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
\(\Z/{5}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(\frac{4}{5} a^{3} - \frac{47}{10} a^{2} - 6 a + 36 : \frac{11}{5} a^{3} - \frac{409}{10} a^{2} + \frac{163}{5} a + 162 : 1\right)$ | $0$ | $5$ |
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: | $j$ | = | \( -\frac{180573546428986317635}{10307264} a^{3} - \frac{464347704777436882827}{10307264} a^{2} + \frac{65134388954518111809}{1288408} a + \frac{311720018028907303419}{2576816} \) | ||
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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: | $\Omega(E/K)$ | ≈ | \( 78.703725329994026729391425184452999576 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 50 \) = \(( 2 \cdot 5 )\cdot5\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(5\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 1.75986879920339 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}1.759868799 \approx L(E/K,1) & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 78.703725 \cdot 1 \cdot 50 } { {5^2 \cdot 89.442719} } \\ & \approx 1.759868799 \end{aligned}$$
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 | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((a^2+a-4)\) | \(4\) | \(10\) | \(I_{10}\) | 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.1.1[2] |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
5.
Its isogeny class
44.1-c
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.