Base field 4.4.11197.1
Generator \(a\), with minimal polynomial \( x^{4} - 2 x^{3} - 4 x^{2} + 3 x + 1 \); class number \(1\).
Weierstrass equation
This is a global minimal model.
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(-26 a^{3} + 30 a^{2} + 129 a + 31 : 286 a^{3} - 330 a^{2} - 1424 a - 345 : 1\right)$ | $1.4799659268206625945465314691941261603$ | $\infty$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((-a^3+a^2+4a+2)\) | = | \((-a-1)\cdot(a^2-2a-2)\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 39 \) | = | \(3\cdot13\) |
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| Discriminant: | $\Delta$ | = | $2a^3-3a^2-9a+2$ | ||
| Discriminant ideal: | $\frak{D}_{\mathrm{min}} = (\Delta)$ | = | \((2a^3-3a^2-9a+2)\) | = | \((-a-1)\cdot(a^2-2a-2)\) |
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| Discriminant norm: | $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ | = | \( 39 \) | = | \(3\cdot13\) |
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| j-invariant: | $j$ | = | \( \frac{1568037508964}{39} a^{3} - \frac{1857017012602}{13} a^{2} + \frac{2379030713467}{39} a + \frac{1009760062646}{39} \) | ||
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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: | $\mathrm{Reg}(E/K)$ | ≈ | \( 1.4799659268206625945465314691941261603 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 5.9198637072826503781861258767765046412 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 50.520755700431678387930669696806348725 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 1 \) = \(1\cdot1\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(1\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 2.82638101465671 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}2.826381015 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 50.520756 \cdot 5.919864 \cdot 1 } { {1^2 \cdot 105.815878} } \\ & \approx 2.826381015 \end{aligned}$$
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 | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((-a-1)\) | \(3\) | \(1\) | \(I_{1}\) | Non-split multiplicative | \(1\) | \(1\) | \(1\) | \(1\) |
| \((a^2-2a-2)\) | \(13\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
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 39.1-b 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.