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
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(-\frac{5}{81} a^{3} + \frac{70}{81} a^{2} + \frac{233}{81} a + \frac{505}{81} : \frac{5786}{729} a^{3} + \frac{158}{729} a^{2} - \frac{17507}{729} a - \frac{17062}{729} : 1\right)$ | $1.5579385919361862006341748848721371057$ | $\infty$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((-a^3+2a^2+2a-3)\) | = | \((a+1)\cdot(-a)^{2}\) |
|
| |||||
| Conductor norm: | $N(\frak{N})$ | = | \( 18 \) | = | \(2\cdot3^{2}\) |
|
| |||||
| Discriminant: | $\Delta$ | = | $-1024a^3+2048a^2+5120a-12288$ | ||
| Discriminant ideal: | $\frak{D}_{\mathrm{min}} = (\Delta)$ | = | \((-1024a^3+2048a^2+5120a-12288)\) | = | \((a+1)^{42}\cdot(-a)^{6}\) |
|
| |||||
| Discriminant norm: | $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ | = | \( 3206175906594816 \) | = | \(2^{42}\cdot3^{6}\) |
|
| |||||
| j-invariant: | $j$ | = | \( \frac{77526197}{1024} a^{3} - \frac{110292741}{2048} a^{2} - \frac{381531341}{1024} a - \frac{361724209}{2048} \) | ||
|
| |||||
| 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: | $\mathrm{Reg}(E/K)$ | ≈ | \( 1.5579385919361862006341748848721371057 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 6.2317543677447448025366995394885484228 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 20.243816487117913085903049436117003559 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 2 \) = \(2\cdot1\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(1\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 2.36891569441451 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}2.368915694 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 20.243816 \cdot 6.231754 \cdot 2 } { {1^2 \cdot 106.508216} } \\ & \approx 2.368915694 \end{aligned}$$
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 | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((a+1)\) | \(2\) | \(2\) | \(I_{42}\) | Non-split multiplicative | \(1\) | \(1\) | \(42\) | \(42\) |
| \((-a)\) | \(3\) | \(1\) | \(I_0^{*}\) | Additive | \(-1\) | \(2\) | \(6\) | \(0\) |
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.1.2 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3.
Its isogeny class
18.1-c
consists of curves linked by isogenies of
degree 3.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.