Minimal Weierstrass equation
\(y^2=x^3+x\)
Mordell-Weil group structure
\(\Z/{2}\Z\)
Torsion generators
\( \left(0, 0\right) \)
Integral points
\( \left(0, 0\right) \)
Invariants
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sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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| Conductor: | \( 64 \) | = | \(2^{6}\) |
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sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
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| Discriminant: | \(-64 \) | = | \(-1 \cdot 2^{6} \) |
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sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
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| j-invariant: | \( 1728 \) | = | \(2^{6} \cdot 3^{3}\) |
| Endomorphism ring: | \(\Z\) | ||
| Geometric endomorphism ring: | \(\Z[\sqrt{-1}]\) | (potential complex multiplication) | |
| Sato-Tate group: | $N(\mathrm{U}(1))$ | ||
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: | \(3.7081493546027438368677006944\) | ||
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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: | \( 1 \) | ||
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sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
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| Torsion order: | \(2\) | ||
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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: | 4 | ||
| \( \Gamma_0(N) \)-optimal: | no | ||
| Manin constant: | 2 | ||
Special L-value
\( L(E,1) \) ≈ \( 0.92703733865068595921692517359767012944 \)
Local data
This elliptic curve is not semistable. There is only one prime of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(N\)) | ord(\(\Delta\)) | ord\((j)_{-}\) |
|---|---|---|---|---|---|---|---|
| \(2\) | \(1\) | \(II\) | Additive | -1 | 6 | 6 | 0 |
Galois representations
The mod \( p \) Galois representation has maximal image for all primes \( p < 1000 \) .
The image is a Borel subgroup if \(p=2\), 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 and 4.
Its isogeny class 64a
consists of 3 curves linked by isogenies of
degrees dividing 4.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-1}) \) | \(\Z/2\Z \times \Z/2\Z\) | 2.0.4.1-256.1-CMa1 |
| $2$ | \(\Q(\sqrt{2}) \) | \(\Z/4\Z\) | 2.2.8.1-32.1-a3 |
| $2$ | \(\Q(\sqrt{-2}) \) | \(\Z/4\Z\) | 2.0.8.1-32.1-a2 |
| $4$ | \(\Q(\zeta_{8})\) | \(\Z/4\Z \times \Z/4\Z\) | Not in database |
| $4$ | 4.2.1024.1 | \(\Z/8\Z\) | Not in database |
| $8$ | 8.0.4194304.1 | \(\Z/4\Z \times \Z/8\Z\) | Not in database |
| $8$ | 8.2.573308928.1 | \(\Z/6\Z\) | Not in database |
| $8$ | 8.0.32768000.1 | \(\Z/2\Z \times \Z/10\Z\) | Not in database |
| $16$ | 16.0.18014398509481984.1 | \(\Z/8\Z \times \Z/8\Z\) | Not in database |
| $16$ | 16.4.4611686018427387904.2 | \(\Z/16\Z\) | Not in database |
| $16$ | 16.0.328683126924509184.1 | \(\Z/6\Z \times \Z/6\Z\) | Not in database |
| $16$ | 16.4.16777216000000000000.2 | \(\Z/10\Z\) | Not in database |
| $16$ | 16.4.5258930030792146944.1 | \(\Z/12\Z\) | Not in database |
| $16$ | 16.0.17179869184000000.1 | \(\Z/4\Z \times \Z/20\Z\) | Not in database |
| $16$ | 16.0.5258930030792146944.3 | \(\Z/12\Z\) | Not in database |
We only show fields where the torsion growth is primitive.