Base field \(\Q(\sqrt{-2}) \)
Generator \(a\), with minimal polynomial \( x^{2} + 2 \); class number \(1\).
Weierstrass equation
This is a global minimal model.
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
$P$ | $\hat{h}(P)$ | Order |
---|---|---|
$\left(-2 a - 1 : 0 : 1\right)$ | $0$ | $2$ |
Invariants
Conductor: | $\frak{N}$ | = | \((96)\) | = | \((a)^{10}\cdot(-a-1)\cdot(a-1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | $N(\frak{N})$ | = | \( 9216 \) | = | \(2^{10}\cdot3\cdot3\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | $\Delta$ | = | $6463488a-4841472$ | ||
Discriminant ideal: | $\frak{D}_{\mathrm{min}} = (\Delta)$ | = | \((6463488a-4841472)\) | = | \((a)^{26}\cdot(-a-1)\cdot(a-1)^{12}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ | = | \( 106993205379072 \) | = | \(2^{26}\cdot3\cdot3^{12}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | $j$ | = | \( \frac{715706108}{531441} a + \frac{421307996}{531441} \) | ||
sage: E.j_invariant()
gp: E.j
magma: jInvariant(E);
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Endomorphism ring: | $\mathrm{End}(E)$ | = | \(\Z\) | ||
Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) | ||
sage: E.has_cm(), E.cm_discriminant()
magma: HasComplexMultiplication(E);
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Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ |
BSD invariants
Analytic rank: | $r_{\mathrm{an}}$ | = | \( 0 \) |
sage: E.rank()
magma: Rank(E);
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Mordell-Weil rank: | $r$ | = | \(0\) |
Regulator: | $\mathrm{Reg}(E/K)$ | = | \( 1 \) |
Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | = | \( 1 \) |
Global period: | $\Omega(E/K)$ | ≈ | \( 1.82280853495194049503465436399147106172 \) |
Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 8 \) = \(2^{2}\cdot1\cdot2\) |
Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(2\) |
Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 1.2889202758692330527995154135457454918 \) |
Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$\displaystyle 1.288920276 \approx L(E/K,1) \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \approx \frac{ 1 \cdot 1.822809 \cdot 1 \cdot 8 } { {2^2 \cdot 2.828427} } \approx 1.288920276$
Local data at primes of bad reduction
This elliptic curve is not semistable. There are 3 primes $\frak{p}$ of bad reduction.
$\mathfrak{p}$ | $N(\mathfrak{p})$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
---|---|---|---|---|---|---|---|---|
\((a)\) | \(2\) | \(4\) | \(I_{12}^{*}\) | Additive | \(-1\) | \(10\) | \(26\) | \(0\) |
\((-a-1)\) | \(3\) | \(1\) | \(I_{1}\) | Non-split multiplicative | \(1\) | \(1\) | \(1\) | \(1\) |
\((a-1)\) | \(3\) | \(2\) | \(I_{12}\) | Non-split multiplicative | \(1\) | \(1\) | \(12\) | \(12\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
prime | Image of Galois Representation |
---|---|
\(2\) | 2B |
\(3\) | 3B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3, 4, 6 and 12.
Its isogeny class
9216.2-c
consists of curves linked by isogenies of
degrees dividing 12.
Base change
This elliptic curve is a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.