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
\(\Z \oplus \Z/{8}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(\frac{5}{2} a^{3} - 4 a^{2} - 15 a + \frac{41}{2} : -\frac{9}{2} a^{3} + 30 a^{2} + 23 a - \frac{363}{2} : 1\right)$ | $0.55290560099697775742505983416534392941$ | $\infty$ |
| $\left(\frac{3}{2} a^{3} + \frac{3}{2} a^{2} - \frac{19}{2} a - 12 : \frac{11}{2} a^{3} + \frac{15}{2} a^{2} - \frac{71}{2} a - 51 : 1\right)$ | $0$ | $8$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((1/2a^3-3/2a^2-5/2a+6)\) | = | \((1/2a^3-3/2a^2-5/2a+6)\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 4 \) | = | \(4\) |
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| Discriminant: | $\Delta$ | = | $-9/2a^3+14a^2+21a-129/2$ | ||
| Discriminant ideal: | $\frak{D}_{\mathrm{min}} = (\Delta)$ | = | \((-9/2a^3+14a^2+21a-129/2)\) | = | \((1/2a^3-3/2a^2-5/2a+6)^{8}\) |
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| Discriminant norm: | $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ | = | \( 65536 \) | = | \(4^{8}\) |
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| j-invariant: | $j$ | = | \( -\frac{123165}{512} a^{3} + \frac{131307}{256} a^{2} + \frac{369921}{256} a + \frac{105115}{512} \) | ||
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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: | $\mathrm{Reg}(E/K)$ | ≈ | \( 0.55290560099697775742505983416534392941 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 2.21162240398791102970023933666137571764 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 713.53159130117649670636356812462591988 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 8 \) |
| 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
$$\begin{aligned}1.689919216 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 713.531591 \cdot 2.211622 \cdot 8 } { {8^2 \cdot 116.726175} } \\ & \approx 1.689919216 \end{aligned}$$
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 | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((1/2a^3-3/2a^2-5/2a+6)\) | \(4\) | \(8\) | \(I_{8}\) | Split multiplicative | \(-1\) | \(1\) | \(8\) | \(8\) |
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 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 4 and 8.
Its isogeny class
4.2-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.