Base field 4.4.1957.1
Generator \(a\), with minimal polynomial \( x^{4} - 4 x^{2} - x + 1 \); class number \(1\).
Weierstrass equation
This is a global minimal model.
Mordell-Weil group structure
\(\Z/{2}\Z \oplus \Z/{4}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(-3 a^{2} + a + 1 : -2 a^{3} + 3 a^{2} + 4 a - 1 : 1\right)$ | $0$ | $2$ |
| $\left(a^{3} + 2 a^{2} - a : 2 a^{3} + 4 a^{2} + 3 a + 1 : 1\right)$ | $0$ | $4$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((-2a^3+a^2+7a)\) | = | \((a^3-4a)\cdot(a^2-2)\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 21 \) | = | \(3\cdot7\) |
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| Discriminant: | $\Delta$ | = | $-69a^3-190a^2-85a+425$ | ||
| Discriminant ideal: | $\frak{D}_{\mathrm{min}} = (\Delta)$ | = | \((-69a^3-190a^2-85a+425)\) | = | \((a^3-4a)^{12}\cdot(a^2-2)^{6}\) |
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| Discriminant norm: | $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ | = | \( 62523502209 \) | = | \(3^{12}\cdot7^{6}\) |
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| j-invariant: | $j$ | = | \( -\frac{84413562014875846901}{62523502209} a^{3} + \frac{148907375867724857512}{62523502209} a^{2} + \frac{24992949793645640572}{20841167403} a - \frac{47851964758998446149}{62523502209} \) | ||
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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: | $\Omega(E/K)$ | ≈ | \( 525.49285223579858306090972175503041554 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 4 \) = \(2\cdot2\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(8\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 0.742422999880055 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}0.742423000 \approx L(E/K,1) & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 525.492852 \cdot 1 \cdot 4 } { {8^2 \cdot 44.237993} } \\ & \approx 0.742423000 \end{aligned}$$
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 | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((a^3-4a)\) | \(3\) | \(2\) | \(I_{12}\) | Non-split multiplicative | \(1\) | \(1\) | \(12\) | \(12\) |
| \((a^2-2)\) | \(7\) | \(2\) | \(I_{6}\) | Non-split multiplicative | \(1\) | \(1\) | \(6\) | \(6\) |
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, 4, 6 and 12.
Its isogeny class
21.1-a
consists of curves linked by isogenies of
degrees dividing 24.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.