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
\(\Z \oplus \Z/{3}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(\frac{177}{19} a^{5} - \frac{232}{19} a^{4} - \frac{1318}{19} a^{3} + \frac{1051}{19} a^{2} + \frac{430}{19} a - \frac{180}{19} : -\frac{1931}{19} a^{5} + \frac{2269}{19} a^{4} + \frac{14739}{19} a^{3} - \frac{9626}{19} a^{2} - \frac{6256}{19} a + \frac{1709}{19} : 1\right)$ | $0.011032955638540121004931146789541747477$ | $\infty$ |
| $\left(\frac{2}{57} a^{5} - \frac{14}{57} a^{4} + \frac{4}{57} a^{3} + \frac{26}{19} a^{2} - \frac{24}{19} a - \frac{1}{19} : \frac{173}{57} a^{5} - \frac{185}{57} a^{4} - \frac{461}{19} a^{3} + \frac{781}{57} a^{2} + \frac{318}{19} a - \frac{115}{19} : 1\right)$ | $0$ | $3$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((-20/19a^5+7/19a^4+169/19a^3+18/19a^2-116/19a+11/19)\) | = | \((-20/19a^5+7/19a^4+169/19a^3+18/19a^2-116/19a+11/19)\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 19 \) | = | \(19\) |
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| Discriminant: | $\Delta$ | = | $-6/19a^5+4/19a^4+64/19a^3-25/19a^2-69/19a-48/19$ | ||
| Discriminant ideal: | $\frak{D}_{\mathrm{min}} = (\Delta)$ | = | \((-6/19a^5+4/19a^4+64/19a^3-25/19a^2-69/19a-48/19)\) | = | \((-20/19a^5+7/19a^4+169/19a^3+18/19a^2-116/19a+11/19)^{3}\) |
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| Discriminant norm: | $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ | = | \( -6859 \) | = | \(-19^{3}\) |
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| j-invariant: | $j$ | = | \( -\frac{38489878116}{6859} a^{5} - \frac{8133128325}{6859} a^{4} + \frac{344847564441}{6859} a^{3} + \frac{226851839055}{6859} a^{2} - \frac{299489451885}{6859} a - \frac{180082842453}{6859} \) | ||
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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: | $\mathrm{Reg}(E/K)$ | ≈ | \( 0.011032955638540121004931146789541747477 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 0.06619773383124072602958688073725048486200 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 105782.05314360112768937513853994427453 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 3 \) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(3\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 2.57747 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}2.577470000 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 105782.053144 \cdot 0.066198 \cdot 3 } { {3^2 \cdot 905.607531} } \\ & \approx 2.577471277 \end{aligned}$$
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 | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((-20/19a^5+7/19a^4+169/19a^3+18/19a^2-116/19a+11/19)\) | \(19\) | \(3\) | \(I_{3}\) | 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 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3.
Its isogeny class
19.1-a
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.