Base field 3.3.169.1
Generator \(a\), with minimal polynomial \( x^{3} - x^{2} - 4 x - 1 \); 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 |
|---|---|---|
| $\left(2 a^{2} - 3 a + 1 : -6 a : 1\right)$ | $0.15732221389775571213925769454806618432$ | $\infty$ |
| $\left(\frac{7}{4} a^{2} - \frac{5}{2} a + \frac{3}{4} : -a^{2} - \frac{17}{8} a + \frac{1}{4} : 1\right)$ | $0$ | $2$ |
| $\left(a^{2} - 7 a - 4 : 2 a^{2} + 4 a + 3 : 1\right)$ | $0$ | $2$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((-2a^2+8a+9)\) | = | \((-a^2+2a+3)^{2}\cdot(-a+1)^{2}\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 625 \) | = | \(5^{2}\cdot5^{2}\) |
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| Discriminant: | $\Delta$ | = | $1081a^2-2099a+3073$ | ||
| Discriminant ideal: | $\frak{D}_{\mathrm{min}} = (\Delta)$ | = | \((1081a^2-2099a+3073)\) | = | \((-a^2+2a+3)^{8}\cdot(-a+1)^{8}\) |
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| Discriminant norm: | $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ | = | \( 152587890625 \) | = | \(5^{8}\cdot5^{8}\) |
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| j-invariant: | $j$ | = | \( \frac{217317341}{5} a^{2} - \frac{2574843704}{25} a - \frac{785560232}{25} \) | ||
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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: | $\mathrm{Reg}(E/K)$ | ≈ | \( 0.15732221389775571213925769454806618432 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 0.471966641693267136417773083644198552960 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 69.801133175285460706965586134532881220 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 16 \) = \(2^{2}\cdot2^{2}\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(4\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 2.5341389547018442170115234589052769627 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}2.534138955 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 69.801133 \cdot 0.471967 \cdot 16 } { {4^2 \cdot 13.000000} } \\ & \approx 2.534138955 \end{aligned}$$
Local data at primes of bad reduction
This elliptic curve is not semistable. There are 2 primes $\frak{p}$ of bad reduction.
| $\mathfrak{p}$ | $N(\mathfrak{p})$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((-a^2+2a+3)\) | \(5\) | \(4\) | \(I_{2}^{*}\) | Additive | \(1\) | \(2\) | \(8\) | \(2\) |
| \((-a+1)\) | \(5\) | \(4\) | \(I_{2}^{*}\) | Additive | \(1\) | \(2\) | \(8\) | \(2\) |
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 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2.
Its isogeny class
625.2-i
consists of curves linked by isogenies of
degrees dividing 4.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.