Minimal Weierstrass equation
\(y^2+xy+y=x^3-x^2-8867874164x+321376081297106\) 
Mordell-Weil group structure
\(\Z^2 \times \Z/{2}\Z\)
Infinite order Mordell-Weil generators and heights
| \(P\) | = | \(\left(55858, 533742\right)\) | \(\left(\frac{2464107}{49}, \frac{544353256}{343}\right)\) |
| \(\hat{h}(P)\) | ≈ | $3.6431915450333852838614983484$ | $5.2982143287281042019572876311$ |
Torsion generators
\( \left(\frac{219675}{4}, -\frac{219679}{8}\right) \)
Integral points
\( \left(55858, 533742\right) \), \( \left(55858, -589601\right) \), \( \left(246828, 114638317\right) \), \( \left(246828, -114885146\right) \)
Invariants
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sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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| Conductor: | \( 439569 \) | = | \(3^{2} \cdot 13^{2} \cdot 17^{2}\) |
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sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
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| Discriminant: | \(13736068281063240698341006173 \) | = | \(3^{22} \cdot 13^{7} \cdot 17^{8} \) |
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sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
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| j-invariant: | \( \frac{908031902324522977}{161726530797} \) | = | \(3^{-16} \cdot 13^{-1} \cdot 17^{-2} \cdot 968353^{3}\) |
| 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: | \(19.125419769460858106419147723\) | ||
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sage: E.period_lattice().omega()
gp: E.omega[1]
magma: RealPeriod(E);
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| Real period: | \(0.038480565137320497226161127846\) | ||
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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: | \( 64 \) = \( 2^{2}\cdot2^{2}\cdot2^{2} \) | ||
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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: | \(1\) (rounded) | ||
Modular invariants
Modular form 439569.2.a.z
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: | 396361728 | ||
| \( \Gamma_0(N) \)-optimal: | not computed* (one of 4 curves in this isogeny class which might be optimal) | ||
| Manin constant: | 1 (conditional*) | ||
Special L-value
\( L^{(2)}(E,1)/2! \) ≈ \( 11.775311379477371483540094303954101341 \)
Local data
This elliptic curve is not semistable. There are 3 primes of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(N\)) | ord(\(\Delta\)) | ord\((j)_{-}\) |
|---|---|---|---|---|---|---|---|
| \(3\) | \(4\) | \(I_{16}^{*}\) | Additive | -1 | 2 | 22 | 16 |
| \(13\) | \(4\) | \(I_1^{*}\) | Additive | 1 | 2 | 7 | 1 |
| \(17\) | \(4\) | \(I_2^{*}\) | Additive | 1 | 2 | 8 | 2 |
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 X121.
This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^4\Z_2)$ generated by $\left(\begin{array}{rr} 3 & 3 \\ 0 & 3 \end{array}\right),\left(\begin{array}{rr} 3 & 0 \\ 8 & 3 \end{array}\right),\left(\begin{array}{rr} 7 & 0 \\ 0 & 7 \end{array}\right),\left(\begin{array}{rr} 7 & 0 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 7 & 0 \\ 0 & 3 \end{array}\right)$ 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 439569.z
consists of 4 curves linked by isogenies of
degrees dividing 8.