Minimal Weierstrass equation
\(y^2+xy+y=x^3-x^2-28777352x+62835704039\)
Mordell-Weil group structure
\(\Z\times \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
| \(P\) | = | \( \left(3273, 59203\right) \) |
| \(\hat{h}(P)\) | ≈ | $4.1236301163450474347844743638$ |
Torsion generators
\( \left(-\frac{24933}{4}, \frac{24929}{8}\right) \)
Integral points
\( \left(3273, 59203\right) \), \( \left(3273, -62477\right) \)
Invariants
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sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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| Conductor: | \( 471510 \) | = | \(2 \cdot 3^{2} \cdot 5 \cdot 13^{2} \cdot 31\) |
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sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
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| Discriminant: | \(-180066301008500169336060 \) | = | \(-1 \cdot 2^{2} \cdot 3^{7} \cdot 5 \cdot 13^{6} \cdot 31^{8} \) |
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sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
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| j-invariant: | \( -\frac{749011598724977281}{51173462246460} \) | = | \(-1 \cdot 2^{-2} \cdot 3^{-1} \cdot 5^{-1} \cdot 31^{-8} \cdot 71^{3} \cdot 12791^{3}\) |
| 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: | \(4.1236301163450474347844743638\) | ||
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sage: E.period_lattice().omega()
gp: E.omega[1]
magma: RealPeriod(E);
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| Real period: | \(0.099615744153929840298093210509\) | ||
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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: | \( 128 \) = \( 2\cdot2^{2}\cdot1\cdot2\cdot2^{3} \) | ||
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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 471510.2.a.ep
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: | 62914560 | ||
| \( \Gamma_0(N) \)-optimal: | no | ||
| Manin constant: | 1 (conditional*) | ||
Special L-value
\( L'(E,1) \) ≈ \( 13.144911444968581958039922245036691982 \)
Local data
This elliptic curve is semistable. There are 5 primes of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(N\)) | ord(\(\Delta\)) | ord\((j)_{-}\) |
|---|---|---|---|---|---|---|---|
| \(2\) | \(2\) | \(I_{2}\) | Split multiplicative | -1 | 1 | 2 | 2 |
| \(3\) | \(4\) | \(I_1^{*}\) | Additive | -1 | 2 | 7 | 1 |
| \(5\) | \(1\) | \(I_{1}\) | Split multiplicative | -1 | 1 | 1 | 1 |
| \(13\) | \(2\) | \(I_0^{*}\) | Additive | 1 | 2 | 6 | 0 |
| \(31\) | \(8\) | \(I_{8}\) | 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 X85.
This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 3 & 3 \\ 0 & 3 \end{array}\right),\left(\begin{array}{rr} 7 & 0 \\ 0 & 7 \end{array}\right),\left(\begin{array}{rr} 7 & 0 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 7 & 0 \\ 0 & 3 \end{array}\right)$ and has index 24.
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 471510ep
consists of 4 curves linked by isogenies of
degrees dividing 8.