Base field \(\Q(\sqrt{-3}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x + 1 \); 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 |
|---|---|---|
| $\left(-1 : -2 a + 1 : 1\right)$ | $0.23339247911250027931188927036053817720$ | $\infty$ |
| $\left(2 a : a - 1 : 1\right)$ | $0.23339247911250027931188927036053817720$ | $\infty$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((-225a+17)\) | = | \((3a-2)^{2}\cdot(6a-5)^{2}\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 47089 \) | = | \(7^{2}\cdot31^{2}\) |
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| Discriminant: | $\Delta$ | = | $-1749a+760$ | ||
| Discriminant ideal: | $\frak{D}_{\mathrm{min}} = (\Delta)$ | = | \((-1749a+760)\) | = | \((3a-2)^{4}\cdot(6a-5)^{2}\) |
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| Discriminant norm: | $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ | = | \( 2307361 \) | = | \(7^{4}\cdot31^{2}\) |
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| j-invariant: | $j$ | = | \( 0 \) | ||
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| Endomorphism ring: | $\mathrm{End}(E)$ | = | \(\Z[(1+\sqrt{-3})/2]\) (complex multiplication) | ||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z[(1+\sqrt{-3})/2]\) | ||
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| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{U}(1)$ | ||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | \( 2 \) |
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| Mordell-Weil rank: | $r$ | = | \(2\) |
| Regulator: | $\mathrm{Reg}(E/K)$ | ≈ | \( 0.040854036979709159374683611108360669149 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 0.1634161479188366374987344444334426766 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 8.284730488679183374758807451339221788 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 3 \) = \(3\cdot1\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(1\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 2.3449521291571959599831411399745688706 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}2.344952129 \approx L^{(2)}(E/K,1)/2! & \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 8.284730 \cdot 0.163416 \cdot 3 } { {1^2 \cdot 1.732051} } \\ & \approx 2.344952129 \end{aligned}$$
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 | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((3a-2)\) | \(7\) | \(3\) | \(IV\) | Additive | \(1\) | \(2\) | \(4\) | \(0\) |
| \((6a-5)\) | \(31\) | \(1\) | \(II\) | Additive | \(-1\) | \(2\) | \(2\) | \(0\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(31\) | 31Cs.7.1 |
For all other primes \(p\), the image is a Borel subgroup if \(p=3\), a split Cartan subgroup if \(\left(\frac{ -3 }{p}\right)=+1\) or a nonsplit Cartan subgroup if \(\left(\frac{ -3 }{p}\right)=-1\).
Isogenies and isogeny class
This curve has no rational isogenies other than endomorphisms. Its isogeny class 47089.9-CMa consists of this curve only.
Base change
This elliptic curve is a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.