Minimal Weierstrass equation
\(y^2+xy=x^3-x^2-2696004x-1703114672\)
Mordell-Weil group structure
$\Z\times \Z/{2}\Z$
Infinite order Mordell-Weil generator and height
| $P$ | = |
\(\left(\frac{645576}{49}, \frac{512192708}{343}\right)\)
|
| $\hat{h}(P)$ | ≈ | $9.4961185700546968126609078207$ |
Torsion generators
\( \left(-952, 476\right) \)
Integral points
\( \left(-952, 476\right) \)
Invariants
|
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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| Conductor: | \( 441090 \) | = | $2 \cdot 3^{2} \cdot 5 \cdot 13^{2} \cdot 29$ |
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sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
|
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| Discriminant: | $76410224518867200 $ | = | $2^{8} \cdot 3^{8} \cdot 5^{2} \cdot 13^{7} \cdot 29 $ |
|
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
|
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| j-invariant: | \( \frac{615882348586441}{21715200} \) | = | $2^{-8} \cdot 3^{-2} \cdot 5^{-2} \cdot 13^{-1} \cdot 29^{-1} \cdot 85081^{3}$ |
| Endomorphism ring: | $\Z$ | ||
| Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
| Sato-Tate group: | $\mathrm{SU}(2)$ | ||
| Faltings height: | $2.3293740043896081306343129324\dots$ | ||
| Stable Faltings height: | $0.49759318132478491690994659315\dots$ | ||
BSD invariants
sage: E.rank()
magma: Rank(E);
| |||
| Analytic rank: | $1$ | ||
sage: E.regulator()
magma: Regulator(E);
| |||
| Regulator: | $9.4961185700546968126609078207\dots$ | ||
sage: E.period_lattice().omega()
gp: E.omega[1]
magma: RealPeriod(E);
| |||
| Real period: | $0.11782035659915018638501520435\dots$ | ||
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\cdot2\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) $ ≈ $ 8.9506886098532522329779929916 $ | ||
Modular invariants
Modular form 441090.2.a.bw
For more coefficients, see the Downloads section to the right.
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sage: E.modular_degree()
magma: ModularDegree(E);
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| Modular degree: | 16515072 | ||
| $ \Gamma_0(N) $-optimal: | not computed* (one of 3 curves in this isogeny class which might be optimal) | ||
| Manin constant: | 1 (conditional*) | ||
Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{8}$ | Non-split multiplicative | 1 | 1 | 8 | 8 |
| $3$ | $2$ | $I_{2}^{*}$ | Additive | -1 | 2 | 8 | 2 |
| $5$ | $2$ | $I_{2}$ | Split multiplicative | -1 | 1 | 2 | 2 |
| $13$ | $4$ | $I_{1}^{*}$ | Additive | 1 | 2 | 7 | 1 |
| $29$ | $1$ | $I_{1}$ | Non-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 | 4.6.0.1 |
$p$-adic regulators
$p$-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 non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 441090bw
consists of 4 curves linked by isogenies of
degrees dividing 4.