Base field 4.4.18097.1
Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 7 x^{2} + 6 x + 4 \); 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 |
|---|---|---|
| $\left(a + 1 : -2 : 1\right)$ | $0.19352830942768447863280547906309483493$ | $\infty$ |
| $\left(\frac{1}{4} a^{3} - \frac{1}{2} a : -\frac{3}{8} a^{3} - \frac{3}{4} a^{2} + 2 a - 1 : 1\right)$ | $0$ | $2$ |
| $\left(-\frac{3}{8} a^{3} - \frac{5}{8} a^{2} + \frac{17}{8} a + \frac{5}{4} : \frac{1}{4} a^{3} + \frac{3}{8} a^{2} - \frac{1}{8} a - \frac{17}{8} : 1\right)$ | $0$ | $2$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((-a-2)\) | = | \((1/2a^3-1/2a^2-5/2a+1)\cdot(a)\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 12 \) | = | \(3\cdot4\) |
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| Discriminant: | $\Delta$ | = | $-3a^3+3a^2+10a-4$ | ||
| Discriminant ideal: | $\frak{D}_{\mathrm{min}} = (\Delta)$ | = | \((-3a^3+3a^2+10a-4)\) | = | \((1/2a^3-1/2a^2-5/2a+1)^{4}\cdot(a)^{4}\) |
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| Discriminant norm: | $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ | = | \( 20736 \) | = | \(3^{4}\cdot4^{4}\) |
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| j-invariant: | $j$ | = | \( \frac{5580005647}{2592} a^{3} - \frac{2698752947}{864} a^{2} - \frac{35403039655}{2592} a + \frac{3091018997}{162} \) | ||
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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: | $\mathrm{Reg}(E/K)$ | ≈ | \( 0.19352830942768447863280547906309483493 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 0.774113237710737914531221916252379339720 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 1262.2567220197134531610955832994660600 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 8 \) = \(2\cdot2^{2}\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(4\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 3.63177468817164 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}3.631774688 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 1262.256722 \cdot 0.774113 \cdot 8 } { {4^2 \cdot 134.525091} } \\ & \approx 3.631774688 \end{aligned}$$
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 | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((1/2a^3-1/2a^2-5/2a+1)\) | \(3\) | \(2\) | \(I_{4}\) | Non-split multiplicative | \(1\) | \(1\) | \(4\) | \(4\) |
| \((a)\) | \(4\) | \(4\) | \(I_{4}\) | 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
12.1-a
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.