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 |
|---|---|---|
| $\left(\frac{1429}{4} : \frac{45}{8} : 1\right)$ | $0.55568073570994009832121611547628461980$ | $\infty$ |
| $\left(\frac{10002702256001}{33103613773476} a + \frac{12764969199644209}{33103613773476} : \frac{967206363311804486507}{95232045756556701612} a + \frac{180273057632720568894353}{190464091513113403224} : 1\right)$ | $17.767891171470850966599466794908338652$ | $\infty$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((-98a+49)\) | = | \((-2a+1)\cdot(7)^{2}\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 26411 \) | = | \(11\cdot49^{2}\) |
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| Discriminant: | $\Delta$ | = | $-1294139$ | ||
| Discriminant ideal: | $\frak{D}_{\mathrm{min}} = (\Delta)$ | = | \((-1294139)\) | = | \((-2a+1)^{2}\cdot(7)^{6}\) |
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| Discriminant norm: | $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ | = | \( 1674795751321 \) | = | \(11^{2}\cdot49^{6}\) |
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| j-invariant: | $j$ | = | \( -\frac{52893159101157376}{11} \) | ||
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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: | $\mathrm{Reg}(E/K)$ | ≈ | \( 9.7960795681672868272335793265070810795 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 39.184318272669147308934317306028324318 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 0.1058024927689119610118860978185856507800 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 8 \) = \(2\cdot2^{2}\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(1\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 10.000042362957965076019311812791120485 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}10.000042363 \approx L^{(2)}(E/K,1)/2! & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 0.105802 \cdot 39.184318 \cdot 8 } { {1^2 \cdot 3.316625} } \\ & \approx 10.000042363 \end{aligned}$$
Local data at primes of bad reduction
This elliptic curve is not semistable. There are 2 primes $\frak{p}$ of bad reduction.
| $\mathfrak{p}$ | $N(\mathfrak{p})$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((-2a+1)\) | \(11\) | \(2\) | \(I_{2}\) | Split multiplicative | \(-1\) | \(1\) | \(2\) | \(2\) |
| \((7)\) | \(49\) | \(4\) | \(I_0^{*}\) | Additive | \(1\) | \(2\) | \(6\) | \(0\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(2\) | 2Cn |
| \(5\) | 5B.4.2 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
5 and 25.
Its isogeny class
26411.1-c
consists of curves linked by isogenies of
degrees dividing 25.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
| Base field | Curve |
|---|---|
| \(\Q\) | 539.a1 |
| \(\Q\) | 5929.h1 |