Base field 6.6.820125.1
Generator \(a\), with minimal polynomial \( x^{6} - 9 x^{4} - 4 x^{3} + 9 x^{2} + 3 x - 1 \); class number \(1\).
Weierstrass equation
This is a global minimal model.
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(-\frac{53}{38} a^{5} + \frac{39}{76} a^{4} + \frac{909}{76} a^{3} + \frac{65}{76} a^{2} - \frac{186}{19} a + \frac{7}{76} : \frac{1}{152} a^{5} + \frac{11}{19} a^{4} - \frac{55}{152} a^{3} - \frac{683}{152} a^{2} + \frac{363}{152} a + \frac{58}{19} : 1\right)$ | $0$ | $2$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((-24/19a^5+16/19a^4+199/19a^3-24/19a^2-162/19a-21/19)\) | = | \((-24/19a^5+16/19a^4+199/19a^3-24/19a^2-162/19a-21/19)\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 71 \) | = | \(71\) |
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| Discriminant: | $\Delta$ | = | $781/19a^5-33/19a^4-6931/19a^3-2601/19a^2+5438/19a+1118/19$ | ||
| Discriminant ideal: | $\frak{D}_{\mathrm{min}} = (\Delta)$ | = | \((781/19a^5-33/19a^4-6931/19a^3-2601/19a^2+5438/19a+1118/19)\) | = | \((-24/19a^5+16/19a^4+199/19a^3-24/19a^2-162/19a-21/19)^{6}\) |
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| Discriminant norm: | $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ | = | \( 128100283921 \) | = | \(71^{6}\) |
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| j-invariant: | $j$ | = | \( -\frac{156831929359541405397}{2433905394499} a^{5} - \frac{150395003262395600328}{2433905394499} a^{4} + \frac{1274931496457544507273}{2433905394499} a^{3} + \frac{1855892975254178590359}{2433905394499} a^{2} + \frac{329513252750777598723}{2433905394499} a - \frac{168438420445546290933}{2433905394499} \) | ||
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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: | $\Omega(E/K)$ | ≈ | \( 1046.6006091913915317720046427088967971 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 2 \) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(2\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 2.31138 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 4 \) (rounded) |
BSD formula
$$\begin{aligned}2.311380000 \approx L(E/K,1) & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 4 \cdot 1046.600609 \cdot 1 \cdot 2 } { {2^2 \cdot 905.607531} } \\ & \approx 2.311377884 \end{aligned}$$
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 | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((-24/19a^5+16/19a^4+199/19a^3-24/19a^2-162/19a-21/19)\) | \(71\) | \(2\) | \(I_{6}\) | Non-split multiplicative | \(1\) | \(1\) | \(6\) | \(6\) |
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
71.6-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.