Base field \(\Q(\sqrt{5}, \sqrt{17})\)
Generator \(a\), with minimal polynomial \( x^{4} - 11 x^{2} + 9 \); class number \(1\).
Weierstrass equation
This is a global minimal model.
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(\frac{23}{24} a^{3} + \frac{7}{8} a^{2} - \frac{175}{24} a - \frac{11}{2} : -\frac{17}{48} a^{3} - \frac{13}{8} a^{2} + \frac{19}{12} a + \frac{57}{16} : 1\right)$ | $0$ | $2$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((2)\) | = | \((1/6a^3-1/2a^2-5/6a+2)\cdot(-1/6a^3-1/2a^2+5/6a+2)\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 16 \) | = | \(4\cdot4\) |
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| Discriminant: | $\Delta$ | = | $-4/3a^3+56/3a+28$ | ||
| Discriminant ideal: | $\frak{D}_{\mathrm{min}} = (\Delta)$ | = | \((-4/3a^3+56/3a+28)\) | = | \((1/6a^3-1/2a^2-5/6a+2)^{3}\cdot(-1/6a^3-1/2a^2+5/6a+2)^{6}\) |
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| Discriminant norm: | $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ | = | \( 262144 \) | = | \(4^{3}\cdot4^{6}\) |
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| j-invariant: | $j$ | = | \( \frac{2134939673743248165}{128} a^{3} - \frac{2014355762928143019}{128} a^{2} - \frac{21583753823445138297}{128} a + \frac{2545584743833170707}{16} \) | ||
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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: | $\Omega(E/K)$ | ≈ | \( 200.97924934729210783361842191666967269 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 2 \) = \(1\cdot2\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(2\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 1.18223087851348 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}1.182230879 \approx L(E/K,1) & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 200.979249 \cdot 1 \cdot 2 } { {2^2 \cdot 85.000000} } \\ & \approx 1.182230879 \end{aligned}$$
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 | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((1/6a^3-1/2a^2-5/6a+2)\) | \(4\) | \(1\) | \(I_{3}\) | Non-split multiplicative | \(1\) | \(1\) | \(3\) | \(3\) |
| \((-1/6a^3-1/2a^2+5/6a+2)\) | \(4\) | \(2\) | \(I_{6}\) | Non-split multiplicative | \(1\) | \(1\) | \(6\) | \(6\) |
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 |
| \(3\) | 3B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3, 6, 9 and 18.
Its isogeny class
16.1-a
consists of curves linked by isogenies of
degrees dividing 18.
Base change
This elliptic curve is a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.