Base field 4.4.9909.1
Generator \(a\), with minimal polynomial \( x^{4} - 6 x^{2} - 3 x + 3 \); 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 |
|---|---|---|
| $\left(\frac{81}{25} a^{3} - \frac{99}{25} a^{2} - \frac{73}{5} a + \frac{442}{25} : -\frac{3467}{125} a^{3} + \frac{4518}{125} a^{2} + \frac{2901}{25} a - \frac{11619}{125} : 1\right)$ | $3.3091189068442430371077242172717261987$ | $\infty$ |
| $\left(2 a^{3} - a^{2} - 10 a + 7 : -6 a^{3} + 6 a^{2} + 20 a - 12 : 1\right)$ | $0$ | $2$ |
| $\left(-\frac{5}{4} a^{3} - \frac{3}{2} a^{2} + \frac{23}{4} a - \frac{9}{4} : \frac{45}{8} a^{3} - 3 a^{2} - \frac{61}{4} a + 7 : 1\right)$ | $0$ | $2$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((a-3)\) | = | \((-a^3+a^2+4a)\cdot(a^3-a^2-4a+1)\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 21 \) | = | \(3\cdot7\) |
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| Discriminant: | $\Delta$ | = | $-2487a^3+2531a^2+9684a-1314$ | ||
| Discriminant ideal: | $\frak{D}_{\mathrm{min}} = (\Delta)$ | = | \((-2487a^3+2531a^2+9684a-1314)\) | = | \((-a^3+a^2+4a)^{2}\cdot(a^3-a^2-4a+1)^{16}\) |
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| Discriminant norm: | $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ | = | \( 299096375126409 \) | = | \(3^{2}\cdot7^{16}\) |
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| j-invariant: | $j$ | = | \( \frac{3075730896093395126238965002}{99698791708803} a^{3} - \frac{3714007662409537173380188813}{99698791708803} a^{2} - \frac{4656548508714567651606714255}{33232930569601} a + \frac{2547146210989604914629326570}{33232930569601} \) | ||
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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: | $\mathrm{Reg}(E/K)$ | ≈ | \( 3.3091189068442430371077242172717261987 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 13.236475627376972148430896869086904795 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 22.084485286496948867431462576997268447 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 4 \) = \(2\cdot2\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(4\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 2.93659957703088 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 4 \) (rounded) |
BSD formula
$$\begin{aligned}2.936599577 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 4 \cdot 22.084485 \cdot 13.236476 \cdot 4 } { {4^2 \cdot 99.543960} } \\ & \approx 2.936599577 \end{aligned}$$
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 | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((-a^3+a^2+4a)\) | \(3\) | \(2\) | \(I_{2}\) | Split multiplicative | \(-1\) | \(1\) | \(2\) | \(2\) |
| \((a^3-a^2-4a+1)\) | \(7\) | \(2\) | \(I_{16}\) | Non-split multiplicative | \(1\) | \(1\) | \(16\) | \(16\) |
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 and 4.
Its isogeny class
21.1-d
consists of curves linked by isogenies of
degrees dividing 8.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.