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
\(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(2 a + 2 : -5 a + 2 : 1\right)$ | $1.0002955069741953013651874570726776137$ | $\infty$ |
| $\left(a - 3 : 0 : 1\right)$ | $0$ | $2$ |
| $\left(-2 a : 0 : 1\right)$ | $0$ | $2$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((144)\) | = | \((a)^{8}\cdot(-a-1)^{2}\cdot(a-1)^{2}\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 20736 \) | = | \(2^{8}\cdot3^{2}\cdot3^{2}\) |
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| Discriminant: | $\Delta$ | = | $419904$ | ||
| Discriminant ideal: | $\frak{D}_{\mathrm{min}} = (\Delta)$ | = | \((419904)\) | = | \((a)^{12}\cdot(-a-1)^{8}\cdot(a-1)^{8}\) |
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| Discriminant norm: | $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ | = | \( 176319369216 \) | = | \(2^{12}\cdot3^{8}\cdot3^{8}\) |
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| j-invariant: | $j$ | = | \( \frac{64}{9} \) | ||
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| Endomorphism ring: | $\mathrm{End}(E)$ | = | \(\Z\) | ||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) | ||
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| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | ||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | \( 1 \) |
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| Mordell-Weil rank: | $r$ | = | \(1\) |
| Regulator: | $\mathrm{Reg}(E/K)$ | ≈ | \( 1.0002955069741953013651874570726776137 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 2.0005910139483906027303749141453552274 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 3.2431309741884706942554121444821260636 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 32 \) = \(2\cdot2^{2}\cdot2^{2}\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(4\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 4.5878351450199469952191564144023463567 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}4.587835145 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 3.243131 \cdot 2.000591 \cdot 32 } { {4^2 \cdot 2.828427} } \\ & \approx 4.587835145 \end{aligned}$$
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 | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((a)\) | \(2\) | \(2\) | \(I_0^{*}\) | Additive | \(1\) | \(8\) | \(12\) | \(0\) |
| \((-a-1)\) | \(3\) | \(4\) | \(I_{2}^{*}\) | Additive | \(-1\) | \(2\) | \(8\) | \(2\) |
| \((a-1)\) | \(3\) | \(4\) | \(I_{2}^{*}\) | Additive | \(-1\) | \(2\) | \(8\) | \(2\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(2\) | 2Cs |
| \(5\) | 5B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 5 and 10.
Its isogeny class
20736.3-k
consists of curves linked by isogenies of
degrees dividing 20.
Base change
This elliptic curve is a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.