Base field \(\Q(\sqrt{-11}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x + 3 \); class number \(1\).
Weierstrass equation
This is a global minimal model.
Mordell-Weil group structure
\(\Z \oplus \Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(0 : -1 : 1\right)$ | $0.37697290799757927808201735066819872402$ | $\infty$ |
| $\left(a - 1 : -1 : 1\right)$ | $0.44387823796945581388786320183330004990$ | $\infty$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((64)\) | = | \((2)^{6}\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 4096 \) | = | \(4^{6}\) |
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| Discriminant: | $\Delta$ | = | $64$ | ||
| Discriminant ideal: | $\frak{D}_{\mathrm{min}} = (\Delta)$ | = | \((64)\) | = | \((2)^{6}\) |
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| Discriminant norm: | $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ | = | \( 4096 \) | = | \(4^{6}\) |
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| j-invariant: | $j$ | = | \( 1024 a + 1536 \) | ||
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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: | $r_{\mathrm{an}}$ | = | \( 2 \) |
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| Mordell-Weil rank: | $r$ | = | \(2\) |
| Regulator: | $\mathrm{Reg}(E/K)$ | ≈ | \( 0.14930460543801541037894921371415420152 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 0.5972184217520616415157968548566168061 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 13.40029994282671684927629895356514765 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 1 \) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(1\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 2.4129669434213104333499031420519961981 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}2.412966943 \approx L^{(2)}(E/K,1)/2! & \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 13.400300 \cdot 0.597218 \cdot 1 } { {1^2 \cdot 3.316625} } \\ & \approx 2.412966943 \end{aligned}$$
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 | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((2)\) | \(4\) | \(1\) | \(II\) | Additive | \(-1\) | \(6\) | \(6\) | \(0\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(2\) | 2Cn |
| \(3\) | 3Nn |
Isogenies and isogeny class
This curve has no rational isogenies. Its isogeny class 4096.1-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.