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
trivial
Invariants
| Conductor: | $\frak{N}$ | = | \((-56a+104)\) | = | \((a-1)^{2}\cdot(2)^{3}\cdot(-a-1)^{2}\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 14400 \) | = | \(3^{2}\cdot4^{3}\cdot5^{2}\) |
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| Discriminant: | $\Delta$ | = | $7190064128a-5778148352$ | ||
| Discriminant ideal: | $\frak{D}_{\mathrm{min}} = (\Delta)$ | = | \((7190064128a-5778148352)\) | = | \((a-1)^{15}\cdot(2)^{10}\cdot(-a-1)^{10}\) |
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| Discriminant norm: | $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ | = | \( 146932807680000000000 \) | = | \(3^{15}\cdot4^{10}\cdot5^{10}\) |
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| j-invariant: | $j$ | = | \( \frac{10718164}{19683} a - \frac{2692192}{6561} \) | ||
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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: | $r_{\mathrm{an}}$ | = | \( 0 \) |
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| Mordell-Weil rank: | $r$ | = | \(0\) |
| Regulator: | $\mathrm{Reg}(E/K)$ | = | \( 1 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | = | \( 1 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 0.570374193157192688894499124508085364720 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 4 \) = \(2\cdot2\cdot1\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(1\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 6.1910744360861603159390332565008771153 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 9 \) (rounded) |
BSD formula
$$\begin{aligned}6.191074436 \approx L(E/K,1) & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 9 \cdot 0.570374 \cdot 1 \cdot 4 } { {1^2 \cdot 3.316625} } \\ & \approx 6.191074436 \end{aligned}$$
Local data at primes of bad reduction
This elliptic curve is not semistable. There are 3 primes $\frak{p}$ of bad reduction.
| $\mathfrak{p}$ | $N(\mathfrak{p})$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((a-1)\) | \(3\) | \(2\) | \(I_{9}^{*}\) | Additive | \(-1\) | \(2\) | \(15\) | \(9\) |
| \((2)\) | \(4\) | \(2\) | \(III^{*}\) | Additive | \(1\) | \(3\) | \(10\) | \(0\) |
| \((-a-1)\) | \(5\) | \(1\) | \(II^{*}\) | Additive | \(1\) | \(2\) | \(10\) | \(0\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) .
Isogenies and isogeny class
This curve has no rational isogenies. Its isogeny class 14400.7-l consists of this curve only.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.