Base field \(\Q(\sqrt{-663}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x + 166 \); class number \(16\).
Weierstrass equation
This is not a global minimal model: it is minimal at all primes except \((17,a+8)\). No global minimal model exists.
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(-242 : 289 a - 24 : 1\right)$ | $1.2430917263210118730614958106028541014$ | $\infty$ |
| $\left(-\frac{131}{3} : -\frac{136}{9} a + \frac{260}{9} : 1\right)$ | $1.9633404895605087414040278188198604020$ | $\infty$ |
| $\left(-\frac{589}{25} : -\frac{774}{125} : 1\right)$ | $6.0705123554606643216641072065434946721$ | $\infty$ |
| $\left(-\frac{101}{4} : \frac{97}{8} : 1\right)$ | $0$ | $2$ |
| $\left(-21 : 10 : 1\right)$ | $0$ | $2$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((17,a+8)\) | = | \((17,a+8)\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 17 \) | = | \(17\) |
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| Discriminant: | $\Delta$ | = | $6975757441$ | ||
| Discriminant ideal: | $(\Delta)$ | = | \((6975757441)\) | = | \((17,a+8)^{16}\) |
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| Discriminant norm: | $N(\Delta)$ | = | \( 48661191875666868481 \) | = | \(17^{16}\) |
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| Minimal discriminant: | $\frak{D}_{\mathrm{min}}$ | = | \((289)\) | = | \((17,a+8)^{4}\) |
| Minimal discriminant norm: | $N(\frak{D}_{\mathrm{min}})$ | = | \( 83521 \) | = | \(17^{4}\) |
| j-invariant: | $j$ | = | \( \frac{20346417}{289} \) | ||
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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: | $r_{\mathrm{an}}$ | = | \( 3 \) |
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| Mordell-Weil rank: | $r$ | = | \(3\) |
| Regulator: | $\mathrm{Reg}(E/K)$ | ≈ | \( 14.577061286926798671229554729497357426 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 116.61649029541438936983643783597885941 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 16.991509592519864644445744556309832632 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 2 \) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(4\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 4.8096689498337259687770352865702038078 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}4.809668950 \approx L^{(3)}(E/K,1)/3! & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 8.495755 \cdot 116.616490 \cdot 2 } { {4^2 \cdot 25.748786} } \\ & \approx 4.809668950 \end{aligned}$$
Local data at primes of bad reduction
This elliptic curve is semistable. There is only one prime $\frak{p}$ of bad reduction. Primes of good reduction for the curve but which divide the discriminant of the model above (if any) are included.
| $\mathfrak{p}$ | $N(\mathfrak{p})$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((17,a+8)\) | \(17\) | \(2\) | \(I_{4}\) | Non-split multiplicative | \(1\) | \(1\) | \(4\) | \(4\) |
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 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2.
Its isogeny class
17.1-a
consists of curves linked by isogenies of
degrees dividing 4.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
| Base field | Curve |
|---|---|
| \(\Q\) | 289.a2 |
| \(\Q\) | 25857.e2 |