Base field \(\Q(\sqrt{-6}) \)
Generator \(a\), with minimal polynomial \( x^{2} + 6 \); class number \(2\).
Weierstrass equation
This is a global minimal model.
Mordell-Weil group structure
\(\Z \oplus \Z/{7}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(-\frac{29}{49} a + \frac{99}{49} : \frac{522}{343} a - \frac{900}{343} : 1\right)$ | $2.7155460888496257210250691199938901949$ | $\infty$ |
| $\left(-1 : -2 : 1\right)$ | $0$ | $7$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((26,13a)\) | = | \((2,a)\cdot(13)\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 338 \) | = | \(2\cdot169\) |
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| Discriminant: | $\Delta$ | = | $1664$ | ||
| Discriminant ideal: | $\frak{D}_{\mathrm{min}} = (\Delta)$ | = | \((1664)\) | = | \((2,a)^{14}\cdot(13)\) |
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| Discriminant norm: | $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ | = | \( 2768896 \) | = | \(2^{14}\cdot169\) |
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| j-invariant: | $j$ | = | \( -\frac{2146689}{1664} \) | ||
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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: | $\mathrm{Reg}(E/K)$ | ≈ | \( 2.7155460888496257210250691199938901949 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 5.4310921776992514420501382399877803898 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 7.8417990398626970909740354626865420542 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 14 \) = \(( 2 \cdot 7 )\cdot1\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(7\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 2.4838720303143851022930889080941294825 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}2.483872030 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 7.841799 \cdot 5.431092 \cdot 14 } { {7^2 \cdot 4.898979} } \\ & \approx 2.483872030 \end{aligned}$$
Local data at primes of bad reduction
This elliptic curve is semistable. There are 2 primes $\frak{p}$ of bad reduction.
| $\mathfrak{p}$ | $N(\mathfrak{p})$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((2,a)\) | \(2\) | \(14\) | \(I_{14}\) | Split multiplicative | \(-1\) | \(1\) | \(14\) | \(14\) |
| \((13)\) | \(169\) | \(1\) | \(I_{1}\) | 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 |
|---|---|
| \(7\) | 7B.1.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
7.
Its isogeny class
338.1-c
consists of curves linked by isogenies of
degree 7.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
| Base field | Curve |
|---|---|
| \(\Q\) | 26.b2 |
| \(\Q\) | 7488.w2 |