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
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(-5 a + 8 : 14 a + 49 : 1\right)$ | $2.6504363042870374111046289987478889329$ | $\infty$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((-20a+17)\) | = | \((a-1)^{2}\cdot(a+3)^{2}\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 1089 \) | = | \(3^{2}\cdot11^{2}\) |
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| Discriminant: | $\Delta$ | = | $-864073559a-1362467251$ | ||
| Discriminant ideal: | $\frak{D}_{\mathrm{min}} = (\Delta)$ | = | \((-864073559a-1362467251)\) | = | \((a-1)^{17}\cdot(a+3)^{10}\) |
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| Discriminant norm: | $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ | = | \( 3349563240773349963 \) | = | \(3^{17}\cdot11^{10}\) |
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| j-invariant: | $j$ | = | \( -\frac{56125081}{177147} a - \frac{30950923}{177147} \) | ||
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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: | $\mathrm{Reg}(E/K)$ | ≈ | \( 2.6504363042870374111046289987478889329 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 5.3008726085740748222092579974957778658 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 0.788716808846139659495715532197507758480 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 2 \) = \(2\cdot1\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(1\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 2.9563337809593589324481202678935673851 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}2.956333781 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 0.788717 \cdot 5.300873 \cdot 2 } { {1^2 \cdot 2.828427} } \\ & \approx 2.956333781 \end{aligned}$$
Local data at primes of bad reduction
This elliptic curve is not semistable. There are 2 primes $\frak{p}$ of bad reduction.
| $\mathfrak{p}$ | $N(\mathfrak{p})$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((a-1)\) | \(3\) | \(2\) | \(I_{11}^{*}\) | Additive | \(-1\) | \(2\) | \(17\) | \(11\) |
| \((a+3)\) | \(11\) | \(1\) | \(II^{*}\) | Additive | \(-1\) | \(2\) | \(10\) | \(0\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(11\) | 11B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
11.
Its isogeny class
1089.7-a
consists of curves linked by isogenies of
degree 11.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.