Minimal Weierstrass equation
\(y^2=x^3+x^2+10102320x-3213585900\)
Mordell-Weil group structure
\(\Z\times \Z/{4}\Z\)
Infinite order Mordell-Weil generator and height
| \(P\) | = | \( \left(\frac{56490}{121}, \frac{53319000}{1331}\right) \) |
| \(\hat{h}(P)\) | ≈ | $7.4990864877313457484390307495$ |
Torsion generators
\( \left(3540, 277350\right) \)
Integral points
\( \left(315, 0\right) \), \((3540,\pm 277350)\)
Invariants
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sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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| Conductor: | \( 443760 \) | = | \(2^{4} \cdot 3 \cdot 5 \cdot 43^{2}\) |
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sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
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| Discriminant: | \(-70455383998934400000000 \) | = | \(-1 \cdot 2^{13} \cdot 3^{4} \cdot 5^{8} \cdot 43^{7} \) |
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sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
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| j-invariant: | \( \frac{4403686064471}{2721093750} \) | = | \(2^{-1} \cdot 3^{-4} \cdot 5^{-8} \cdot 37^{3} \cdot 43^{-1} \cdot 443^{3}\) |
| 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: | \(7.4990864877313457484390307495\) | ||
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sage: E.period_lattice().omega()
gp: E.omega[1]
magma: RealPeriod(E);
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| Real period: | \(0.063268494844984976499309928778\) | ||
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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: | \( 512 \) = \( 2^{2}\cdot2^{2}\cdot2^{3}\cdot2^{2} \) | ||
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sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
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| Torsion order: | \(4\) | ||
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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 443760.2.a.df
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: | 34062336 | ||
| \( \Gamma_0(N) \)-optimal: | no | ||
| Manin constant: | 1 (conditional*) | ||
Special L-value
\( L'(E,1) \) ≈ \( 15.182589273316068534368132739845895818 \)
Local data
This elliptic curve is not semistable. There are 4 primes of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(N\)) | ord(\(\Delta\)) | ord\((j)_{-}\) |
|---|---|---|---|---|---|---|---|
| \(2\) | \(4\) | \(I_5^{*}\) | Additive | -1 | 4 | 13 | 1 |
| \(3\) | \(4\) | \(I_{4}\) | Split multiplicative | -1 | 1 | 4 | 4 |
| \(5\) | \(8\) | \(I_{8}\) | Split multiplicative | -1 | 1 | 8 | 8 |
| \(43\) | \(4\) | \(I_1^{*}\) | Additive | -1 | 2 | 7 | 1 |
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 X13h.
This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^2\Z_2)$ generated by $\left(\begin{array}{rr} 3 & 0 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 0 & 3 \end{array}\right)$ and has index 12.
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 and 4.
Its isogeny class 443760.df
consists of 3 curves linked by isogenies of
degrees dividing 4.