Properties

Label 405720gn2
Conductor $405720$
Discriminant $4.149\times 10^{19}$
j-invariant \( \frac{168591300897604}{472410225} \)
CM no
Rank $0$
Torsion structure \(\Z/{2}\Z \times \Z/{2}\Z\)

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Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([0, 0, 0, -5115747, -4442809714])
 
gp: E = ellinit([0, 0, 0, -5115747, -4442809714])
 
magma: E := EllipticCurve([0, 0, 0, -5115747, -4442809714]);
 

\(y^2=x^3-5115747x-4442809714\)  Toggle raw display

Mordell-Weil group structure

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

Torsion generators

sage: E.torsion_subgroup().gens()
 
gp: elltors(E)
 
magma: TorsionSubgroup(E);
 

\( \left(-1358, 0\right) \), \( \left(2611, 0\right) \)  Toggle raw display

Integral points

sage: E.integral_points()
 
magma: IntegralPoints(E);
 

\( \left(-1358, 0\right) \), \( \left(-1253, 0\right) \), \( \left(2611, 0\right) \)  Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 405720 \)  =  $2^{3} \cdot 3^{2} \cdot 5 \cdot 7^{2} \cdot 23$
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: $41489195539442918400 $  =  $2^{10} \cdot 3^{12} \cdot 5^{2} \cdot 7^{8} \cdot 23^{2} $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{168591300897604}{472410225} \)  =  $2^{2} \cdot 3^{-6} \cdot 5^{-2} \cdot 7^{-2} \cdot 13^{3} \cdot 23^{-2} \cdot 2677^{3}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $2.6365916841917022965016345112\dots$
Stable Faltings height: $0.53670781486336970707030875313\dots$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: $0$
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: $1$
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: $0.10040271025701693477128943254\dots$
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: $ 128 $  = $ 2\cdot2^{2}\cdot2\cdot2^{2}\cdot2 $
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
Torsion order: $4$
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 
Analytic order of Ш: $4$ = $2^2$ (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) $ ≈ $ 3.2128867282245419126812618412452695751 $

Modular invariants

Modular form 405720.2.a.gn

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);
 

\( q + q^{5} + 4 q^{11} - 2 q^{13} - 2 q^{17} + 4 q^{19} + O(q^{20}) \)  Toggle raw display

For more coefficients, see the Downloads section to the right.

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 11796480
$ \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$(j)_{-}$
$2$ $2$ $III^{*}$ Additive -1 3 10 0
$3$ $4$ $I_{6}^{*}$ Additive -1 2 12 6
$5$ $2$ $I_{2}$ Split multiplicative -1 1 2 2
$7$ $4$ $I_{2}^{*}$ Additive -1 2 8 2
$23$ $2$ $I_{2}$ Split multiplicative -1 1 2 2

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
$2$ 2Cs 4.12.0.1

$p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(5,20) if E.conductor().valuation(p)<2]
 

All $p$-adic regulators are identically $1$ since the rank is $0$.

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 405720gn consists of 2 curves linked by isogenies of degrees dividing 4.