# Properties

 Label 433200.jp2 Conductor $433200$ Discriminant $-1.338\times 10^{23}$ j-invariant $$-\frac{186169411}{6480}$$ CM no Rank $1$ Torsion structure $$\Z/{2}\Z$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([0, 1, 0, -32637408, -73904092812])

gp: E = ellinit([0, 1, 0, -32637408, -73904092812])

magma: E := EllipticCurve([0, 1, 0, -32637408, -73904092812]);

## Simplified equation

 $$y^2=x^3+x^2-32637408x-73904092812$$ y^2=x^3+x^2-32637408x-73904092812 (homogenize, simplify) $$y^2z=x^3+x^2z-32637408xz^2-73904092812z^3$$ y^2z=x^3+x^2z-32637408xz^2-73904092812z^3 (dehomogenize, simplify) $$y^2=x^3-2643630075x-53868152769750$$ y^2=x^3-2643630075x-53868152769750 (homogenize, minimize)

## Mordell-Weil group structure

$$\Z \oplus \Z/{2}\Z$$

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(\frac{944014903002}{17648401}, \frac{911748529413389472}{74140932601}\right)$$ (944014903002/17648401, 911748529413389472/74140932601) $\hat{h}(P)$ ≈ $24.730151292119084305406357155$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(6618, 0\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(6618, 0\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$433200$$ = $2^{4} \cdot 3 \cdot 5^{2} \cdot 19^{2}$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $-133825042022906880000000$ = $-1 \cdot 2^{16} \cdot 3^{4} \cdot 5^{7} \cdot 19^{9}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{186169411}{6480}$$ = $-1 \cdot 2^{-4} \cdot 3^{-4} \cdot 5^{-1} \cdot 571^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $3.2105115593319404460896208200\dots$ Stable Faltings height: $-0.49568381181988539563476154199\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $24.730151292119084305406357155\dots$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $0.031517589873179802110961370599\dots$ 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: $64$  = $2^{2}\cdot2^{2}\cdot2\cdot2$ 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: $1$ (exact) 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: $L'(E,1)$ ≈ $12.470956254827149593933211512$

## Modular invariants

Modular form 433200.2.a.jp

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);

$$q + q^{3} + 2 q^{7} + q^{9} - 2 q^{13} + 6 q^{17} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 42024960 $\Gamma_0(N)$-optimal: not computed* (one of 2 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 this curve 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$(j)_{-}$
$2$ $4$ $I_{8}^{*}$ Additive -1 4 16 4
$3$ $4$ $I_{4}$ Split multiplicative -1 1 4 4
$5$ $2$ $I_{1}^{*}$ Additive 1 2 7 1
$19$ $2$ $III^{*}$ Additive 1 2 9 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 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 433200.jp consists of 2 curves linked by isogenies of degree 2.