Minimal Weierstrass equation
\(y^2+xy=x^3-41608x-90515392\)
Mordell-Weil group structure
\(\Z/{6}\Z\)
Torsion generators
\( \left(2864, 151160\right) \)
Integral points
\( \left(752, 17048\right) \), \( \left(752, -17800\right) \), \( \left(2864, 151160\right) \), \( \left(2864, -154024\right) \)
Invariants
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sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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| Conductor: | \( 1122 \) | = | \(2 \cdot 3 \cdot 11 \cdot 17\) |
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sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
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| Discriminant: | \(-3534510366354604032 \) | = | \(-1 \cdot 2^{15} \cdot 3^{6} \cdot 11^{6} \cdot 17^{4} \) |
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sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
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| j-invariant: | \( -\frac{7966267523043306625}{3534510366354604032} \) | = | \(-1 \cdot 2^{-15} \cdot 3^{-6} \cdot 5^{3} \cdot 11^{-6} \cdot 17^{-4} \cdot 19^{3} \cdot 21023^{3}\) |
| 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: | \(0\) | ||
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sage: E.regulator()
magma: Regulator(E);
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| Regulator: | \(1\) | ||
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sage: E.period_lattice().omega()
gp: E.omega[1]
magma: RealPeriod(E);
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| Real period: | \(0.11220167125559804079938946905\) | ||
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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: | \( 1080 \) = \( ( 3 \cdot 5 )\cdot( 2 \cdot 3 )\cdot( 2 \cdot 3 )\cdot2 \) | ||
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sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
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| Torsion order: | \(6\) | ||
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sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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| Analytic order of Ш: | \(1\) (exact) | ||
Modular invariants
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: | 23040 | ||
| \( \Gamma_0(N) \)-optimal: | no | ||
| Manin constant: | 1 | ||
Special L-value
\( L(E,1) \) ≈ \( 3.3660501376679412239816840714991795686 \)
Local data
This elliptic curve is semistable. There are 4 primes of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(N\)) | ord(\(\Delta\)) | ord\((j)_{-}\) |
|---|---|---|---|---|---|---|---|
| \(2\) | \(15\) | \(I_{15}\) | Split multiplicative | -1 | 1 | 15 | 15 |
| \(3\) | \(6\) | \(I_{6}\) | Split multiplicative | -1 | 1 | 6 | 6 |
| \(11\) | \(6\) | \(I_{6}\) | Split multiplicative | -1 | 1 | 6 | 6 |
| \(17\) | \(2\) | \(I_{4}\) | Non-split multiplicative | 1 | 1 | 4 | 4 |
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 X19.
This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 0 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 0 & 5 \end{array}\right)$ 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 |
| \(3\) | B.1.1 |
$p$-adic data
$p$-adic regulators
All \(p\)-adic regulators are identically \(1\) since the rank is \(0\).
Iwasawa invariants
| $p$ | 2 | 3 | 5 | 11 | 17 |
|---|---|---|---|---|---|
| Reduction type | split | split | ss | split | nonsplit |
| $\lambda$-invariant(s) | 6 | 3 | 2,2 | 1 | 0 |
| $\mu$-invariant(s) | 1 | 0 | 0,0 | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 7$ of good reduction are zero.
Isogenies
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3 and 6.
Its isogeny class 1122.j
consists of 4 curves linked by isogenies of
degrees dividing 6.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{6}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-2}) \) | \(\Z/2\Z \times \Z/6\Z\) | Not in database |
| $4$ | 4.2.34848.2 | \(\Z/12\Z\) | Not in database |
| $6$ | 6.0.2255067.2 | \(\Z/3\Z \times \Z/6\Z\) | Not in database |
| $8$ | 8.0.77720518656.13 | \(\Z/2\Z \times \Z/12\Z\) | Not in database |
| $8$ | 8.0.4567597056.5 | \(\Z/2\Z \times \Z/12\Z\) | Not in database |
| $9$ | 9.3.135878452228432951488.2 | \(\Z/18\Z\) | Not in database |
| $12$ | Deg 12 | \(\Z/6\Z \times \Z/6\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/24\Z\) | Not in database |
| $18$ | 18.0.38719620445623100574568675231701844967594917888.1 | \(\Z/2\Z \times \Z/18\Z\) | 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.