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Properties

Label 411840i3
Conductor $411840$
Discriminant $7.815\times 10^{24}$
j-invariant \( \frac{287849398425814280018}{81784533026485575} \)
CM no
Rank $0$
Torsion structure \(\Z/{2}\Z\)

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Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([0, 0, 0, -62886828, -136949964752])
 
gp: E = ellinit([0, 0, 0, -62886828, -136949964752])
 
magma: E := EllipticCurve([0, 0, 0, -62886828, -136949964752]);
 

Minimal equation

Minimal equation

Simplified equation

\(y^2=x^3-62886828x-136949964752\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z=x^3-62886828xz^2-136949964752z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3-62886828x-136949964752\) Copy content Toggle raw display (homogenize, minimize)

Mordell-Weil group structure

\(\Z/{2}\Z\)

Torsion generators

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

\( \left(8852, 0\right) \) Copy content Toggle raw display

Integral points

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

\( \left(8852, 0\right) \) Copy content Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 411840 \)  =  $2^{6} \cdot 3^{2} \cdot 5 \cdot 11 \cdot 13$
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: $7814633826065840101785600 $  =  $2^{17} \cdot 3^{34} \cdot 5^{2} \cdot 11 \cdot 13 $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{287849398425814280018}{81784533026485575} \)  =  $2 \cdot 3^{-28} \cdot 5^{-2} \cdot 11^{-1} \cdot 13^{-1} \cdot 37^{3} \cdot 141637^{3}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $3.4832258685258237328760982412\dots$
Stable Faltings height: $1.9519612183985130321707301173\dots$

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: $0.054769167401059818433634481912\dots$
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: $ 32 $  = $ 2^{2}\cdot2^{2}\cdot2\cdot1\cdot1 $
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) $ ≈ $ 0.43815333920847854746907585529 $

Modular invariants

Modular form 411840.2.a.i

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);
 

\( q - q^{5} - 4 q^{7} - q^{11} - q^{13} + 2 q^{17} - 4 q^{19} + O(q^{20}) \) Copy content Toggle raw display

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 88080384
$ \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 411840i1 is optimal.

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$(j)_{-}$
$2$ $4$ $I_{7}^{*}$ Additive 1 6 17 0
$3$ $4$ $I_{28}^{*}$ Additive -1 2 34 28
$5$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2
$11$ $1$ $I_{1}$ Non-split multiplicative 1 1 1 1
$13$ $1$ $I_{1}$ Non-split multiplicative 1 1 1 1

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 8.12.0.12

$p$-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 and 4.
Its isogeny class 411840i consists of 4 curves linked by isogenies of degrees dividing 4.