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
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(-\frac{232}{25} a - \frac{1637}{25} : -\frac{10031}{125} a - \frac{3071}{125} : 1\right)$ | $0.84456702395889367518514171910959871792$ | $\infty$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((66a-198)\) | = | \((-a)^{3}\cdot(a-1)\cdot(2)\cdot(-2a+1)^{2}\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 39204 \) | = | \(3^{3}\cdot3\cdot4\cdot11^{2}\) |
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| Discriminant: | $\Delta$ | = | $-6547689456a-22449220992$ | ||
| Discriminant ideal: | $\frak{D}_{\mathrm{min}} = (\Delta)$ | = | \((-6547689456a-22449220992)\) | = | \((-a)^{5}\cdot(a-1)\cdot(2)^{4}\cdot(-2a+1)^{15}\) |
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| Discriminant norm: | $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ | = | \( 779574762369026452224 \) | = | \(3^{5}\cdot3\cdot4^{4}\cdot11^{15}\) |
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| j-invariant: | $j$ | = | \( -\frac{44349865310515}{7730448} a - \frac{78923474734103}{2576816} \) | ||
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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: | $\mathrm{Reg}(E/K)$ | ≈ | \( 0.84456702395889367518514171910959871792 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 1.68913404791778735037028343821919743584 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 0.219249904754221952520280222973818535180 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 48 \) = \(3\cdot1\cdot2^{2}\cdot2^{2}\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(1\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 5.3597980240639053072288460553086528407 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}5.359798024 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 0.219250 \cdot 1.689134 \cdot 48 } { {1^2 \cdot 3.316625} } \\ & \approx 5.359798024 \end{aligned}$$
Local data at primes of bad reduction
This elliptic curve is not semistable. There are 4 primes $\frak{p}$ of bad reduction.
| $\mathfrak{p}$ | $N(\mathfrak{p})$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((-a)\) | \(3\) | \(3\) | \(IV\) | Additive | \(1\) | \(3\) | \(5\) | \(0\) |
| \((a-1)\) | \(3\) | \(1\) | \(I_{1}\) | Non-split multiplicative | \(1\) | \(1\) | \(1\) | \(1\) |
| \((2)\) | \(4\) | \(4\) | \(I_{4}\) | Split multiplicative | \(-1\) | \(1\) | \(4\) | \(4\) |
| \((-2a+1)\) | \(11\) | \(4\) | \(I_{9}^{*}\) | Additive | \(-1\) | \(2\) | \(15\) | \(9\) |
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 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3 and 9.
Its isogeny class
39204.2-b
consists of curves linked by isogenies of
degrees dividing 9.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.