# Properties

 Label 450840.h2 Conductor $450840$ Discriminant $-2.694\times 10^{12}$ j-invariant $$-\frac{8100185168}{2142075}$$ CM no Rank $2$ Torsion structure $$\Z/{2}\Z$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([0, -1, 0, -4516, 142516])

gp: E = ellinit([0, -1, 0, -4516, 142516])

magma: E := EllipticCurve([0, -1, 0, -4516, 142516]);

$$y^2=x^3-x^2-4516x+142516$$

## Mordell-Weil group structure

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

### Infinite order Mordell-Weil generators and heights

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(6, 340\right)$$ $$\left(38, 156\right)$$ $\hat{h}(P)$ ≈ $1.3664068060317464614411058993$ $1.4460344036563470610337051473$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(-79, 0\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-79, 0\right)$$, $$(-27,\pm 494)$$, $$(6,\pm 340)$$, $$(38,\pm 156)$$, $$(90,\pm 676)$$, $$(129,\pm 1300)$$, $$(2981,\pm 162690)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$450840$$ = $2^{3} \cdot 3 \cdot 5 \cdot 13 \cdot 17^{2}$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $-2694147705600$ = $-1 \cdot 2^{8} \cdot 3 \cdot 5^{2} \cdot 13^{4} \cdot 17^{3}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{8100185168}{2142075}$$ = $-1 \cdot 2^{4} \cdot 3^{-1} \cdot 5^{-2} \cdot 13^{-4} \cdot 797^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $1.1018645849621527291334240654\dots$ Stable Faltings height: $-0.068536871425198163873781003374\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $2$ sage: E.regulator()  magma: Regulator(E); Regulator: $1.9639147464239291299411059243\dots$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $0.76869076105094638260985317797\dots$ 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: $32$  = $2\cdot1\cdot2\cdot2^{2}\cdot2$ 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: $L^{(2)}(E,1)/2!$ ≈ $12.077144968542291708711542998940620762$

## Modular invariants

Modular form 450840.2.a.h

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);

$$q - q^{3} - q^{5} + q^{9} - 4q^{11} + q^{13} + q^{15} + 4q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 819200 $\Gamma_0(N)$-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$(j)_{-}$
$2$ $2$ $I_1^{*}$ Additive 1 3 8 0
$3$ $1$ $I_{1}$ Non-split multiplicative 1 1 1 1
$5$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2
$13$ $4$ $I_{4}$ Split multiplicative -1 1 4 4
$17$ $2$ $III$ Additive 1 2 3 0

## 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 450840.h consists of 2 curves linked by isogenies of degree 2.