Base field \(\Q(\sqrt{-1}) \)
Generator \(i\), with minimal polynomial \( x^{2} + 1 \); class number \(1\).
Weierstrass equation
This is a global minimal model.
Mordell-Weil group structure
\(\Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
$P$ | $\hat{h}(P)$ | Order |
---|---|---|
$\left(-15 i - 9 : 0 : 1\right)$ | $0$ | $2$ |
$\left(31 i : 0 : 1\right)$ | $0$ | $2$ |
Invariants
Conductor: | $\frak{N}$ | = | \((312)\) | = | \((i+1)^{6}\cdot(3)\cdot(-3i-2)\cdot(2i+3)\) |
| |||||
Conductor norm: | $N(\frak{N})$ | = | \( 97344 \) | = | \(2^{6}\cdot9\cdot13\cdot13\) |
| |||||
Discriminant: | $\Delta$ | = | $25022177856$ | ||
Discriminant ideal: | $\frak{D}_{\mathrm{min}} = (\Delta)$ | = | \((25022177856)\) | = | \((i+1)^{12}\cdot(3)^{4}\cdot(-3i-2)^{6}\cdot(2i+3)^{6}\) |
| |||||
Discriminant norm: | $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ | = | \( 626109384657296756736 \) | = | \(2^{12}\cdot9^{4}\cdot13^{6}\cdot13^{6}\) |
| |||||
j-invariant: | $j$ | = | \( -\frac{420526439488}{390971529} \) | ||
| |||||
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: | $r_{\mathrm{an}}$ | = | \( 0 \) |
|
|||
Mordell-Weil rank: | $r$ | = | \(0\) |
Regulator: | $\mathrm{Reg}(E/K)$ | = | \( 1 \) |
Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | = | \( 1 \) |
Global period: | $\Omega(E/K)$ | ≈ | \( 0.501352930208176441715359716908669240180 \) |
Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 64 \) = \(2^{2}\cdot2^{2}\cdot2\cdot2\) |
Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(4\) |
Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 4.0108234416654115337228777352693539215 \) |
Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 4 \) (rounded) |
BSD formula
$$\begin{aligned}4.010823442 \approx L(E/K,1) & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 4 \cdot 0.501353 \cdot 1 \cdot 64 } { {4^2 \cdot 2.000000} } \\ & \approx 4.010823442 \end{aligned}$$
Local data at primes of bad reduction
This elliptic curve is not semistable. There are 4 primes $\frak{p}$ of bad reduction.
$\mathfrak{p}$ | $N(\mathfrak{p})$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
---|---|---|---|---|---|---|---|---|
\((i+1)\) | \(2\) | \(4\) | \(I_{2}^{*}\) | Additive | \(1\) | \(6\) | \(12\) | \(0\) |
\((3)\) | \(9\) | \(4\) | \(I_{4}\) | Split multiplicative | \(-1\) | \(1\) | \(4\) | \(4\) |
\((-3i-2)\) | \(13\) | \(2\) | \(I_{6}\) | Non-split multiplicative | \(1\) | \(1\) | \(6\) | \(6\) |
\((2i+3)\) | \(13\) | \(2\) | \(I_{6}\) | Non-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 |
---|---|
\(2\) | 2Cs |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2.
Its isogeny class
97344.2-e
consists of curves linked by isogenies of
degrees dividing 4.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
Base field | Curve |
---|---|
\(\Q\) | 1248.j2 |
\(\Q\) | 1248.e2 |