Base field \(\Q(\sqrt{2}, \sqrt{17})\)
Generator \(a\), with minimal polynomial \( x^{4} - 2 x^{3} - 11 x^{2} + 12 x + 2 \); class number \(1\).
Weierstrass equation
This is a global minimal model.
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(-\frac{1}{9} a^{3} - \frac{1}{3} a^{2} + \frac{14}{9} a + \frac{40}{9} : \frac{11}{9} a^{3} - \frac{7}{3} a^{2} - \frac{118}{9} a + \frac{100}{9} : 1\right)$ | $0.068789810019964630063027109881296664046$ | $\infty$ |
| $\left(-\frac{1}{3} a^{3} + \frac{1}{2} a^{2} + \frac{11}{3} a - \frac{19}{6} : -\frac{13}{36} a^{3} + \frac{2}{3} a^{2} + \frac{137}{36} a - \frac{73}{18} : 1\right)$ | $0$ | $2$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((2/9a^3-1/3a^2-28/9a+1/9)\) | = | \((4/9a^3-2/3a^2-47/9a+29/9)^{2}\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 4 \) | = | \(2^{2}\) |
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| Discriminant: | $\Delta$ | = | $2/9a^3-1/3a^2-28/9a+55/9$ | ||
| Discriminant ideal: | $\frak{D}_{\mathrm{min}} = (\Delta)$ | = | \((2/9a^3-1/3a^2-28/9a+55/9)\) | = | \((4/9a^3-2/3a^2-47/9a+29/9)^{8}\) |
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| Discriminant norm: | $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ | = | \( 256 \) | = | \(2^{8}\) |
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| j-invariant: | $j$ | = | \( 128 a^{3} - 192 a^{2} - 1792 a + 2496 \) | ||
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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: | $\mathrm{Reg}(E/K)$ | ≈ | \( 0.068789810019964630063027109881296664046 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 0.2751592400798585202521084395251866561840 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 942.87015599959480067398350535607697652 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 3 \) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(2\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 1.43073218172147 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}1.430732182 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 942.870156 \cdot 0.275159 \cdot 3 } { {2^2 \cdot 136.000000} } \\ & \approx 1.430732182 \end{aligned}$$
Local data at primes of bad reduction
This elliptic curve is not semistable. There is only one prime $\frak{p}$ of bad reduction.
| $\mathfrak{p}$ | $N(\mathfrak{p})$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((4/9a^3-2/3a^2-47/9a+29/9)\) | \(2\) | \(3\) | \(IV^{*}\) | Additive | \(-1\) | \(2\) | \(8\) | \(0\) |
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
4.3-b
consists of curves linked by isogenies of
degree 2.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.