Base field 3.3.361.1
Generator \(a\), with minimal polynomial \( x^{3} - x^{2} - 6 x + 7 \); class number \(1\).
Weierstrass equation
This is a global minimal model.
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{9}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(\frac{73}{49} a^{2} + \frac{19}{49} a - \frac{44}{7} : \frac{226}{343} a^{2} - \frac{13}{343} a - \frac{272}{49} : 1\right)$ | $1.1770068430849075450129949620794194453$ | $\infty$ |
| $\left(8 a^{2} + 10 a - 24 : 48 a^{2} + 60 a - 150 : 1\right)$ | $1.1770068430849075450129949620794194453$ | $\infty$ |
| $\left(a + 1 : a^{2} - a - 2 : 1\right)$ | $0$ | $9$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((2a^2-2a-8)\) | = | \((2)\cdot(a^2-a-4)\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 152 \) | = | \(8\cdot19\) |
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| Discriminant: | $\Delta$ | = | $-152$ | ||
| Discriminant ideal: | $\frak{D}_{\mathrm{min}} = (\Delta)$ | = | \((-152)\) | = | \((2)^{3}\cdot(a^2-a-4)^{3}\) |
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| Discriminant norm: | $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ | = | \( -3511808 \) | = | \(-8^{3}\cdot19^{3}\) |
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| j-invariant: | $j$ | = | \( -\frac{413493625}{152} \) | ||
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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: | $r_{\mathrm{an}}$ | = | \( 2 \) |
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| Mordell-Weil rank: | $r$ | = | \(2\) |
| Regulator: | $\mathrm{Reg}(E/K)$ | ≈ | \( 1.0390088315015251290093453598381036192 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 9.3510794835137261610841082385429325728 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 182.46725367387427957523565450721944330 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 3 \) = \(1\cdot3\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(9\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 3.3260541759120084807228687381942275689 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}3.326054176 \approx L^{(2)}(E/K,1)/2! & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 182.467254 \cdot 9.351079 \cdot 3 } { {9^2 \cdot 19.000000} } \\ & \approx 3.326054176 \end{aligned}$$
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 | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((2)\) | \(8\) | \(1\) | \(I_{3}\) | Non-split multiplicative | \(1\) | \(1\) | \(3\) | \(3\) |
| \((a^2-a-4)\) | \(19\) | \(3\) | \(I_{3}\) | 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 and 9.
Its isogeny class
152.1-b
consists of curves linked by isogenies of
degrees dividing 9.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following elliptic curve:
| Base field | Curve |
|---|---|
| \(\Q\) | 38.a2 |