This is the minimal-conductor curve with torsion $(\Z/2\Z) \times (\Z/6\Z)$.
Minimal Weierstrass equation
\( y^2 + x y + y = x^{3} - 19 x + 26 \)
Mordell-Weil group structure
Torsion generators
\( \left(3, -2\right) \), \( \left(-2, 8\right) \)
Integral points
\( \left(-5, 2\right) \), \( \left(-2, 8\right) \), \( \left(-2, -7\right) \), \( \left(1, 2\right) \), \( \left(1, -4\right) \), \( \left(3, -2\right) \), \( \left(4, 2\right) \), \( \left(4, -7\right) \), \( \left(13, 38\right) \), \( \left(13, -52\right) \)
Invariants
|
magma: Conductor(E);
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
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| Conductor: | \( 30 \) | = | \(2 \cdot 3 \cdot 5\) | ||
|
magma: Discriminant(E);
sage: E.discriminant().factor()
gp: E.disc
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| Discriminant: | \(72900 \) | = | \(2^{2} \cdot 3^{6} \cdot 5^{2} \) | ||
|
magma: jInvariant(E);
sage: E.j_invariant().factor()
gp: E.j
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| j-invariant: | \( \frac{702595369}{72900} \) | = | \(2^{-2} \cdot 3^{-6} \cdot 5^{-2} \cdot 7^{3} \cdot 127^{3}\) | ||
| Endomorphism ring: | \(\Z\) | (no Complex Multiplication) | |||
| Sato-Tate Group: | $\mathrm{SU}(2)$ | ||||
BSD invariants
|
magma: Rank(E);
sage: E.rank()
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| Rank: | \(0\) | ||
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magma: Regulator(E);
sage: E.regulator()
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| Regulator: | \(1\) | ||
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magma: RealPeriod(E);
sage: E.period_lattice().omega()
gp: E.omega[1]
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| Real period: | \(3.35194825924\) | ||
|
magma: TamagawaNumbers(E);
sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
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| Tamagawa product: | \( 24 \) = \( 2\cdot( 2 \cdot 3 )\cdot2 \) | ||
|
magma: Order(TorsionSubgroup(E));
sage: E.torsion_order()
gp: elltors(E)[1]
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| Torsion order: | \(12\) | ||
|
magma: MordellWeilShaInformation(E);
sage: E.sha().an_numerical()
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| Analytic order of Ш: | \(1\) (exact) | ||
Modular invariants
Modular form 30.2.a.a
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magma: ModularDegree(E);
sage: E.modular_degree()
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| Modular degree: | 4 | ||
| \( \Gamma_0(N) \)-optimal: | no | ||
| Manin constant: | 1 | ||
Special L-value
\( L(E,1) \) ≈ \( 0.558658043207 \)
Local data
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(N\)) | ord(\(\Delta\)) | ord\((j)_{-}\) |
|---|---|---|---|---|---|---|---|
| \(2\) | \(2\) | \( I_{2} \) | Non-split multiplicative | 1 | 1 | 2 | 2 |
| \(3\) | \(6\) | \( I_{6} \) | Split multiplicative | -1 | 1 | 6 | 6 |
| \(5\) | \(2\) | \( I_{2} \) | 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 X8c.
This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 0 & 7 \end{array}\right),\left(\begin{array}{rr} 3 & 6 \\ 0 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 5 \end{array}\right)$ and has index 12.
The mod \( p \) Galois representation has maximal image \(\GL(2,\F_p)\) for all primes \( p \) except those listed.
| prime | Image of Galois representation |
|---|---|
| \(2\) | Cs |
| \(3\) | 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 |
|---|---|---|---|
| Reduction type | nonsplit | split | nonsplit |
| $\lambda$-invariant(s) | 0 | 1 | 0 |
| $\mu$-invariant(s) | 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 30a
consists of 8 curves linked by isogenies of
degrees dividing 12.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \times \Z/{6}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base-change curve |
|---|---|---|---|
| 4 | \(\Q(\sqrt{2}, \sqrt{5})\) | \(\Z/2\Z \times \Z/12\Z\) | Not in database |
| \(\Q(\sqrt{3}, \sqrt{-5})\) | \(\Z/2\Z \times \Z/12\Z\) | Not in database | |
| \(\Q(\sqrt{-2}, \sqrt{-3})\) | \(\Z/2\Z \times \Z/12\Z\) | Not in database | |
| 6 | 6.0.270000.1 | \(\Z/6\Z \times \Z/6\Z\) | 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.
Additional information
This is the curve of minimal conductor and torsion $(\Z/2\Z) \times (\Z/6\Z)$. Every elliptic curve $E/\Q$ with this torsion group must have conductor divisible by $30$ (for instance, if $E$ had good reduction at $5$ then the reduction mod $5$ would have at least $12$ points, which exceeds the Weil bound $(\sqrt5+1)^2 < 11$.