Properties

 Label 405600z1 Conductor $405600$ Discriminant $-3.806\times 10^{16}$ j-invariant $$-\frac{150061288}{729}$$ CM no Rank $1$ Torsion structure trivial

Related objects

Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([0, -1, 0, -413768, -102734568])

gp: E = ellinit([0, -1, 0, -413768, -102734568])

magma: E := EllipticCurve([0, -1, 0, -413768, -102734568]);

$$y^2=x^3-x^2-413768x-102734568$$

Mordell-Weil group structure

$\Z$

Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(\frac{111017}{121}, \frac{22632480}{1331}\right)$$ $\hat{h}(P)$ ≈ $6.0847463578744377682536868893$

Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

None

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: $-38058732518976000$ = $-1 \cdot 2^{9} \cdot 3^{6} \cdot 5^{3} \cdot 13^{8}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{150061288}{729}$$ = $-1 \cdot 2^{3} \cdot 3^{-6} \cdot 13 \cdot 113^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $2.0301249860551195200955831315\dots$ Stable Faltings height: $-0.60206111578105571298652242061\dots$

BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $6.0847463578744377682536868893\dots$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $0.094092291518750731599373585606\dots$ 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: $12$  = $1\cdot2\cdot2\cdot3$ 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: $1$ (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: $L'(E,1)$ ≈ $6.8703327374733403763508589095363235309$

Modular invariants

Modular form 405600.2.a.z

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);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 4432896 $\Gamma_0(N)$-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$(j)_{-}$
$2$ $1$ $I_0^{*}$ Additive -1 5 9 0
$3$ $2$ $I_{6}$ Non-split multiplicative 1 1 6 6
$5$ $2$ $III$ Additive -1 2 3 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 405600z consists of this curve only.