Base field 4.4.10273.1
Generator \(a\), with minimal polynomial \( x^{4} - 2 x^{3} - 5 x^{2} + x + 2 \); 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 |
|---|---|---|
| $\left(-7 a^{3} + 25 a^{2} + 2 a - 11 : 44 a^{3} - 147 a^{2} - 13 a + 63 : 1\right)$ | $0.18052722975354376496195611129336396631$ | $\infty$ |
| $\left(a - 1 : -a^{2} + a + 1 : 1\right)$ | $0$ | $2$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((a^3-3a^2-2a+4)\) | = | \((a)^{2}\cdot(-a^2+1)\) |
|
| |||||
| Conductor norm: | $N(\frak{N})$ | = | \( 12 \) | = | \(2^{2}\cdot3\) |
|
| |||||
| Discriminant: | $\Delta$ | = | $-15a^3+37a^2+61a-38$ | ||
| Discriminant ideal: | $\frak{D}_{\mathrm{min}} = (\Delta)$ | = | \((-15a^3+37a^2+61a-38)\) | = | \((a)^{8}\cdot(-a^2+1)^{6}\) |
|
| |||||
| Discriminant norm: | $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ | = | \( -186624 \) | = | \(-2^{8}\cdot3^{6}\) |
|
| |||||
| j-invariant: | $j$ | = | \( -\frac{82012674512}{729} a^{3} + \frac{216490672625}{729} a^{2} + \frac{90475703044}{243} a - \frac{255370359212}{729} \) | ||
|
| |||||
| 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: | $\mathrm{Reg}(E/K)$ | ≈ | \( 0.18052722975354376496195611129336396631 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 0.722108919014175059847824445173455865240 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 597.19274002794471388218883450450408598 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 2 \) = \(1\cdot2\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(2\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 2.12734824268151 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}2.127348243 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 597.192740 \cdot 0.722109 \cdot 2 } { {2^2 \cdot 101.355809} } \\ & \approx 2.127348243 \end{aligned}$$
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 | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((a)\) | \(2\) | \(1\) | \(IV^{*}\) | Additive | \(-1\) | \(2\) | \(8\) | \(0\) |
| \((-a^2+1)\) | \(3\) | \(2\) | \(I_{6}\) | 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 |
|---|---|
| \(2\) | 2B |
| \(3\) | 3B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3 and 6.
Its isogeny class
12.1-a
consists of curves linked by isogenies of
degrees dividing 6.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.