# Properties

 Label 405600.n1 Conductor $405600$ Discriminant $2.255\times 10^{15}$ j-invariant $$\frac{520000}{27}$$ CM no Rank $2$ Torsion structure trivial

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([0, -1, 0, -73233, 7302177])

gp: E = ellinit([0, -1, 0, -73233, 7302177])

magma: E := EllipticCurve([0, -1, 0, -73233, 7302177]);

$$y^2=x^3-x^2-73233x+7302177$$

## Mordell-Weil group structure

$\Z^2$

### Infinite order Mordell-Weil generators and heights

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(113, 676\right)$$ $$\left(281, 2972\right)$$ $\hat{h}(P)$ ≈ $0.50607925911371752453483256828$ $4.4242785760375394073091886506$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$(-79,\pm 3548)$$, $$(113,\pm 676)$$, $$(281,\pm 2972)$$, $$(451,\pm 8112)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$405600$$ = $2^{5} \cdot 3 \cdot 5^{2} \cdot 13^{2}$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $2255332297420800$ = $2^{12} \cdot 3^{3} \cdot 5^{2} \cdot 13^{8}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{520000}{27}$$ = $2^{6} \cdot 3^{-3} \cdot 5^{4} \cdot 13$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $1.7034856778057221004312509889\dots$ Stable Faltings height: $-0.96786739313426442878843264914\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $2$ sage: E.regulator()  magma: Regulator(E); Regulator: $2.2329385046691763042738109101\dots$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $0.45531187686720820608112163639\dots$ 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: $12$  = $2^{2}\cdot1\cdot1\cdot3$ 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: $L^{(2)}(E,1)/2!$ ≈ $12.200201057879760208767291539279522763$

## Modular invariants

Modular form 405600.2.a.n

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);

$$q - q^{3} - 4q^{7} + q^{9} + q^{11} + 2q^{17} - 2q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 2156544 $\Gamma_0(N)$-optimal: yes Manin constant: 1

## 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$(j)_{-}$
$2$ $4$ $I_3^{*}$ Additive -1 5 12 0
$3$ $1$ $I_{3}$ Non-split multiplicative 1 1 3 3
$5$ $1$ $II$ Additive 1 2 2 0
$13$ $3$ $IV^{*}$ Additive 1 2 8 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 $\ell$.

## $p$-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 no rational isogenies. Its isogeny class 405600.n consists of this curve only.