Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x\)
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(homogenize, simplify) |
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\(y^2z=x^3-xz^2\)
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(dehomogenize, simplify) |
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\(y^2=x^3-x\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z \oplus \Z/{2}\Z\)
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
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sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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| Conductor: | \( 32 \) | = | $2^{5}$ |
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sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
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| Discriminant: | $64 $ | = | $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: | $N(\mathrm{U}(1))$ | ||
| Faltings height: | $-0.96395933563153686354365902258\dots$ | ||
| Stable Faltings height: | $-1.3105329259115095182522750833\dots$ | ||
BSD invariants
sage: E.rank()
magma: Rank(E);
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| Analytic rank: | $0$ | ||
sage: E.regulator()
magma: Regulator(E);
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| Regulator: | $1$ | ||
sage: E.period_lattice().omega()
gp: if(E.disc>0,2,1)*E.omega[1]
magma: (Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E);
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| Real period: | $5.2441151085842396209296791798\dots$ | ||
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: | $ 2 $ = $ 2 $ | ||
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
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| Torsion order: | $4$ | ||
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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| Analytic order of Ш: | $1$ (exact) | ||
sage: r = E.rank();
gp: ar = ellanalyticrank(E);
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
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| Special value: | $ L(E,1) $ ≈ $ 0.65551438857302995261620989747 $ | ||
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: | 2 | ||
| $ \Gamma_0(N) $-optimal: | no | ||
| Manin constant: | 2 | ||
Local data
This elliptic curve is not semistable. There is only one prime of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $III$ | Additive | -1 | 5 | 6 | 0 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$ except those listed in the table below.
| prime $\ell$ | mod-$\ell$ image | $\ell$-adic image |
|---|---|---|
| $2$ | 2Cs | 16.384.9.568 |
$p$-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 less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-1}) \) | \(\Z/2\Z \oplus \Z/4\Z\) | 2.0.4.1-64.1-CMa1 |
| $2$ | \(\Q(\sqrt{2}) \) | \(\Z/2\Z \oplus \Z/4\Z\) | 2.2.8.1-32.1-a4 |
| $4$ | \(\Q(\zeta_{8})\) | \(\Z/4\Z \oplus \Z/4\Z\) | Not in database |
| $8$ | 8.0.4194304.1 | \(\Z/4\Z \oplus \Z/8\Z\) | Not in database |
| $8$ | 8.4.67108864.1 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
| $8$ | 8.2.143327232.1 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
| $8$ | 8.0.8192000.1 | \(\Z/2\Z \oplus \Z/20\Z\) | Not in database |
| $16$ | 16.0.18014398509481984.1 | \(\Z/8\Z \oplus \Z/8\Z\) | Not in database |
| $16$ | 16.0.20542695432781824.1 | \(\Z/6\Z \oplus \Z/12\Z\) | Not in database |
| $16$ | 16.4.1048576000000000000.1 | \(\Z/2\Z \oplus \Z/10\Z\) | Not in database |
| $16$ | 16.4.5258930030792146944.1 | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
| $16$ | 16.0.17179869184000000.1 | \(\Z/4\Z \oplus \Z/20\Z\) | Not in database |
We only show fields where the torsion growth is primitive.