Base field 4.4.17609.1
Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 7 x^{2} + 10 x - 1 \); class number \(1\).
Weierstrass equation
This is a global minimal model.
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(-a^{3} - a^{2} + 5 a : a^{3} + 2 a^{2} - 2 a - 1 : 1\right)$ | $0.018230905140861081562677478933907477184$ | $\infty$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((a^3+a^2-6a)\) | = | \((a^3+a^2-5a+1)\cdot(-a^3+6a-2)\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 14 \) | = | \(2\cdot7\) |
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| Discriminant: | $\Delta$ | = | $-3a^2-4a+15$ | ||
| Discriminant ideal: | $\frak{D}_{\mathrm{min}} = (\Delta)$ | = | \((-3a^2-4a+15)\) | = | \((a^3+a^2-5a+1)^{4}\cdot(-a^3+6a-2)^{2}\) |
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| Discriminant norm: | $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ | = | \( 784 \) | = | \(2^{4}\cdot7^{2}\) |
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| j-invariant: | $j$ | = | \( -\frac{3112729}{784} a^{3} + \frac{825441}{392} a^{2} + \frac{26502365}{784} a - \frac{32146199}{784} \) | ||
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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: | $\mathrm{Reg}(E/K)$ | ≈ | \( 0.018230905140861081562677478933907477184 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 0.07292362056344432625070991573562990873600 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 1191.1471896677131459338857380305414565 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 8 \) = \(2^{2}\cdot2\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(1\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 5.23668310212968 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}5.236683102 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 1191.147190 \cdot 0.072924 \cdot 8 } { {1^2 \cdot 132.698907} } \\ & \approx 5.236683102 \end{aligned}$$
Local data at primes of bad reduction
This elliptic curve is semistable. There are 2 primes $\frak{p}$ of bad reduction.
| $\mathfrak{p}$ | $N(\mathfrak{p})$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((a^3+a^2-5a+1)\) | \(2\) | \(4\) | \(I_{4}\) | Split multiplicative | \(-1\) | \(1\) | \(4\) | \(4\) |
| \((-a^3+6a-2)\) | \(7\) | \(2\) | \(I_{2}\) | Non-split multiplicative | \(1\) | \(1\) | \(2\) | \(2\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) .
Isogenies and isogeny class
This curve has no rational isogenies. Its isogeny class 14.1-c consists of this curve only.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.