Base field 4.4.8525.1
Generator \(a\), with minimal polynomial \( x^{4} - 2 x^{3} - 8 x^{2} + 9 x + 19 \); class number \(1\).
Weierstrass equation
This is a global minimal model.
Mordell-Weil group structure
\(\Z \oplus \Z/{5}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(-\frac{427}{361} a^{3} + \frac{47}{361} a^{2} + \frac{2760}{361} a + \frac{104}{19} : -\frac{23327}{6859} a^{3} - \frac{14046}{6859} a^{2} + \frac{162291}{6859} a + \frac{10502}{361} : 1\right)$ | $1.3845725157713669891851077809692300582$ | $\infty$ |
| $\left(-a^{3} + 6 a + 4 : a^{3} - 5 a - 3 : 1\right)$ | $0$ | $5$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((a^3-2a^2-4a+3)\) | = | \((-a^3+5a+3)^{2}\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 25 \) | = | \(5^{2}\) |
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| Discriminant: | $\Delta$ | = | $-a^3+7a-1$ | ||
| Discriminant ideal: | $\frak{D}_{\mathrm{min}} = (\Delta)$ | = | \((-a^3+7a-1)\) | = | \((-a^3+5a+3)^{4}\) |
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| Discriminant norm: | $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ | = | \( -625 \) | = | \(-5^{4}\) |
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| j-invariant: | $j$ | = | \( 12049 a^{3} - 41176 a^{2} - 38199 a + 162580 \) | ||
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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: | $\mathrm{Reg}(E/K)$ | ≈ | \( 1.3845725157713669891851077809692300582 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 5.5382900630854679567404311238769202328 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 1233.2784557814021153519159750233859786 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 1 \) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(5\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 2.95903185245472 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}2.959031852 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 1233.278456 \cdot 5.538290 \cdot 1 } { {5^2 \cdot 92.330927} } \\ & \approx 2.959031852 \end{aligned}$$
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 | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((-a^3+5a+3)\) | \(5\) | \(1\) | \(IV\) | Additive | \(-1\) | \(2\) | \(4\) | \(0\) |
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
25.3-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.