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
\(\Z \oplus \Z/{6}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(-6 : -5 : 1\right)$ | $1.1704641535947222311163008003251896398$ | $\infty$ |
| $\left(-1 : -10 : 1\right)$ | $0$ | $6$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((130)\) | = | \((2)\cdot(-a-1)\cdot(a-2)\cdot(13)\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 16900 \) | = | \(4\cdot5\cdot5\cdot169\) |
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| Discriminant: | $\Delta$ | = | $26000$ | ||
| Discriminant ideal: | $\frak{D}_{\mathrm{min}} = (\Delta)$ | = | \((26000)\) | = | \((2)^{4}\cdot(-a-1)^{3}\cdot(a-2)^{3}\cdot(13)\) |
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| Discriminant norm: | $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ | = | \( 676000000 \) | = | \(4^{4}\cdot5^{3}\cdot5^{3}\cdot169\) |
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| j-invariant: | $j$ | = | \( \frac{3803721481}{26000} \) | ||
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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: | $\mathrm{Reg}(E/K)$ | ≈ | \( 1.1704641535947222311163008003251896398 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 2.3409283071894444622326016006503792796 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 3.7842989973186296467954633365637712828 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 36 \) = \(2^{2}\cdot3\cdot3\cdot1\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(6\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 2.6710204517112505396144721213021072057 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}2.671020452 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 3.784299 \cdot 2.340928 \cdot 36 } { {6^2 \cdot 3.316625} } \\ & \approx 2.671020452 \end{aligned}$$
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 | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((2)\) | \(4\) | \(4\) | \(I_{4}\) | Split multiplicative | \(-1\) | \(1\) | \(4\) | \(4\) |
| \((-a-1)\) | \(5\) | \(3\) | \(I_{3}\) | Split multiplicative | \(-1\) | \(1\) | \(3\) | \(3\) |
| \((a-2)\) | \(5\) | \(3\) | \(I_{3}\) | Split multiplicative | \(-1\) | \(1\) | \(3\) | \(3\) |
| \((13)\) | \(169\) | \(1\) | \(I_{1}\) | 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 |
|---|---|
| \(2\) | 2B |
| \(3\) | 3B.1.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3 and 6.
Its isogeny class
16900.2-b
consists of curves linked by isogenies of
degrees dividing 6.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
| Base field | Curve |
|---|---|
| \(\Q\) | 130.a2 |
| \(\Q\) | 15730.s2 |