Base field 6.6.1995125.1
Generator \(a\), with minimal polynomial \( x^{6} - 2 x^{5} - 6 x^{4} + 6 x^{3} + 12 x^{2} + x - 1 \); class number \(1\).
Weierstrass equation
This is a global minimal model.
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(\frac{86563}{841} a^{5} - \frac{272450}{841} a^{4} - \frac{192797}{841} a^{3} + \frac{700848}{841} a^{2} + \frac{200001}{841} a - \frac{41466}{841} : \frac{37595317}{24389} a^{5} - \frac{124355976}{24389} a^{4} - \frac{65373015}{24389} a^{3} + \frac{318569637}{24389} a^{2} + \frac{41027618}{24389} a - \frac{35391933}{24389} : 1\right)$ | $3.9575521479894492090225436958826619313$ | $\infty$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((a^2-2a-3)\) | = | \((a^2-2a-3)\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 29 \) | = | \(29\) |
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| Discriminant: | $\Delta$ | = | $-a^4+4a^3-9a$ | ||
| Discriminant ideal: | $\frak{D}_{\mathrm{min}} = (\Delta)$ | = | \((-a^4+4a^3-9a)\) | = | \((a^2-2a-3)\) |
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| Discriminant norm: | $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ | = | \( -29 \) | = | \(-29\) |
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| j-invariant: | $j$ | = | \( -\frac{61552291988378339328827}{29} a^{5} + \frac{108272845536099489867930}{29} a^{4} + \frac{395403349463381380906095}{29} a^{3} - \frac{274036730257245363907994}{29} a^{2} - \frac{804659831415094141348378}{29} a - \frac{255444408231731420144507}{29} \) | ||
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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: | $\mathrm{Reg}(E/K)$ | ≈ | \( 3.9575521479894492090225436958826619313 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 23.745312887936695254135262175295971588 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 3.1300984271796062170130256185564388845 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 1 \) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(1\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 4.26222 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 81 \) (rounded) |
BSD formula
$$\begin{aligned}4.262220000 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 81 \cdot 3.130098 \cdot 23.745313 \cdot 1 } { {1^2 \cdot 1412.488938} } \\ & \approx 4.262219920 \end{aligned}$$
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 | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((a^2-2a-3)\) | \(29\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(3\) | 3B.1.2 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3.
Its isogeny class
29.4-c
consists of curves linked by isogenies of
degree 3.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.