Base field 4.4.13625.1
Generator \(a\), with minimal polynomial \( x^{4} - 2 x^{3} - 11 x^{2} + 12 x + 31 \); class number \(1\).
Weierstrass equation
This is a global minimal model.
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z \oplus \Z/{4}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(\frac{29}{8} a^{3} + \frac{19}{4} a^{2} - \frac{91}{4} a - \frac{255}{8} : -36 a^{3} - \frac{809}{16} a^{2} + \frac{3563}{16} a + \frac{5219}{16} : 1\right)$ | $1.1058112019939555148501196683306878588$ | $\infty$ |
| $\left(\frac{1}{8} a^{3} - \frac{1}{4} a^{2} - \frac{7}{4} a - \frac{11}{8} : \frac{5}{16} a^{2} + \frac{1}{16} a - \frac{23}{16} : 1\right)$ | $0$ | $2$ |
| $\left(-\frac{1}{2} a^{2} - \frac{1}{2} a + \frac{3}{2} : \frac{1}{2} a^{2} + \frac{1}{2} a - \frac{3}{2} : 1\right)$ | $0$ | $4$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((-1/2a^3-a^2+3a+11/2)\) | = | \((-1/2a^3-a^2+3a+11/2)\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 4 \) | = | \(4\) |
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| Discriminant: | $\Delta$ | = | $-1/2a^3+2a^2+4a-27/2$ | ||
| Discriminant ideal: | $\frak{D}_{\mathrm{min}} = (\Delta)$ | = | \((-1/2a^3+2a^2+4a-27/2)\) | = | \((-1/2a^3-a^2+3a+11/2)^{4}\) |
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| Discriminant norm: | $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ | = | \( 256 \) | = | \(4^{4}\) |
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| j-invariant: | $j$ | = | \( \frac{10570617}{32} a^{3} - \frac{9262425}{8} a^{2} - \frac{30217737}{16} a + \frac{218496889}{32} \) | ||
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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: | $\mathrm{Reg}(E/K)$ | ≈ | \( 1.1058112019939555148501196683306878588 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 4.4232448079758220594004786733227514352 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 713.53159130117649670636356812462591989 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 4 \) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(8\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 1.68991921609341 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}1.689919216 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 713.531591 \cdot 4.423245 \cdot 4 } { {8^2 \cdot 116.726175} } \\ & \approx 1.689919216 \end{aligned}$$
Local data at primes of bad reduction
This elliptic curve is semistable. There is only one prime $\frak{p}$ of bad reduction.
| $\mathfrak{p}$ | $N(\mathfrak{p})$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((-1/2a^3-a^2+3a+11/2)\) | \(4\) | \(4\) | \(I_{4}\) | 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
4.1-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.