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
\(\Z \oplus \Z/{2}\Z \oplus \Z/{4}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(5 a^{3} + 10 a^{2} - 2 : 35 a^{3} + 68 a^{2} - 9 a - 18 : 1\right)$ | $0.16895046680151041653672650899403658506$ | $\infty$ |
| $\left(0 : 0 : 1\right)$ | $0$ | $2$ |
| $\left(a^{3} + 2 a^{2} - a - 1 : 2 a^{3} + 3 a^{2} - 3 a - 2 : 1\right)$ | $0$ | $4$ |
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: | $\Delta$ | = | $72$ | ||
| Discriminant ideal: | $\frak{D}_{\mathrm{min}} = (\Delta)$ | = | \((72)\) | = | \((a^3-4a+1)^{12}\cdot(a^2-2)^{4}\) |
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| Discriminant norm: | $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ | = | \( 26873856 \) | = | \(2^{12}\cdot9^{4}\) |
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| j-invariant: | $j$ | = | \( \frac{21952}{9} \) | ||
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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: | $\mathrm{Reg}(E/K)$ | ≈ | \( 0.16895046680151041653672650899403658506 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 0.675801867206041666146906035976146340240 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 964.09676174925286871614685221123068292 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 16 \) = \(2^{2}\cdot2^{2}\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(8\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 3.39342912373668 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}3.393429124 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 964.096762 \cdot 0.675802 \cdot 16 } { {8^2 \cdot 48.000000} } \\ & \approx 3.393429124 \end{aligned}$$
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 | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((a^3-4a+1)\) | \(2\) | \(4\) | \(I_{2}^{*}\) | Additive | \(1\) | \(6\) | \(12\) | \(0\) |
| \((a^2-2)\) | \(9\) | \(4\) | \(I_{4}\) | Split multiplicative | \(-1\) | \(1\) | \(4\) | \(4\) |
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
576.1-c
consists of curves linked by isogenies of
degrees dividing 16.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 10 elliptic curves:
| Base field | Curve |
|---|---|
| \(\Q\) | 96.b3 |
| \(\Q\) | 192.a2 |
| \(\Q\) | 288.c3 |
| \(\Q\) | 576.h2 |
| \(\Q(\sqrt{3}) \) | 2.2.12.1-768.1-j3 |
| \(\Q(\sqrt{3}) \) | 2.2.12.1-192.1-a3 |
| \(\Q(\sqrt{6}) \) | 2.2.24.1-96.1-a3 |
| \(\Q(\sqrt{6}) \) | 2.2.24.1-96.1-d3 |
| \(\Q(\sqrt{2}) \) | 2.2.8.1-288.1-b2 |
| \(\Q(\sqrt{2}) \) | 2.2.8.1-2592.1-e2 |