# Properties

 Label 435344o2 Conductor $435344$ Discriminant $-7.315\times 10^{27}$ j-invariant $$-\frac{449167881463536812032}{369990050199923699}$$ CM no Rank $2$ Torsion structure trivial

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([0, -1, 0, -431421397, 5369381586269])

gp: E = ellinit([0, -1, 0, -431421397, 5369381586269])

magma: E := EllipticCurve([0, -1, 0, -431421397, 5369381586269]);

$$y^2=x^3-x^2-431421397x+5369381586269$$

## Mordell-Weil group structure

$\Z^2$

### Infinite order Mordell-Weil generators and heights

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(\frac{252613}{4}, \frac{121324931}{8}\right)$$ $$\left(-7076, 2840383\right)$$ $\hat{h}(P)$ ≈ $0.77637743141740108923319790091$ $2.0686958934939223901449190237$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$(-25276,\pm 353717)$$, $$(-9820,\pm 2942597)$$, $$(-7076,\pm 2840383)$$, $$(23668,\pm 2901059)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$435344$$ = $2^{4} \cdot 7 \cdot 13^{2} \cdot 23$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $-7314928862066456615512027136$ = $-1 \cdot 2^{12} \cdot 7^{12} \cdot 13^{9} \cdot 23^{3}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{449167881463536812032}{369990050199923699}$$ = $-1 \cdot 2^{21} \cdot 7^{-12} \cdot 13^{-3} \cdot 19^{3} \cdot 23^{-3} \cdot 47^{3} \cdot 67^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $4.0446361587135560808042583294\dots$ Stable Faltings height: $2.0690142994228424033602824872\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $2$ sage: E.regulator()  magma: Regulator(E); Regulator: $1.5020440351039714414642532521\dots$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $0.038343991646730109331084367854\dots$ 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: $144$  = $1\cdot( 2^{2} \cdot 3 )\cdot2^{2}\cdot3$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $1$ 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: $L^{(2)}(E,1)/2!$ ≈ $8.2935884066468354004465979259311305577$

## Modular invariants

Modular form 435344.2.a.o

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);

$$q - q^{3} - 3q^{5} + q^{7} - 2q^{9} - 3q^{11} + 3q^{15} - 6q^{17} - 7q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 282175488 $\Gamma_0(N)$-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 435344o1 is optimal.

## Local data

This elliptic curve is not semistable. There are 4 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$(j)_{-}$
$2$ $1$ $II^{*}$ Additive -1 4 12 0
$7$ $12$ $I_{12}$ Split multiplicative -1 1 12 12
$13$ $4$ $I_3^{*}$ Additive 1 2 9 3
$23$ $3$ $I_{3}$ Split multiplicative -1 1 3 3

## 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
$3$ 3Cs 3.12.0.1

## $p$-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 $d=$ 3.
Its isogeny class 435344o consists of 2 curves linked by isogenies of degrees dividing 9.