Minimal Weierstrass equation
\(y^2+xy=x^3+x^2+98688x-3064914\)
Mordell-Weil group structure
\(\Z^2 \times \Z/{2}\Z\)
Infinite order Mordell-Weil generators and heights
| \(P\) | = | \( \left(137, 3544\right) \) | \( \left(\frac{1023}{4}, \frac{48927}{8}\right) \) |
| \(\hat{h}(P)\) | ≈ | $0.70625380499887367913285701272$ | $4.0380490183809089941934496631$ |
Torsion generators
\( \left(\frac{123}{4}, -\frac{123}{8}\right) \)
Integral points
\( \left(137, 3544\right) \), \( \left(137, -3681\right) \), \( \left(187, 4594\right) \), \( \left(187, -4781\right) \), \( \left(681, 19167\right) \), \( \left(681, -19848\right) \), \( \left(987, 32019\right) \), \( \left(987, -33006\right) \), \( \left(1565, 62364\right) \), \( \left(1565, -63929\right) \), \( \left(6495, 520854\right) \), \( \left(6495, -527349\right) \), \( \left(24837, 3902244\right) \), \( \left(24837, -3927081\right) \)
Invariants
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sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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| Conductor: | \( 372810 \) | = | \(2 \cdot 3 \cdot 5 \cdot 17^{2} \cdot 43\) |
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sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
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| Discriminant: | \(-65680588146093750 \) | = | \(-1 \cdot 2 \cdot 3^{4} \cdot 5^{8} \cdot 17^{6} \cdot 43 \) |
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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: | \(2\) | ||
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sage: E.regulator()
magma: Regulator(E);
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| Regulator: | \(2.8479819646676937912126672845\) | ||
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sage: E.period_lattice().omega()
gp: E.omega[1]
magma: RealPeriod(E);
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| Real period: | \(0.20124600393129931571667705576\) | ||
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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: | \( 64 \) = \( 1\cdot2\cdot2^{3}\cdot2^{2}\cdot1 \) | ||
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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 372810.2.a.n
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: | 3932160 | ||
| \( \Gamma_0(N) \)-optimal: | no | ||
| Manin constant: | 1 | ||
Special L-value
\( L^{(2)}(E,1)/2! \) ≈ \( 9.1703198345245480572468251787045877949 \)
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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|
| \(2\) | \(1\) | \(I_{1}\) | Non-split multiplicative | 1 | 1 | 1 | 1 |
| \(3\) | \(2\) | \(I_{4}\) | Non-split multiplicative | 1 | 1 | 4 | 4 |
| \(5\) | \(8\) | \(I_{8}\) | Split multiplicative | -1 | 1 | 8 | 8 |
| \(17\) | \(4\) | \(I_0^{*}\) | Additive | 1 | 2 | 6 | 0 |
| \(43\) | \(1\) | \(I_{1}\) | Non-split multiplicative | 1 | 1 | 1 | 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 X13.
This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^2\Z_2)$ generated by $\left(\begin{array}{rr} 3 & 0 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 0 \\ 0 & 3 \end{array}\right)$ and has index 6.
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 372810n
consists of 3 curves linked by isogenies of
degrees dividing 4.
Growth of torsion in number fields
The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-86}) \) | \(\Z/2\Z \times \Z/2\Z\) | Not in database |
| $2$ | \(\Q(\sqrt{731}) \) | \(\Z/4\Z\) | Not in database |
| $2$ | \(\Q(\sqrt{-34}) \) | \(\Z/4\Z\) | Not in database |
| $4$ | \(\Q(\sqrt{-34}, \sqrt{-86})\) | \(\Z/2\Z \times \Z/4\Z\) | Not in database |
| $8$ | Deg 8 | \(\Z/2\Z \times \Z/4\Z\) | Not in database |
| $8$ | Deg 8 | \(\Z/8\Z\) | Not in database |
| $8$ | Deg 8 | \(\Z/8\Z\) | Not in database |
| $8$ | Deg 8 | \(\Z/6\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/4\Z \times \Z/4\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/2\Z \times \Z/8\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/2\Z \times \Z/8\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/2\Z \times \Z/6\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/12\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/12\Z\) | Not in database |
We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.