Base field 6.6.1528713.1
Generator \(a\), with minimal polynomial \( x^{6} - 3 x^{5} - 3 x^{4} + 7 x^{3} + 3 x^{2} - 3 x - 1 \); class number \(1\).
Weierstrass equation
This is a global minimal model.
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $\frak{N}$ | = | \((2)\) | = | \((2a^5-6a^4-5a^3+12a^2+a-3)\cdot(-a^5+3a^4+3a^3-6a^2-2a+1)\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 64 \) | = | \(8\cdot8\) |
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| Discriminant: | $\Delta$ | = | $-128$ | ||
| Discriminant ideal: | $\frak{D}_{\mathrm{min}} = (\Delta)$ | = | \((-128)\) | = | \((2a^5-6a^4-5a^3+12a^2+a-3)^{7}\cdot(-a^5+3a^4+3a^3-6a^2-2a+1)^{7}\) |
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| Discriminant norm: | $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ | = | \( 4398046511104 \) | = | \(8^{7}\cdot8^{7}\) |
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| j-invariant: | $j$ | = | \( -\frac{189613868625}{128} \) | ||
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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: | $r_{\mathrm{an}}$ | = | \( 0 \) |
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| Mordell-Weil rank: | $r$ | = | \(0\) |
| Regulator: | $\mathrm{Reg}(E/K)$ | = | \( 1 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | = | \( 1 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 0.14130061597090587323346007386220416794 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 49 \) = \(7\cdot7\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(1\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 2.46954 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 441 \) (rounded) |
BSD formula
$$\begin{aligned}2.469540000 \approx L(E/K,1) & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 441 \cdot 0.141301 \cdot 1 \cdot 49 } { {1^2 \cdot 1236.411339} } \\ & \approx 2.469538182 \end{aligned}$$
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 | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((2a^5-6a^4-5a^3+12a^2+a-3)\) | \(8\) | \(7\) | \(I_{7}\) | Split multiplicative | \(-1\) | \(1\) | \(7\) | \(7\) |
| \((-a^5+3a^4+3a^3-6a^2-2a+1)\) | \(8\) | \(7\) | \(I_{7}\) | Split multiplicative | \(-1\) | \(1\) | \(7\) | \(7\) |
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 |
| \(7\) | 7B.1.3 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3, 7 and 21.
Its isogeny class
64.1-g
consists of curves linked by isogenies of
degrees dividing 21.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 3 elliptic curves:
| Base field | Curve |
|---|---|
| \(\Q\) | 162.c1 |
| \(\Q(\zeta_{9})^+\) | 3.3.81.1-8.1-a4 |
| \(\Q(\zeta_{9})^+\) | a curve with conductor norm 434312 (not in the database) |