Base field \(\Q(\sqrt{-7}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x + 2 \); class number \(1\).
Weierstrass equation
This is a global minimal model.
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(1 : -8 : 1\right)$ | $0.43158104789063939599289247250504465551$ | $\infty$ |
| $\left(a - 4 : -14 : 1\right)$ | $0.86316209578127879198578494501008931103$ | $\infty$ |
| $\left(-7 : 0 : 1\right)$ | $0$ | $2$ |
| $\left(-a + 4 : 0 : 1\right)$ | $0$ | $2$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((112)\) | = | \((a)^{4}\cdot(-a+1)^{4}\cdot(-2a+1)^{2}\) |
|
| |||||
| Conductor norm: | $N(\frak{N})$ | = | \( 12544 \) | = | \(2^{4}\cdot2^{4}\cdot7^{2}\) |
|
| |||||
| Discriminant: | $\Delta$ | = | $-1404928$ | ||
| Discriminant ideal: | $\frak{D}_{\mathrm{min}} = (\Delta)$ | = | \((-1404928)\) | = | \((a)^{12}\cdot(-a+1)^{12}\cdot(-2a+1)^{6}\) |
|
| |||||
| Discriminant norm: | $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ | = | \( 1973822685184 \) | = | \(2^{12}\cdot2^{12}\cdot7^{6}\) |
|
| |||||
| j-invariant: | $j$ | = | \( -3375 \) | ||
|
| |||||
| Endomorphism ring: | $\mathrm{End}(E)$ | = | \(\Z[(1+\sqrt{-7})/2]\) (complex multiplication) | ||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z[(1+\sqrt{-7})/2]\) | ||
|
| |||||
| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{U}(1)$ | ||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | \( 2 \) |
|
|
|||
| Mordell-Weil rank: | $r$ | = | \(2\) |
| Regulator: | $\mathrm{Reg}(E/K)$ | ≈ | \( 0.32595885157216915771537791641303317225 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 1.30383540628867663086151166565213268900 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 2.4722523001412733682490984840921888222 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 64 \) = \(2^{2}\cdot2^{2}\cdot2^{2}\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(4\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 4.8733379720496633476074996934952393066 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}4.873337972 \approx L^{(2)}(E/K,1)/2! & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 2.472252 \cdot 1.303835 \cdot 64 } { {4^2 \cdot 2.645751} } \\ & \approx 4.873337972 \end{aligned}$$
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 | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((a)\) | \(2\) | \(4\) | \(I_{4}^{*}\) | Additive | \(-1\) | \(4\) | \(12\) | \(0\) |
| \((-a+1)\) | \(2\) | \(4\) | \(I_{4}^{*}\) | Additive | \(-1\) | \(4\) | \(12\) | \(0\) |
| \((-2a+1)\) | \(7\) | \(4\) | \(I_0^{*}\) | Additive | \(-1\) | \(2\) | \(6\) | \(0\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) .
The image is a Borel subgroup if \(p=7\), a split Cartan subgroup if \(\left(\frac{ -7 }{p}\right)=+1\) or a nonsplit Cartan subgroup if \(\left(\frac{ -7 }{p}\right)=-1\).
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies (excluding endomorphisms) of degree \(d\) for \(d=\)
2.
Its isogeny class
12544.5-CMa
consists of curves linked by isogenies of
degree 2.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
| Base field | Curve |
|---|---|
| \(\Q\) | 784.f2 |
| \(\Q\) | 784.f4 |