Base field \(\Q(\sqrt{-7}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x + 2 \); class number \(1\).
Weierstrass equation
This is a global minimal model.
Mordell-Weil group structure
\(\Z \oplus \Z/{3}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(4 a - 2 : -8 a + 11 : 1\right)$ | $0.096548555477260134650442281716011619029$ | $\infty$ |
| $\left(2 : -5 : 1\right)$ | $0$ | $3$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((-52a+26)\) | = | \((a)\cdot(-a+1)\cdot(-2a+1)\cdot(13)\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 4732 \) | = | \(2\cdot2\cdot7\cdot169\) |
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| Discriminant: | $\Delta$ | = | $-46592$ | ||
| Discriminant ideal: | $\frak{D}_{\mathrm{min}} = (\Delta)$ | = | \((-46592)\) | = | \((a)^{9}\cdot(-a+1)^{9}\cdot(-2a+1)^{2}\cdot(13)\) |
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| Discriminant norm: | $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ | = | \( 2170814464 \) | = | \(2^{9}\cdot2^{9}\cdot7^{2}\cdot169\) |
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| j-invariant: | $j$ | = | \( \frac{37595375}{46592} \) | ||
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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: | $\mathrm{Reg}(E/K)$ | ≈ | \( 0.096548555477260134650442281716011619029 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 0.1930971109545202693008845634320232380580 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 4.6156717322095548817724952519399673902 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 162 \) = \(3^{2}\cdot3^{2}\cdot2\cdot1\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(3\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 6.0636506960356360511178890766224687454 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}6.063650696 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 4.615672 \cdot 0.193097 \cdot 162 } { {3^2 \cdot 2.645751} } \\ & \approx 6.063650696 \end{aligned}$$
Local data at primes of bad reduction
This elliptic curve is semistable. There are 4 primes $\frak{p}$ of bad reduction.
| $\mathfrak{p}$ | $N(\mathfrak{p})$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((a)\) | \(2\) | \(9\) | \(I_{9}\) | Split multiplicative | \(-1\) | \(1\) | \(9\) | \(9\) |
| \((-a+1)\) | \(2\) | \(9\) | \(I_{9}\) | Split multiplicative | \(-1\) | \(1\) | \(9\) | \(9\) |
| \((-2a+1)\) | \(7\) | \(2\) | \(I_{2}\) | Split multiplicative | \(-1\) | \(1\) | \(2\) | \(2\) |
| \((13)\) | \(169\) | \(1\) | \(I_{1}\) | 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 |
|---|---|
| \(3\) | 3B.1.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3 and 9.
Its isogeny class
4732.2-g
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 |
|---|---|
| \(\Q\) | 182.d3 |
| \(\Q\) | 1274.j3 |