Minimal Weierstrass equation
\( y^2 = x^{3} - x \)
Mordell-Weil group structure
Torsion generators
\( \left(0, 0\right) \), \( \left(1, 0\right) \)
Integral points
\( \left(-1, 0\right) \), \( \left(0, 0\right) \), \( \left(1, 0\right) \)
Invariants
|
magma: Conductor(E);
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
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| Conductor: | \( 32 \) | = | \(2^{5}\) | ||
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magma: Discriminant(E);
sage: E.discriminant().factor()
gp: E.disc
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| Discriminant: | \(64 \) | = | \(2^{6} \) | ||
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magma: jInvariant(E);
sage: E.j_invariant().factor()
gp: E.j
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| j-invariant: | \( 1728 \) | = | \(2^{6} \cdot 3^{3}\) | ||
| Endomorphism ring: | \(\Z[\sqrt{-1}]\) | ( Complex Multiplication) | |||
| Sato-Tate Group: | $N(\mathrm{U}(1))$ | ||||
BSD invariants
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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: | \(5.24411510858\) | ||
|
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: | \( 2 \) = \( 2 \) | ||
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magma: Order(TorsionSubgroup(E));
sage: E.torsion_order()
gp: elltors(E)[1]
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| Torsion order: | \(4\) | ||
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magma: MordellWeilShaInformation(E);
sage: E.sha().an_numerical()
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| Analytic order of Ш: | \(1\) (exact) | ||
Modular invariants
Modular form 32.2.a.a
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magma: ModularDegree(E);
sage: E.modular_degree()
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| Modular degree: | 2 | ||
| \( \Gamma_0(N) \)-optimal: | no | ||
| Manin constant: | 2 | ||
Special L-value
\( L(E,1) \) ≈ \( 0.655514388573 \)
Local data
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(N\)) | ord(\(\Delta\)) | ord\((j)_{-}\) |
|---|---|---|---|---|---|---|---|
| \(2\) | \(2\) | \( III \) | Additive | -1 | 5 | 6 | 0 |
Galois representations
The mod \( p \) Galois representation has maximal image for all primes \( p \) except those listed.
| prime | Image of Galois representation |
|---|---|
| \(2\) | Cs |
For all other primes \(p\), the image is the normalizer of a split Cartan subgroup if \(\left(\frac{ -1 }{p}\right)=+1\) or the normalizer of a nonsplit Cartan subgroup if \(\left(\frac{ -1 }{p}\right)=-1\).
$p$-adic data
$p$-adic regulators
All \(p\)-adic regulators are identically \(1\) since the rank is \(0\).
Iwasawa invariants
| $p$ | 2 |
|---|---|
| Reduction type | add |
| $\lambda$-invariant(s) | - |
| $\mu$-invariant(s) | - |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ of good reduction are zero.
An entry - indicates that the invariants are not computed because the reduction is additive.
Isogenies
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2.
Its isogeny class 32.a
consists of 4 curves linked by isogenies of
degrees dividing 4.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \times \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base-change curve |
|---|---|---|---|
| 2 | \(\Q(\sqrt{2}) \) | \(\Z/2\Z \times \Z/4\Z\) | 2.2.8.1-32.1-a4 |
| \(\Q(\sqrt{-1}) \) | \(\Z/2\Z \times \Z/4\Z\) | 2.0.4.1-64.1-CMa1 | |
| 4 | \(\Q(\zeta_{8})\) | \(\Z/4\Z \times \Z/4\Z\) | 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.