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
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(2 a : 2 a - 4 : 1\right)$ | $0.79631880308534916454009320540535352713$ | $\infty$ |
| $\left(-\frac{2}{9} a + \frac{8}{9} : \frac{26}{27} a - \frac{32}{27} : 1\right)$ | $1.3781292359956185528310500762882951211$ | $\infty$ |
| $\left(0 : 0 : 1\right)$ | $0$ | $2$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((128)\) | = | \((a)^{14}\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 16384 \) | = | \(2^{14}\) |
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| Discriminant: | $\Delta$ | = | $-1024a$ | ||
| Discriminant ideal: | $\frak{D}_{\mathrm{min}} = (\Delta)$ | = | \((-1024a)\) | = | \((a)^{21}\) |
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| Discriminant norm: | $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ | = | \( 2097152 \) | = | \(2^{21}\) |
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| j-invariant: | $j$ | = | \( 1728 \) | ||
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| Endomorphism ring: | $\mathrm{End}(E)$ | = | \(\Z\) | ||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z[\sqrt{-1}]\) (potential complex multiplication) | ||
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| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $N(\mathrm{U}(1))$ | ||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | \( 2 \) |
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| Mordell-Weil rank: | $r$ | = | \(2\) |
| Regulator: | $\mathrm{Reg}(E/K)$ | ≈ | \( 1.0128043787441488889294007764758939103 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 4.0512175149765955557176031059035756412 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 8.1760198917556304207099325392178141642 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 2 \) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(2\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 5.8553453080835073840774539536478604662 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}5.855345308 \approx L^{(2)}(E/K,1)/2! & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 8.176020 \cdot 4.051218 \cdot 2 } { {2^2 \cdot 2.828427} } \\ & \approx 5.855345308 \end{aligned}$$
Local data at primes of bad reduction
This elliptic curve is not semistable. There is only one prime $\frak{p}$ of bad reduction.
| $\mathfrak{p}$ | $N(\mathfrak{p})$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((a)\) | \(2\) | \(2\) | \(III^{*}\) | Additive | \(-1\) | \(14\) | \(21\) | \(0\) |
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 \(\left(\frac{ -1 }{p}\right)=-1\).
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2.
Its isogeny class
16384.1-a
consists of curves linked by isogenies of
degree 2.
Base change
This elliptic curve is a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.