Base field \(\Q(\sqrt{17}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x - 4 \); class number \(1\).
Weierstrass equation
This is a global minimal model.
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(-\frac{105}{16} a + \frac{185}{16} : \frac{1575}{64} a - \frac{3899}{64} : 1\right)$ | $3.0352561777303321608320536032717473360$ | $\infty$ |
| $\left(-\frac{21}{4} : \frac{17}{8} : 1\right)$ | $0$ | $2$ |
| $\left(-4 a + 5 : 2 a - 3 : 1\right)$ | $0$ | $2$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((-2a+1)\) | = | \((-2a+1)\) |
|
| |||||
| Conductor norm: | $N(\frak{N})$ | = | \( 17 \) | = | \(17\) |
|
| |||||
| Discriminant: | $\Delta$ | = | $17$ | ||
| Discriminant ideal: | $\frak{D}_{\mathrm{min}} = (\Delta)$ | = | \((17)\) | = | \((-2a+1)^{2}\) |
|
| |||||
| Discriminant norm: | $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ | = | \( 289 \) | = | \(17^{2}\) |
|
| |||||
| j-invariant: | $j$ | = | \( \frac{82483294977}{17} \) | ||
|
| |||||
| Endomorphism ring: | $\mathrm{End}(E)$ | = | \(\Z\) | ||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) | ||
|
| |||||
| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | ||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | \( 1 \) |
|
|
|||
| Mordell-Weil rank: | $r$ | = | \(1\) |
| Regulator: | $\mathrm{Reg}(E/K)$ | ≈ | \( 3.0352561777303321608320536032717473360 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 6.0705123554606643216641072065434946720 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 2.3934557638477947321885188538136114417 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 2 \) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(4\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 0.44049025497952142301979342445866619099 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}0.440490255 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 2.393456 \cdot 6.070512 \cdot 2 } { {4^2 \cdot 4.123106} } \\ & \approx 0.440490255 \end{aligned}$$
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 | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((-2a+1)\) | \(17\) | \(2\) | \(I_{2}\) | Split multiplicative | \(-1\) | \(1\) | \(2\) | \(2\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(2\) | 2Cs |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 4 and 8.
Its isogeny class
17.1-a
consists of curves linked by isogenies of
degrees dividing 16.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
| Base field | Curve |
|---|---|
| \(\Q\) | 17.a1 |
| \(\Q\) | 289.a1 |