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
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(\frac{2468608}{621075} a^{3} - \frac{5032882}{621075} a^{2} - \frac{759858}{41405} a + \frac{4240877}{207025} : -\frac{2293749282}{94196375} a^{3} + \frac{2929287028}{94196375} a^{2} + \frac{6141520538}{56517825} a - \frac{8340027949}{94196375} : 1\right)$ | $6.6182378136884860742154484345434523973$ | $\infty$ |
| $\left(\frac{11}{4} a^{3} - \frac{11}{2} a^{2} - \frac{57}{4} a + \frac{39}{4} : -\frac{19}{8} a^{3} + 7 a^{2} + \frac{67}{4} a - 11 : 1\right)$ | $0$ | $2$ |
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: | $\Delta$ | = | $-7703906a^3+9164061a^2+34244423a-23060691$ | ||
| Discriminant ideal: | $\frak{D}_{\mathrm{min}} = (\Delta)$ | = | \((-7703906a^3+9164061a^2+34244423a-23060691)\) | = | \((-a^3+a^2+4a)\cdot(a^3-a^2-4a+1)^{32}\) |
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| Discriminant norm: | $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ | = | \( 3313283022731761938915897603 \) | = | \(3\cdot7^{32}\) |
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| j-invariant: | $j$ | = | \( -\frac{8726799587448504883954432291309058}{3313283022731761938915897603} a^{3} + \frac{3512609295216958802166282509619697}{1104427674243920646305299201} a^{2} + \frac{13211649577996599292098856645855219}{1104427674243920646305299201} a - \frac{7226027092698629340082787287567899}{1104427674243920646305299201} \) | ||
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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: | $\mathrm{Reg}(E/K)$ | ≈ | \( 6.6182378136884860742154484345434523973 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 26.472951254753944296861793738173809589 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 1.3802803304060593042144664110623292779 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 2 \) = \(1\cdot2\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(2\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 2.93659957703088 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 16 \) (rounded) |
BSD formula
$$\begin{aligned}2.936599577 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 16 \cdot 1.380280 \cdot 26.472951 \cdot 2 } { {2^2 \cdot 99.543960} } \\ & \approx 2.936599577 \end{aligned}$$
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 | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((-a^3+a^2+4a)\) | \(3\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
| \((a^3-a^2-4a+1)\) | \(7\) | \(2\) | \(I_{32}\) | Non-split multiplicative | \(1\) | \(1\) | \(32\) | \(32\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(2\) | 2B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 4 and 8.
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.