Base field 4.4.9248.1
Generator \(a\), with minimal polynomial \( x^{4} - 5 x^{2} + 2 \); class number \(1\).
Weierstrass equation
This is a global minimal model.
Mordell-Weil group structure
\(\Z/{2}\Z \oplus \Z/{8}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(a^{2} - 8 : 3 a^{3} + 3 a^{2} - 13 a - 11 : 1\right)$ | $0$ | $2$ |
| $\left(22 a^{3} - 13 a^{2} - 100 a + 60 : 227 a^{3} - 149 a^{2} - 1037 a + 677 : 1\right)$ | $0$ | $8$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((-a^2+a+2)\) | = | \((a)\cdot(-a^3+4a+1)\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 4 \) | = | \(2\cdot2\) |
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| Discriminant: | $\Delta$ | = | $320a^2-384$ | ||
| Discriminant ideal: | $\frak{D}_{\mathrm{min}} = (\Delta)$ | = | \((320a^2-384)\) | = | \((a)^{24}\cdot(-a^3+4a+1)^{12}\) |
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| Discriminant norm: | $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ | = | \( 68719476736 \) | = | \(2^{24}\cdot2^{12}\) |
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| j-invariant: | $j$ | = | \( \frac{203862548967}{4096} a^{2} - \frac{89321178913}{4096} \) | ||
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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: | $\Omega(E/K)$ | ≈ | \( 241.58896489952769229560320805531588814 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 288 \) = \(( 2^{3} \cdot 3 )\cdot( 2^{2} \cdot 3 )\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(16\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 2.82621830526223 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}2.826218305 \approx L(E/K,1) & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 241.588965 \cdot 1 \cdot 288 } { {16^2 \cdot 96.166522} } \\ & \approx 2.826218305 \end{aligned}$$
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 | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((a)\) | \(2\) | \(24\) | \(I_{24}\) | Split multiplicative | \(-1\) | \(1\) | \(24\) | \(24\) |
| \((-a^3+4a+1)\) | \(2\) | \(12\) | \(I_{12}\) | Split multiplicative | \(-1\) | \(1\) | \(12\) | \(12\) |
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 |
| \(3\) | 3B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3, 4, 6, 8, 12 and 24.
Its isogeny class
4.2-b
consists of curves linked by isogenies of
degrees dividing 48.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
| Base field | Curve |
|---|---|
| \(\Q(\sqrt{17}) \) | 2.2.17.1-32.3-a10 |
| \(\Q(\sqrt{17}) \) | 2.2.17.1-128.6-c10 |