Base field \(\Q(\zeta_{13})^+\)
Generator \(a\), with minimal polynomial \( x^{6} - x^{5} - 5 x^{4} + 4 x^{3} + 6 x^{2} - 3 x - 1 \); class number \(1\).
Weierstrass equation
This is a global minimal model.
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(a^{5} + 2 a^{4} - 2 a^{3} - 7 a^{2} - 2 a + 5 : -8 a^{5} - 5 a^{4} + 27 a^{3} + 8 a^{2} - 19 a + 2 : 1\right)$ | $0.019263609030894381206827834445555706402$ | $\infty$ |
| $\left(-a^{2} - a + 1 : a^{5} + a^{4} - 4 a^{3} - 3 a^{2} + 3 a : 1\right)$ | $0$ | $2$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((-a^4+a^3+3a^2-3a-2)\) | = | \((-a^4+a^3+3a^2-3a-2)\) |
|
| |||||
| Conductor norm: | $N(\frak{N})$ | = | \( 79 \) | = | \(79\) |
|
| |||||
| Discriminant: | $\Delta$ | = | $2a^5-a^4-9a^3+3a^2+7a-1$ | ||
| Discriminant ideal: | $\frak{D}_{\mathrm{min}} = (\Delta)$ | = | \((2a^5-a^4-9a^3+3a^2+7a-1)\) | = | \((-a^4+a^3+3a^2-3a-2)\) |
|
| |||||
| Discriminant norm: | $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ | = | \( -79 \) | = | \(-79\) |
|
| |||||
| j-invariant: | $j$ | = | \( \frac{22488882826}{79} a^{5} - \frac{6539496375}{79} a^{4} - 1482055738 a^{3} + \frac{6919343568}{79} a^{2} + \frac{139840964971}{79} a + \frac{31710174030}{79} \) | ||
|
| |||||
| Endomorphism ring: | $\mathrm{End}(E)$ | = | \(\Z\) | ||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) | ||
|
| |||||
| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | ||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | \( 1 \) |
|
|
|||
| Mordell-Weil rank: | $r$ | = | \(1\) |
| Regulator: | $\mathrm{Reg}(E/K)$ | ≈ | \( 0.019263609030894381206827834445555706402 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 0.1155816541853662872409670066733342384120 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 53060.199981557337172027960751972554689 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 1 \) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(2\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 2.51617 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}2.516170000 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 53060.199982 \cdot 0.115582 \cdot 1 } { {2^2 \cdot 609.338166} } \\ & \approx 2.516166733 \end{aligned}$$
Local data at primes of bad reduction
This elliptic curve is semistable. There is only one prime $\frak{p}$ of bad reduction.
| $\mathfrak{p}$ | $N(\mathfrak{p})$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((-a^4+a^3+3a^2-3a-2)\) | \(79\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
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 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2.
Its isogeny class
79.4-a
consists of curves linked by isogenies of
degree 2.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.