Minimal Weierstrass equation
\(y^2=x^3-387193879884x+92734389270274192\)
Mordell-Weil group structure
\(\Z\times \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
| \(P\) | = | \( \left(\frac{1911978621}{5329}, \frac{186999965915}{389017}\right) \) |
| \(\hat{h}(P)\) | ≈ | $12.877320762438196554847403376$ |
Torsion generators
\( \left(359324, 0\right) \)
Integral points
\( \left(359324, 0\right) \)
Invariants
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sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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| Conductor: | \( 479808 \) | = | \(2^{6} \cdot 3^{2} \cdot 7^{2} \cdot 17\) |
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sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
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| Discriminant: | \(403369601585106552706100625408 \) | = | \(2^{21} \cdot 3^{14} \cdot 7^{8} \cdot 17^{8} \) |
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sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
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| j-invariant: | \( \frac{285531136548675601769470657}{17941034271597192} \) | = | \(2^{-3} \cdot 3^{-8} \cdot 7^{-2} \cdot 17^{-8} \cdot 658492993^{3}\) |
| Endomorphism ring: | \(\Z\) | ||
| Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
| Sato-Tate group: | $\mathrm{SU}(2)$ | ||
BSD invariants
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sage: E.rank()
magma: Rank(E);
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| Analytic rank: | \(1\) | ||
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sage: E.regulator()
magma: Regulator(E);
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| Regulator: | \(12.877320762438196554847403376\) | ||
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sage: E.period_lattice().omega()
gp: E.omega[1]
magma: RealPeriod(E);
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| Real period: | \(0.022627173936815808722729652090\) | ||
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sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
magma: TamagawaNumbers(E);
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| Tamagawa product: | \( 128 \) = \( 2\cdot2\cdot2^{2}\cdot2^{3} \) | ||
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sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
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| Torsion order: | \(2\) | ||
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sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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| Analytic order of Ш: | \(1\) (exact) | ||
Modular invariants
Modular form 479808.2.a.ni
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: | 2264924160 | ||
| \( \Gamma_0(N) \)-optimal: | unknown* (one of 4 curves in this isogeny class which might be optimal) | ||
| Manin constant: | 1 (conditional*) | ||
Special L-value
\( L'(E,1) \) ≈ \( 9.3240760554194764640023326361805245781 \)
Local data
This elliptic curve is semistable. There are 4 primes of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(N\)) | ord(\(\Delta\)) | ord\((j)_{-}\) |
|---|---|---|---|---|---|---|---|
| \(2\) | \(2\) | \(I_{11}^{*}\) | Additive | -1 | 6 | 21 | 3 |
| \(3\) | \(2\) | \(I_8^{*}\) | Additive | -1 | 2 | 14 | 8 |
| \(7\) | \(4\) | \(I_2^{*}\) | Additive | -1 | 2 | 8 | 2 |
| \(17\) | \(8\) | \(I_{8}\) | Split multiplicative | -1 | 1 | 8 | 8 |
Galois representations
The image of the 2-adic representation attached to this elliptic curve is the subgroup of $\GL(2,\Z_2)$ with Rouse label X202.
This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 7 & 0 \\ 0 & 7 \end{array}\right),\left(\begin{array}{rr} 7 & 0 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 5 \\ 0 & 1 \end{array}\right)$ and has index 48.
The mod \( p \) Galois representation has maximal image \(\GL(2,\F_p)\) for all primes \( p \) except those listed.
| prime | Image of Galois representation |
|---|---|
| \(2\) | B |
$p$-adic data
$p$-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, 4 and 8.
Its isogeny class 479808.ni
consists of 4 curves linked by isogenies of
degrees dividing 8.