 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) 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); Special value: $L^{(2)}(E,1)/2!$ ≈ $12.808557571457231200568517739620399812$

## Modular invariants

Modular form 468270.2.a.v

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);

$$q - q^{2} + q^{4} - q^{5} + 2q^{7} - q^{8} + q^{10} + 2q^{13} - 2q^{14} + q^{16} - 4q^{17} + 6q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 6451200 $\Gamma_0(N)$-optimal: no Manin constant: 1

## 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$(j)_{-}$
$2$ $1$ $I_{3}$ Non-split multiplicative 1 1 3 3
$3$ $4$ $I_6^{*}$ Additive -1 2 12 6
$5$ $1$ $I_{1}$ Non-split multiplicative 1 1 1 1
$11$ $2$ $I_0^{*}$ Additive -1 2 6 0
$43$ $2$ $I_{2}$ Split multiplicative -1 1 2 2

## Galois representations

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 $\ell$-adic Galois representation has maximal image $\GL(2,\Z_\ell)$ for all primes $\ell$ except those listed in the table below.

prime $\ell$ mod-$\ell$ image $\ell$-adic image
$2$ 2B 2.3.0.1

## $p$-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.v consists of 2 curves linked by isogenies of degree 2.