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
\(\Z \oplus \Z \oplus \Z/{3}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(-1 : 3 : 1\right)$ | $1.0592450864091505454931536917617730576$ | $\infty$ |
| $\left(i - 2 : 4 : 1\right)$ | $1.3405658872857124120850655418360259110$ | $\infty$ |
| $\left(1 : -1 : 1\right)$ | $0$ | $3$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((91)\) | = | \((-3i-2)\cdot(2i+3)\cdot(7)\) |
|
| |||||
| Conductor norm: | $N(\frak{N})$ | = | \( 8281 \) | = | \(13\cdot13\cdot49\) |
|
| |||||
| Discriminant: | $\Delta$ | = | $-91$ | ||
| Discriminant ideal: | $\frak{D}_{\mathrm{min}} = (\Delta)$ | = | \((-91)\) | = | \((-3i-2)\cdot(2i+3)\cdot(7)\) |
|
| |||||
| Discriminant norm: | $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ | = | \( 8281 \) | = | \(13\cdot13\cdot49\) |
|
| |||||
| j-invariant: | $j$ | = | \( -\frac{43614208}{91} \) | ||
|
| |||||
| 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: | $r_{\mathrm{an}}$ | = | \( 2 \) |
|
|
|||
| Mordell-Weil rank: | $r$ | = | \(2\) |
| Regulator: | $\mathrm{Reg}(E/K)$ | ≈ | \( 1.1394877908446318131470032201204083351 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 4.5579511633785272525880128804816333404 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 8.7597211711408433558570225751647355782 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 1 \) = \(1\cdot1\cdot1\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(3\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 2.2181322946040512332765740514834917924 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}2.218132295 \approx L^{(2)}(E/K,1)/2! & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 8.759721 \cdot 4.557951 \cdot 1 } { {3^2 \cdot 2.000000} } \\ & \approx 2.218132295 \end{aligned}$$
Local data at primes of bad reduction
This elliptic curve is semistable. There are 3 primes $\frak{p}$ of bad reduction.
| $\mathfrak{p}$ | $N(\mathfrak{p})$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((-3i-2)\) | \(13\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
| \((2i+3)\) | \(13\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
| \((7)\) | \(49\) | \(1\) | \(I_{1}\) | 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 |
|---|---|
| \(3\) | 3B.1.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3 and 9.
Its isogeny class
8281.2-b
consists of curves linked by isogenies of
degrees dividing 9.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
| Base field | Curve |
|---|---|
| \(\Q\) | 91.b2 |
| \(\Q\) | 1456.k2 |