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
\(\Z \oplus \Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(10 : -41 : 1\right)$ | $0.10197829462292110456679008894485744001$ | $\infty$ |
| $\left(-8 : 18 a - 5 : 1\right)$ | $0.46540767515234654924340731134483364767$ | $\infty$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((162)\) | = | \((-a)^{4}\cdot(a-1)^{4}\cdot(2)\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 26244 \) | = | \(3^{4}\cdot3^{4}\cdot4\) |
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| Discriminant: | $\Delta$ | = | $-3779136$ | ||
| Discriminant ideal: | $\frak{D}_{\mathrm{min}} = (\Delta)$ | = | \((-3779136)\) | = | \((-a)^{10}\cdot(a-1)^{10}\cdot(2)^{6}\) |
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| Discriminant norm: | $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ | = | \( 14281868906496 \) | = | \(3^{10}\cdot3^{10}\cdot4^{6}\) |
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| j-invariant: | $j$ | = | \( \frac{109503}{64} \) | ||
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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: | $r_{\mathrm{an}}$ | = | \( 2 \) |
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| Mordell-Weil rank: | $r$ | = | \(2\) |
| Regulator: | $\mathrm{Reg}(E/K)$ | ≈ | \( 0.047461481016454754268391206598584446764 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 0.1898459240658190170735648263943377870560 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 2.2037222528621653565908812240956224956 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 54 \) = \(3\cdot3\cdot( 2 \cdot 3 )\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(1\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 6.8117006149033037391336263069045156559 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}6.811700615 \approx L^{(2)}(E/K,1)/2! & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 2.203722 \cdot 0.189846 \cdot 54 } { {1^2 \cdot 3.316625} } \\ & \approx 6.811700615 \end{aligned}$$
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 | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((-a)\) | \(3\) | \(3\) | \(IV^{*}\) | Additive | \(1\) | \(4\) | \(10\) | \(0\) |
| \((a-1)\) | \(3\) | \(3\) | \(IV^{*}\) | Additive | \(1\) | \(4\) | \(10\) | \(0\) |
| \((2)\) | \(4\) | \(6\) | \(I_{6}\) | Split multiplicative | \(-1\) | \(1\) | \(6\) | \(6\) |
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.1.2 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3.
Its isogeny class
26244.5-a
consists of curves linked by isogenies of
degree 3.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
| Base field | Curve |
|---|---|
| \(\Q\) | 162.a2 |
| \(\Q\) | 19602.s2 |