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
\(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(-\frac{35635}{1369} i - \frac{232051}{1369} : \frac{142828402}{50653} i - \frac{12041706}{50653} : 1\right)$ | $6.3565329383842135140260193618589013796$ | $\infty$ |
| $\left(-\frac{267}{2} i - 37 : \frac{341}{4} i - \frac{193}{4} : 1\right)$ | $0$ | $2$ |
| $\left(86 i + 14 : -50 i + 36 : 1\right)$ | $0$ | $2$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((100i+240)\) | = | \((i+1)^{4}\cdot(-i-2)\cdot(2i+1)\cdot(-3i-2)^{2}\) |
|
| |||||
| Conductor norm: | $N(\frak{N})$ | = | \( 67600 \) | = | \(2^{4}\cdot5\cdot5\cdot13^{2}\) |
|
| |||||
| Discriminant: | $\Delta$ | = | $30981787017600i-37012665203200$ | ||
| Discriminant ideal: | $\frak{D}_{\mathrm{min}} = (\Delta)$ | = | \((30981787017600i-37012665203200)\) | = | \((i+1)^{14}\cdot(-i-2)^{2}\cdot(2i+1)^{12}\cdot(-3i-2)^{12}\) |
|
| |||||
| Discriminant norm: | $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ | = | \( 2329808512248100000000000000 \) | = | \(2^{14}\cdot5^{2}\cdot5^{12}\cdot13^{12}\) |
|
| |||||
| j-invariant: | $j$ | = | \( -\frac{12415547946147007137}{2356840332031250} i + \frac{5474429230691529908}{1178420166015625} \) | ||
|
| |||||
| 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: | $\mathrm{Reg}(E/K)$ | ≈ | \( 6.3565329383842135140260193618589013796 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 12.713065876768427028052038723717802759 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 0.1337220056625785824434343349793645380480 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 64 \) = \(2^{2}\cdot2\cdot2\cdot2^{2}\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(4\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 3.4000333343239243000530617954080781598 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}3.400033334 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 0.133722 \cdot 12.713066 \cdot 64 } { {4^2 \cdot 2.000000} } \\ & \approx 3.400033334 \end{aligned}$$
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 | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((i+1)\) | \(2\) | \(4\) | \(I_{6}^{*}\) | Additive | \(1\) | \(4\) | \(14\) | \(2\) |
| \((-i-2)\) | \(5\) | \(2\) | \(I_{2}\) | Non-split multiplicative | \(1\) | \(1\) | \(2\) | \(2\) |
| \((2i+1)\) | \(5\) | \(2\) | \(I_{12}\) | Non-split multiplicative | \(1\) | \(1\) | \(12\) | \(12\) |
| \((-3i-2)\) | \(13\) | \(4\) | \(I_{6}^{*}\) | Additive | \(1\) | \(2\) | \(12\) | \(6\) |
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 |
| \(3\) | 3B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3 and 6.
Its isogeny class
67600.4-a
consists of curves linked by isogenies of
degrees dividing 12.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.