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 |
|---|---|---|
| $\left(0 : 0 : 1\right)$ | $0.12241051743729606694204853282334433150$ | $\infty$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((-19a-11)\) | = | \((-19a-11)\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 601 \) | = | \(601\) |
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| Discriminant: | $\Delta$ | = | $-30a-49$ | ||
| Discriminant ideal: | $\frak{D}_{\mathrm{min}} = (\Delta)$ | = | \((-30a-49)\) | = | \((-19a-11)\) |
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| Discriminant norm: | $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ | = | \( 601 \) | = | \(601\) |
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| j-invariant: | $j$ | = | \( -\frac{283022}{601} a + \frac{405615}{601} \) | ||
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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: | $\mathrm{Reg}(E/K)$ | ≈ | \( 0.12241051743729606694204853282334433150 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 0.244821034874592133884097065646688663000 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 13.695669344935855179777941373730780724 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 1 \) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(1\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 1.1854602556282248063654918686358555340 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}1.185460256 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 13.695669 \cdot 0.244821 \cdot 1 } { {1^2 \cdot 2.828427} } \\ & \approx 1.185460256 \end{aligned}$$
Local data at primes of bad reduction
This elliptic curve is semistable. There is only one prime $\frak{p}$ of bad reduction.
| $\mathfrak{p}$ | $N(\mathfrak{p})$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((-19a-11)\) | \(601\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) .
Isogenies and isogeny class
This curve has no rational isogenies. Its isogeny class 601.1-a consists of this curve only.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.