Base field 4.4.19025.1
Generator \(a\), with minimal polynomial \( x^{4} - 2 x^{3} - 13 x^{2} + 14 x + 44 \); class number \(1\).
Weierstrass equation
This is a global minimal model.
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(-\frac{3639}{1210} a^{3} + \frac{3221}{605} a^{2} + \frac{21391}{1210} a + \frac{324}{121} : -\frac{623707}{66550} a^{3} + \frac{482267}{13310} a^{2} + \frac{757973}{33275} a - \frac{3521633}{33275} : 1\right)$ | $3.8495530509208297746029141114450957595$ | $\infty$ |
| $\left(-a^{3} - \frac{17}{4} a^{2} + 17 a + 35 : -\frac{7}{8} a^{3} - \frac{27}{4} a^{2} + \frac{47}{2} a + 55 : 1\right)$ | $0$ | $2$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((-a-1)\) | = | \((1/2a^3-7/2a-1)\cdot(-1/2a^3+3/2a^2+3a-9)\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 20 \) | = | \(4\cdot5\) |
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| Discriminant: | $\Delta$ | = | $-13/2a^3+11a^2+93/2a-81$ | ||
| Discriminant ideal: | $\frak{D}_{\mathrm{min}} = (\Delta)$ | = | \((-13/2a^3+11a^2+93/2a-81)\) | = | \((1/2a^3-7/2a-1)^{6}\cdot(-1/2a^3+3/2a^2+3a-9)^{3}\) |
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| Discriminant norm: | $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ | = | \( -512000 \) | = | \(-4^{6}\cdot5^{3}\) |
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| j-invariant: | $j$ | = | \( -\frac{193066453992715621}{1600} a^{3} + \frac{442116976490500419}{800} a^{2} + \frac{11429398180028347}{80} a - \frac{51448046581033861}{25} \) | ||
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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: | $\mathrm{Reg}(E/K)$ | ≈ | \( 3.8495530509208297746029141114450957595 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 15.398212203683319098411656445780383038 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 17.841628351168177837105113396511790217 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 6 \) = \(( 2 \cdot 3 )\cdot1\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(2\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 2.98767749196749 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}2.987677492 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 17.841628 \cdot 15.398212 \cdot 6 } { {2^2 \cdot 137.931142} } \\ & \approx 2.987677492 \end{aligned}$$
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 | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((1/2a^3-7/2a-1)\) | \(4\) | \(6\) | \(I_{6}\) | Split multiplicative | \(-1\) | \(1\) | \(6\) | \(6\) |
| \((-1/2a^3+3/2a^2+3a-9)\) | \(5\) | \(1\) | \(I_{3}\) | Non-split multiplicative | \(1\) | \(1\) | \(3\) | \(3\) |
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 |
| \(3\) | 3B.1.2 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3 and 6.
Its isogeny class
20.1-a
consists of curves linked by isogenies of
degrees dividing 6.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.