Minimal Weierstrass equation
\( y^2 + x y = x^{3} - 363 x + 1305 \)
Mordell-Weil group structure
Torsion generators
\( \left(-12, 69\right) \)
Integral points
\( \left(-12, 69\right) \), \( \left(-12, -57\right) \), \( \left(24, 69\right) \), \( \left(24, -93\right) \)
Invariants
|
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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| Conductor: | \( 462 \) | = | \(2 \cdot 3 \cdot 7 \cdot 11\) | ||
|
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
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| Discriminant: | \(2291573592 \) | = | \(2^{3} \cdot 3^{12} \cdot 7^{2} \cdot 11 \) | ||
|
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
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| j-invariant: | \( \frac{5290763640625}{2291573592} \) | = | \(2^{-3} \cdot 3^{-12} \cdot 5^{6} \cdot 7^{-2} \cdot 11^{-1} \cdot 17^{3} \cdot 41^{3}\) | ||
| Endomorphism ring: | \(\Z\) | (no Complex Multiplication) | |||
| Sato-Tate Group: | $\mathrm{SU}(2)$ | ||||
BSD invariants
|
sage: E.rank()
magma: Rank(E);
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| Rank: | \(0\) | ||
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sage: E.regulator()
magma: Regulator(E);
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| Regulator: | \(1\) | ||
|
sage: E.period_lattice().omega()
gp: E.omega[1]
magma: RealPeriod(E);
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| Real period: | \(1.31360313742\) | ||
|
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: | \( 72 \) = \( 3\cdot( 2^{2} \cdot 3 )\cdot2\cdot1 \) | ||
|
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
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| Torsion order: | \(6\) | ||
|
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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| Analytic order of Ш: | \(1\) (exact) | ||
Modular invariants
Modular form 462.2.a.f
|
sage: E.modular_degree()
magma: ModularDegree(E);
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| Modular degree: | 192 | ||
| \( \Gamma_0(N) \)-optimal: | no | ||
| Manin constant: | 1 | ||
Special L-value
\( L(E,1) \) ≈ \( 2.62720627483 \)
Local data
This elliptic curve is semistable.
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(N\)) | ord(\(\Delta\)) | ord\((j)_{-}\) |
|---|---|---|---|---|---|---|---|
| \(2\) | \(3\) | \( I_{3} \) | Split multiplicative | -1 | 1 | 3 | 3 |
| \(3\) | \(12\) | \( I_{12} \) | Split multiplicative | -1 | 1 | 12 | 12 |
| \(7\) | \(2\) | \( I_{2} \) | Split multiplicative | -1 | 1 | 2 | 2 |
| \(11\) | \(1\) | \( I_{1} \) | 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 X6.
This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^1\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right)$ and has index 3.
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 | 7 | 11 |
|---|---|---|---|---|
| Reduction type | split | split | split | nonsplit |
| $\lambda$-invariant(s) | 2 | 1 | 1 | 0 |
| $\mu$-invariant(s) | 0 | 0 | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ 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 462.f
consists of 4 curves linked by isogenies of
degrees dividing 6.
Growth of torsion in number fields
The number fields $K$ of degree up to 7 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 | \(\Q(\sqrt{22}) \) | \(\Z/2\Z \times \Z/6\Z\) | Not in database |
| 4 | 4.0.17248.1 | \(\Z/12\Z\) | Not in database |
| 6 | 6.0.949132107.1 | \(\Z/3\Z \times \Z/6\Z\) | Not in database |
We only show fields where the torsion growth is primitive. For each field $K$ we either show its label, or a defining polynomial when $K$ is not in the database.