Minimal Weierstrass equation
\(y^2+xy=x^3-x^2+2262456x-2602551600\)
Mordell-Weil group structure
$\Z/{2}\Z$
Torsion generators
\( \left(\frac{3459}{4}, -\frac{3459}{8}\right) \)
Integral points
None
Invariants
|
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
|
|||
| Conductor: | \( 441090 \) | = | $2 \cdot 3^{2} \cdot 5 \cdot 13^{2} \cdot 29$ |
|
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
|
|||
| Discriminant: | $-3665957219936853800400 $ | = | $-1 \cdot 2^{4} \cdot 3^{18} \cdot 5^{2} \cdot 13^{8} \cdot 29 $ |
|
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
|
|||
| j-invariant: | \( \frac{363979050334199}{1041836936400} \) | = | $2^{-4} \cdot 3^{-12} \cdot 5^{-2} \cdot 13^{-2} \cdot 29^{-1} \cdot 71399^{3}$ |
| Endomorphism ring: | $\Z$ | ||
| Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
| Sato-Tate group: | $\mathrm{SU}(2)$ | ||
| Faltings height: | $2.8212503236365113423086067356\dots$ | ||
| Stable Faltings height: | $0.98946950057168812858424039636\dots$ | ||
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: | $0.071929082425747124267150878549\dots$ | ||
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\cdot2^{2}\cdot2\cdot2^{2}\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();
gp: ar = ellanalyticrank(E);
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
| |||
| Special value: | $ L(E,1) $ ≈ $ 1.1508653188119539882744140568 $ | ||
Modular invariants
Modular form 441090.2.a.bh
For more coefficients, see the Downloads section to the right.
|
sage: E.modular_degree()
magma: ModularDegree(E);
|
|||
| Modular degree: | 20643840 | ||
| $ \Gamma_0(N) $-optimal: | no | ||
| Manin constant: | 1 | ||
Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{4}$ | Non-split multiplicative | 1 | 1 | 4 | 4 |
| $3$ | $4$ | $I_{12}^{*}$ | Additive | -1 | 2 | 18 | 12 |
| $5$ | $2$ | $I_{2}$ | Split multiplicative | -1 | 1 | 2 | 2 |
| $13$ | $4$ | $I_{2}^{*}$ | Additive | 1 | 2 | 8 | 2 |
| $29$ | $1$ | $I_{1}$ | Split multiplicative | -1 | 1 | 1 | 1 |
Galois representations
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
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 441090.bh
consists of 2 curves linked by isogenies of
degree 2.