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
\(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(2 a^{3} + 2 a^{2} - a - 5 : -20 a^{3} - 21 a^{2} + 37 a + 31 : 1\right)$ | $0.69947042092302347062953273790003177725$ | $\infty$ |
| $\left(a^{3} - 4 a^{2} - a + 7 : -a^{3} + 3 a^{2} + 5 a - 8 : 1\right)$ | $0$ | $2$ |
| $\left(-\frac{1}{2} a^{3} - \frac{1}{4} a^{2} + \frac{7}{2} a - \frac{9}{4} : \frac{17}{8} a^{3} - \frac{41}{8} a^{2} - \frac{19}{8} a + \frac{65}{8} : 1\right)$ | $0$ | $2$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((a^3-3a^2-a+3)\) | = | \((a+1)^{3}\cdot(-a)\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 24 \) | = | \(2^{3}\cdot3\) |
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| Discriminant: | $\Delta$ | = | $-2a^2+4a+12$ | ||
| Discriminant ideal: | $\frak{D}_{\mathrm{min}} = (\Delta)$ | = | \((-2a^2+4a+12)\) | = | \((a+1)^{4}\cdot(-a)^{4}\) |
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| Discriminant norm: | $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ | = | \( 1296 \) | = | \(2^{4}\cdot3^{4}\) |
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| j-invariant: | $j$ | = | \( \frac{65431535104}{81} a^{3} + \frac{42803086972}{81} a^{2} - \frac{121350390136}{81} a - \frac{54249295880}{81} \) | ||
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| Endomorphism ring: | $\mathrm{End}(E)$ | = | \(\Z\) | ||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) | ||
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| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | ||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | \( 1 \) |
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| Mordell-Weil rank: | $r$ | = | \(1\) |
| Regulator: | $\mathrm{Reg}(E/K)$ | ≈ | \( 0.69947042092302347062953273790003177725 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 2.79788168369209388251813095160012710900 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 61.516313718835972175445699016456957464 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 8 \) = \(2\cdot2^{2}\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(4\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 3.23196415147851 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 4 \) (rounded) |
BSD formula
$$\begin{aligned}3.231964151 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 4 \cdot 61.516314 \cdot 2.797882 \cdot 8 } { {4^2 \cdot 106.508216} } \\ & \approx 3.231964151 \end{aligned}$$
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 | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((a+1)\) | \(2\) | \(2\) | \(III\) | Additive | \(1\) | \(3\) | \(4\) | \(0\) |
| \((-a)\) | \(3\) | \(4\) | \(I_{4}\) | Split multiplicative | \(-1\) | \(1\) | \(4\) | \(4\) |
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 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2 and 4.
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.