Base field \(\Q(\zeta_{24})^+\)
Generator \(a\), with minimal polynomial \( x^{4} - 4 x^{2} + 1 \); class number \(1\).
Weierstrass equation
This is a global minimal model.
Mordell-Weil group structure
\(\Z/{4}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(a^{3} + \frac{3}{2} a^{2} - 8 a + \frac{7}{2} : 4 a^{3} - 3 a^{2} - \frac{21}{2} a + \frac{11}{2} : 1\right)$ | $0$ | $4$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((-2a^3+10a)\) | = | \((a^3-4a+1)^{6}\cdot(a^2-2)\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 576 \) | = | \(2^{6}\cdot9\) |
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| Discriminant: | $\Delta$ | = | $2a^3-10a$ | ||
| Discriminant ideal: | $\frak{D}_{\mathrm{min}} = (\Delta)$ | = | \((2a^3-10a)\) | = | \((a^3-4a+1)^{6}\cdot(a^2-2)\) |
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| Discriminant norm: | $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ | = | \( 576 \) | = | \(2^{6}\cdot9\) |
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| j-invariant: | $j$ | = | \( -\frac{7158018252821364749750}{3} a^{3} - \frac{3705262898043279737500}{3} a^{2} + \frac{26714087801034738628750}{3} a + \frac{13828229390897420342000}{3} \) | ||
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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: | $\Omega(E/K)$ | ≈ | \( 359.86925354739568191093251394719359697 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 1 \) = \(1\cdot1\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(4\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 1.87431902889269 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 4 \) (rounded) |
BSD formula
$$\begin{aligned}1.874319029 \approx L(E/K,1) & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 4 \cdot 359.869254 \cdot 1 \cdot 1 } { {4^2 \cdot 48.000000} } \\ & \approx 1.874319029 \end{aligned}$$
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 | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((a^3-4a+1)\) | \(2\) | \(1\) | \(II\) | Additive | \(-1\) | \(6\) | \(6\) | \(0\) |
| \((a^2-2)\) | \(9\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
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, 8 and 16.
Its isogeny class
576.1-a
consists of curves linked by isogenies of
degrees dividing 16.
Base change
This elliptic curve is a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.