Base field \(\Q(\sqrt{-6}) \)
Generator \(a\), with minimal polynomial \( x^{2} + 6 \); class number \(2\).
Weierstrass equation
This is not a global minimal model: it is minimal at all primes except \((2,a)\). No global minimal model exists.
Mordell-Weil group structure
\(\Z \oplus \Z/{3}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(\frac{224}{25} a - \frac{1037}{25} : -\frac{10046}{125} a - \frac{21952}{125} : 1\right)$ | $2.5043282847408540896739160338518245800$ | $\infty$ |
| $\left(\frac{1}{3} : -\frac{518}{9} a : 1\right)$ | $0$ | $3$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((38,19a)\) | = | \((2,a)\cdot(19)\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 722 \) | = | \(2\cdot361\) |
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| Discriminant: | $\Delta$ | = | $163208757248$ | ||
| Discriminant ideal: | $(\Delta)$ | = | \((163208757248)\) | = | \((2,a)^{66}\cdot(19)\) |
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| Discriminant norm: | $N(\Delta)$ | = | \( 26637098442436592533504 \) | = | \(2^{66}\cdot361\) |
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| Minimal discriminant: | $\frak{D}_{\mathrm{min}}$ | = | \((2550136832)\) | = | \((2,a)^{54}\cdot(19)\) |
| Minimal discriminant norm: | $N(\frak{D}_{\mathrm{min}})$ | = | \( 6503197861922996224 \) | = | \(2^{54}\cdot361\) |
| j-invariant: | $j$ | = | \( -\frac{69173457625}{2550136832} \) | ||
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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: | $\mathrm{Reg}(E/K)$ | ≈ | \( 2.5043282847408540896739160338518245800 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 5.008656569481708179347832067703649160 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 0.7579089332756520662077146399622840176 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 54 \) = \(( 2 \cdot 3^{3} )\cdot1\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(3\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 4.6492608130786390461960983624724424643 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}4.649260813 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 0.757909 \cdot 5.008657 \cdot 54 } { {3^2 \cdot 4.898979} } \\ & \approx 4.649260813 \end{aligned}$$
Local data at primes of bad reduction
This elliptic curve is semistable. There are 2 primes $\frak{p}$ of bad reduction. Primes of good reduction for the curve but which divide the discriminant of the model above (if any) are included.
| $\mathfrak{p}$ | $N(\mathfrak{p})$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((2,a)\) | \(2\) | \(54\) | \(I_{54}\) | Split multiplicative | \(-1\) | \(1\) | \(54\) | \(54\) |
| \((19)\) | \(361\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(3\) | 3Cs.1.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3 and 9.
Its isogeny class
722.1-d
consists of curves linked by isogenies of
degrees dividing 9.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
| Base field | Curve |
|---|---|
| \(\Q\) | 342.e1 |
| \(\Q\) | 1216.e1 |