Base field 3.1.23.1
Generator \(a\), with minimal polynomial \( x^{3} - x^{2} + 1 \); class number \(1\).
Weierstrass equation
This is a global minimal model.
Mordell-Weil group structure
\(\Z/{9}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(10 a^{2} - 21 a + 17 : 74 a^{2} - 131 a + 99 : 1\right)$ | $0$ | $9$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((-4a^2+7a+1)\) | = | \((2a^2-a)\cdot(3a^2-a+1)\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 259 \) | = | \(7\cdot37\) |
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| Discriminant: | $\Delta$ | = | $-11800a^2+4160a-3211$ | ||
| Discriminant ideal: | $\frak{D}_{\mathrm{min}} = (\Delta)$ | = | \((-11800a^2+4160a-3211)\) | = | \((2a^2-a)^{9}\cdot(3a^2-a+1)^{3}\) |
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| Discriminant norm: | $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ | = | \( -2044031255371 \) | = | \(-7^{9}\cdot37^{3}\) |
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| j-invariant: | $j$ | = | \( \frac{533423413596160}{2044031255371} a^{2} + \frac{14942973020037120}{2044031255371} a + \frac{5291457412308992}{2044031255371} \) | ||
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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: | $\Omega(E/K)$ | ≈ | \( 7.4798711597035459106432971586883680384 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 27 \) = \(3^{2}\cdot3\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(9\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 0.51988698402933238686103269625191968865 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}0.519886984 \approx L(E/K,1) & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 7.479871 \cdot 1 \cdot 27 } { {9^2 \cdot 4.795832} } \\ & \approx 0.519886984 \end{aligned}$$
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 | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((2a^2-a)\) | \(7\) | \(9\) | \(I_{9}\) | Split multiplicative | \(-1\) | \(1\) | \(9\) | \(9\) |
| \((3a^2-a+1)\) | \(37\) | \(3\) | \(I_{3}\) | 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 |
|---|---|
| \(3\) | 3Cs.1.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3 and 9.
Its isogeny class
259.1-A
consists of curves linked by isogenies of
degrees dividing 27.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.