Minimal Weierstrass equation
\(y^2+xy=x^3+x^2-208128x+329936832\)
Mordell-Weil group structure
\(\Z^2\)
Infinite order Mordell-Weil generators and heights
| \(P\) | = | \( \left(311, 17032\right) \) | \( \left(\frac{18424}{9}, \frac{2459864}{27}\right) \) |
| \(\hat{h}(P)\) | ≈ | $2.9796867150306503608550272257$ | $4.9525449706547597631240841852$ |
Integral points
\( \left(-496, 17896\right) \), \( \left(-496, -17400\right) \), \( \left(311, 17032\right) \), \( \left(311, -17343\right) \)
Invariants
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sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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| Conductor: | \( 424830 \) | = | \(2 \cdot 3 \cdot 5 \cdot 7^{2} \cdot 17^{2}\) |
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sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
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| Discriminant: | \(-46474516816875000000 \) | = | \(-1 \cdot 2^{6} \cdot 3^{7} \cdot 5^{10} \cdot 7^{6} \cdot 17^{2} \) |
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sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
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| j-invariant: | \( -\frac{29324621982169}{1366875000000} \) | = | \(-1 \cdot 2^{-6} \cdot 3^{-7} \cdot 5^{-10} \cdot 17 \cdot 67^{3} \cdot 179^{3}\) |
| 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: | \(2\) | ||
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sage: E.regulator()
magma: Regulator(E);
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| Regulator: | \(13.875980487229298711431743739\) | ||
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sage: E.period_lattice().omega()
gp: E.omega[1]
magma: RealPeriod(E);
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| Real period: | \(0.16733542947796594068087488745\) | ||
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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: | \( 4 \) = \( 2\cdot1\cdot2\cdot1\cdot1 \) | ||
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sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
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| Torsion order: | \(1\) | ||
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sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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| Analytic order of Ш: | \(1\) (rounded) | ||
Modular invariants
Modular form 424830.2.a.j
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: | 11430720 | ||
| \( \Gamma_0(N) \)-optimal: | yes | ||
| Manin constant: | 1 | ||
Special L-value
\( L^{(2)}(E,1)/2! \) ≈ \( 9.2877726170335591507783777978033917999 \)
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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|
| \(2\) | \(2\) | \(I_{6}\) | Non-split multiplicative | 1 | 1 | 6 | 6 |
| \(3\) | \(1\) | \(I_{7}\) | Non-split multiplicative | 1 | 1 | 7 | 7 |
| \(5\) | \(2\) | \(I_{10}\) | Non-split multiplicative | 1 | 1 | 10 | 10 |
| \(7\) | \(1\) | \(I_0^{*}\) | Additive | -1 | 2 | 6 | 0 |
| \(17\) | \(1\) | \(II\) | Additive | 1 | 2 | 2 | 0 |
Galois representations
The 2-adic representation attached to this elliptic curve is surjective.
The mod \( p \) Galois representation has maximal image \(\GL(2,\F_p)\) for all primes \( p \) .
$p$-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 no rational isogenies. Its isogeny class 424830.j consists of this curve only.