Base field \(\Q(\sqrt{-127}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x + 32 \); class number \(5\).
Weierstrass equation
This is not a global minimal model: it is minimal at all primes except \((11,a+7)\). No global minimal model exists.
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(-\frac{52}{121} a - \frac{5512}{121} : \frac{24395}{1331} a - \frac{84237}{1331} : 1\right)$ | $2.0186930195978509126141962731822382382$ | $\infty$ |
| $\left(\frac{1}{4} a - \frac{217}{4} : -\frac{5}{8} a + \frac{213}{8} : 1\right)$ | $0$ | $2$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((26,2a+8)\) | = | \((2,a)\cdot(2,a+1)\cdot(13,a+4)\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 52 \) | = | \(2\cdot2\cdot13\) |
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| Discriminant: | $\Delta$ | = | $-498832707184a+45101616640$ | ||
| Discriminant ideal: | $(\Delta)$ | = | \((-498832707184a+45101616640)\) | = | \((2,a)^{15}\cdot(2,a+1)^{4}\cdot(11,a+7)^{12}\cdot(13,a+4)^{6}\) |
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| Discriminant norm: | $N(\Delta)$ | = | \( 7942226226505220509663232 \) | = | \(2^{15}\cdot2^{4}\cdot11^{12}\cdot13^{6}\) |
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| Minimal discriminant: | $\frak{D}_{\mathrm{min}}$ | = | \((158164877312,16a+100845764096)\) | = | \((2,a)^{15}\cdot(2,a+1)^{4}\cdot(13,a+4)^{6}\) |
| Minimal discriminant norm: | $N(\frak{D}_{\mathrm{min}})$ | = | \( 2530638036992 \) | = | \(2^{15}\cdot2^{4}\cdot13^{6}\) |
| j-invariant: | $j$ | = | \( \frac{305086149374671}{158164877312} a - \frac{2152282255725807}{158164877312} \) | ||
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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: | $\mathrm{Reg}(E/K)$ | ≈ | \( 2.0186930195978509126141962731822382382 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 4.0373860391957018252283925463644764764 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 2.2326473727664599130178938224134427318 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 24 \) = \(1\cdot2^{2}\cdot1\cdot( 2 \cdot 3 )\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(2\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 4.7992105353757490340615241848250022809 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}4.799210535 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 2.232647 \cdot 4.037386 \cdot 24 } { {2^2 \cdot 11.269428} } \\ & \approx 4.799210535 \end{aligned}$$
Local data at primes of bad reduction
This elliptic curve is semistable. There are 3 primes $\frak{p}$ of bad reduction. Primes of good reduction for the curve but which divide the discriminant of the model above (if any) are included.
| $\mathfrak{p}$ | $N(\mathfrak{p})$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((2,a)\) | \(2\) | \(1\) | \(I_{15}\) | Non-split multiplicative | \(1\) | \(1\) | \(15\) | \(15\) |
| \((2,a+1)\) | \(2\) | \(4\) | \(I_{4}\) | Split multiplicative | \(-1\) | \(1\) | \(4\) | \(4\) |
| \((11,a+7)\) | \(11\) | \(1\) | \(I_0\) | Good | \(1\) | \(0\) | \(0\) | \(0\) |
| \((13,a+4)\) | \(13\) | \(6\) | \(I_{6}\) | 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 |
|---|---|
| \(2\) | 2B |
| \(3\) | 3B.1.2 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3 and 6.
Its isogeny class
52.3-a
consists of curves linked by isogenies of
degrees dividing 6.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.