Base field 4.4.19225.1
Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 15 x^{2} + 2 x + 44 \); class number \(1\).
Weierstrass equation
This is a global minimal model.
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(-\frac{2281}{6962} a^{3} + \frac{13771}{6962} a^{2} + \frac{18739}{6962} a - \frac{36013}{3481} : -\frac{211257}{410758} a^{3} - \frac{597443}{410758} a^{2} + \frac{775217}{410758} a + \frac{1283089}{205379} : 1\right)$ | $1.1921583630523869225272183863522711667$ | $\infty$ |
| $\left(-\frac{1}{4} a^{3} + 2 a^{2} + \frac{3}{2} a - \frac{49}{4} : -\frac{1}{2} a^{3} - \frac{9}{8} a^{2} + \frac{27}{8} a + \frac{65}{8} : 1\right)$ | $0$ | $2$ |
| $\left(-\frac{1}{2} a^{3} + \frac{5}{2} a^{2} + \frac{7}{2} a - 13 : -\frac{1}{2} a^{3} - \frac{1}{2} a^{2} + \frac{5}{2} a + 3 : 1\right)$ | $0$ | $2$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((1/2a^3-1/2a^2-5/2a+4)\) | = | \((1/2a^3-1/2a^2-5/2a+4)\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 9 \) | = | \(9\) |
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| Discriminant: | $\Delta$ | = | $-4a^3+14a^2+26a-73$ | ||
| Discriminant ideal: | $\frak{D}_{\mathrm{min}} = (\Delta)$ | = | \((-4a^3+14a^2+26a-73)\) | = | \((1/2a^3-1/2a^2-5/2a+4)^{2}\) |
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| Discriminant norm: | $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ | = | \( 81 \) | = | \(9^{2}\) |
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| j-invariant: | $j$ | = | \( -\frac{21331331}{18} a^{3} + \frac{73802737}{18} a^{2} + \frac{45673303}{6} a - \frac{176825299}{9} \) | ||
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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: | $\mathrm{Reg}(E/K)$ | ≈ | \( 1.1921583630523869225272183863522711667 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 4.7686334522095476901088735454090846668 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 921.70200455627329638075954801696945030 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 2 \) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(4\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 3.96242734276914 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}3.962427343 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 921.702005 \cdot 4.768633 \cdot 2 } { {4^2 \cdot 138.654246} } \\ & \approx 3.962427343 \end{aligned}$$
Local data at primes of bad reduction
This elliptic curve is semistable. There is only one prime $\frak{p}$ of bad reduction.
| $\mathfrak{p}$ | $N(\mathfrak{p})$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((1/2a^3-1/2a^2-5/2a+4)\) | \(9\) | \(2\) | \(I_{2}\) | Split multiplicative | \(-1\) | \(1\) | \(2\) | \(2\) |
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 and 6.
Its isogeny class
9.1-b
consists of curves linked by isogenies of
degrees dividing 12.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.