Base field 4.4.11344.1
Generator \(a\), with minimal polynomial \( x^{4} - 2 x^{3} - 4 x^{2} + 4 x + 3 \); 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 |
|---|---|---|
| $\left(5 a^{3} - 6 a^{2} - 5 a + 7 : -19 a^{3} - 21 a^{2} + 40 a + 25 : 1\right)$ | $1.3989408418460469412590654758000635545$ | $\infty$ |
| $\left(\frac{3}{2} a^{3} - \frac{33}{4} a^{2} + \frac{3}{2} a + \frac{39}{4} : \frac{9}{8} a^{3} - \frac{1}{8} a^{2} + \frac{37}{8} a - \frac{55}{8} : 1\right)$ | $0$ | $2$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((a^3-3a^2-a+3)\) | = | \((a+1)^{3}\cdot(-a)\) |
|
| |||||
| Conductor norm: | $N(\frak{N})$ | = | \( 24 \) | = | \(2^{3}\cdot3\) |
|
| |||||
| Discriminant: | $\Delta$ | = | $4a^2-4a-12$ | ||
| Discriminant ideal: | $\frak{D}_{\mathrm{min}} = (\Delta)$ | = | \((4a^2-4a-12)\) | = | \((a+1)^{8}\cdot(-a)^{2}\) |
|
| |||||
| Discriminant norm: | $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ | = | \( 2304 \) | = | \(2^{8}\cdot3^{2}\) |
|
| |||||
| j-invariant: | $j$ | = | \( \frac{116344852775823881080}{9} a^{3} + \frac{90941355856610478778}{9} a^{2} - \frac{212412074607426159394}{9} a - \frac{125477369587562062790}{9} \) | ||
|
| |||||
| 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: | $\mathrm{Reg}(E/K)$ | ≈ | \( 1.3989408418460469412590654758000635545 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 5.5957633673841877650362619032002542180 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 3.8447696074272482609653561885285598415 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 4 \) = \(2\cdot2\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(2\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 3.23196415147851 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 16 \) (rounded) |
BSD formula
$$\begin{aligned}3.231964151 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 16 \cdot 3.844770 \cdot 5.595763 \cdot 4 } { {2^2 \cdot 106.508216} } \\ & \approx 3.231964151 \end{aligned}$$
Local data at primes of bad reduction
This elliptic curve is not semistable. There are 2 primes $\frak{p}$ of bad reduction.
| $\mathfrak{p}$ | $N(\mathfrak{p})$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((a+1)\) | \(2\) | \(2\) | \(I_{1}^{*}\) | Additive | \(1\) | \(3\) | \(8\) | \(0\) |
| \((-a)\) | \(3\) | \(2\) | \(I_{2}\) | Split multiplicative | \(-1\) | \(1\) | \(2\) | \(2\) |
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, 4 and 8.
Its isogeny class
24.1-b
consists of curves linked by isogenies of
degrees dividing 8.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.