Minimal Weierstrass equation
\( y^2 + x y + y = x^{3} - 216 x + 2062 \)
Mordell-Weil group structure
\(\Z/{6}\Z\)
Torsion generators
\( \left(20, 66\right) \)
Integral points
\( \left(2, 39\right) \), \( \left(2, -42\right) \), \( \left(20, 66\right) \), \( \left(20, -87\right) \)
Invariants
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sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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| Conductor: | \( 102 \) | = | \(2 \cdot 3 \cdot 17\) |
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sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
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| Discriminant: | \(-1228691592 \) | = | \(-1 \cdot 2^{3} \cdot 3^{12} \cdot 17^{2} \) |
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sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
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| j-invariant: | \( -\frac{1107111813625}{1228691592} \) | = | \(-1 \cdot 2^{-3} \cdot 3^{-12} \cdot 5^{3} \cdot 17^{-2} \cdot 2069^{3}\) |
| 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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| 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: | \(1.39303249219\) | ||
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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: | \( 24 \) = \( 1\cdot( 2^{2} \cdot 3 )\cdot2 \) | ||
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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: | 48 | ||
| \( \Gamma_0(N) \)-optimal: | no | ||
| Manin constant: | 1 | ||
Special L-value
\( L(E,1) \) ≈ \( 0.928688328127 \)
Local data
This elliptic curve is semistable.
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(N\)) | ord(\(\Delta\)) | ord\((j)_{-}\) |
|---|---|---|---|---|---|---|---|
| \(2\) | \(1\) | \( I_{3} \) | Non-split multiplicative | 1 | 1 | 3 | 3 |
| \(3\) | \(12\) | \( I_{12} \) | Split multiplicative | -1 | 1 | 12 | 12 |
| \(17\) | \(2\) | \( I_{2} \) | Non-split multiplicative | 1 | 1 | 2 | 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 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 | 17 |
|---|---|---|---|
| Reduction type | nonsplit | split | nonsplit |
| $\lambda$-invariant(s) | 1 | 1 | 0 |
| $\mu$-invariant(s) | 1 | 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 102.b
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-2}) \) | \(\Z/2\Z \times \Z/6\Z\) | 2.0.8.1-5202.5-d5 |
| $4$ | 4.2.9248.1 | \(\Z/12\Z\) | Not in database |
| $6$ | 6.0.2255067.2 | \(\Z/3\Z \times \Z/6\Z\) | 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.