Base field 4.4.8000.1
Generator \(a\), with minimal polynomial \( x^{4} - 10 x^{2} + 20 \); class number \(1\).
Weierstrass equation
This is a global minimal model.
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(\frac{346}{121} a^{3} - \frac{311}{242} a^{2} - \frac{1941}{121} a + \frac{2812}{121} : -\frac{29938}{1331} a^{3} - \frac{27994}{1331} a^{2} + \frac{133690}{1331} a - \frac{20604}{1331} : 1\right)$ | $2.9451557591917741386095582579993571221$ | $\infty$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((a^2-a-6)\) | = | \((a^2+a-4)\cdot(1/2a^2+a-2)\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 44 \) | = | \(4\cdot11\) |
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| Discriminant: | $\Delta$ | = | $32a^3-32a^2+32a-672$ | ||
| Discriminant ideal: | $\frak{D}_{\mathrm{min}} = (\Delta)$ | = | \((32a^3-32a^2+32a-672)\) | = | \((a^2+a-4)^{10}\cdot(1/2a^2+a-2)^{5}\) |
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| Discriminant norm: | $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ | = | \( 168874213376 \) | = | \(4^{10}\cdot11^{5}\) |
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| j-invariant: | $j$ | = | \( \frac{186806457187065527217}{644204} a^{3} + \frac{1242268729854976999493}{2576816} a^{2} - \frac{10813953778068508135555}{5153632} a - \frac{17978281963641980467639}{5153632} \) | ||
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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: | $\mathrm{Reg}(E/K)$ | ≈ | \( 2.9451557591917741386095582579993571221 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 11.780623036767096554438233031997428488 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 1.4180440277689308108265889917510757028 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 10 \) = \(2\cdot5\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(1\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 1.86772521103803 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}1.867725211 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 1.418044 \cdot 11.780623 \cdot 10 } { {1^2 \cdot 89.442719} } \\ & \approx 1.867725211 \end{aligned}$$
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 | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((a^2+a-4)\) | \(4\) | \(2\) | \(I_{10}\) | Non-split multiplicative | \(1\) | \(1\) | \(10\) | \(10\) |
| \((1/2a^2+a-2)\) | \(11\) | \(5\) | \(I_{5}\) | Split multiplicative | \(-1\) | \(1\) | \(5\) | \(5\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(5\) | 5B.1.4[2] |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
5.
Its isogeny class
44.1-d
consists of curves linked by isogenies of
degree 5.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.