Base field 6.6.1292517.1
Generator \(a\), with minimal polynomial \( x^{6} - 6 x^{4} - x^{3} + 6 x^{2} - 1 \); class number \(1\).
Weierstrass equation
This is a global minimal model.
Mordell-Weil group structure
\(\Z \oplus \Z/{21}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(\frac{1}{3} a^{4} - \frac{5}{3} a^{2} - \frac{1}{3} a + 1 : -\frac{4}{3} a^{5} - \frac{2}{3} a^{4} + 7 a^{3} + 5 a^{2} - 2 a - \frac{4}{3} : 1\right)$ | $0.75565244373894385671955428273778512750$ | $\infty$ |
| $\left(-a^{5} + 6 a^{3} + a^{2} - 5 a : -a^{4} + 5 a^{2} + a : 1\right)$ | $0$ | $21$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((2)\) | = | \((2)\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 64 \) | = | \(64\) |
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| Discriminant: | $\Delta$ | = | $-8$ | ||
| Discriminant ideal: | $\frak{D}_{\mathrm{min}} = (\Delta)$ | = | \((-8)\) | = | \((2)^{3}\) |
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| Discriminant norm: | $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ | = | \( 262144 \) | = | \(64^{3}\) |
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| j-invariant: | $j$ | = | \( -\frac{140625}{8} \) | ||
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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: | $\mathrm{Reg}(E/K)$ | ≈ | \( 0.75565244373894385671955428273778512750 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 4.53391466243366314031732569642671076500 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 149614.88551524994572039009806833012337 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 3 \) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(21\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 4.05894 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}4.058940000 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 149614.885515 \cdot 4.533915 \cdot 3 } { {21^2 \cdot 1136.889177} } \\ & \approx 4.058940448 \end{aligned}$$
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 | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((2)\) | \(64\) | \(3\) | \(I_{3}\) | 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 |
|---|---|
| \(3\) | 3B.1.1 |
| \(7\) | 7B.1.1 |
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-b
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.c3 |
| \(\Q(\zeta_{9})^+\) | 3.3.81.1-8.1-a1 |
| \(\Q(\zeta_{9})^+\) | a curve with conductor norm 310472 (not in the database) |