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
\(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(-2 a^{2} + 6 : -8 a^{3} + 30 a : 1\right)$ | $0.62016967284967037630000347313671101948$ | $\infty$ |
| $\left(-\frac{3}{2} : \frac{1}{4} a^{3} - \frac{3}{4} a : 1\right)$ | $0$ | $2$ |
| $\left(a^{3} - 5 a + 1 : -a^{3} + a^{2} + 3 a - 2 : 1\right)$ | $0$ | $2$ |
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: | $\Delta$ | = | $24$ | ||
| Discriminant ideal: | $\frak{D}_{\mathrm{min}} = (\Delta)$ | = | \((24)\) | = | \((a^3-4a+1)^{12}\cdot(a^2-2)^{2}\) |
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| Discriminant norm: | $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ | = | \( 331776 \) | = | \(2^{12}\cdot9^{2}\) |
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| j-invariant: | $j$ | = | \( \frac{7301384}{3} \) | ||
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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: | $r_{\mathrm{an}}$ | = | \( 1 \) |
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| Mordell-Weil rank: | $r$ | = | \(1\) |
| Regulator: | $\mathrm{Reg}(E/K)$ | ≈ | \( 0.62016967284967037630000347313671101948 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 2.48067869139868150520001389254684407792 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 64.276255444475334886854285444483505230 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 4 \) = \(2\cdot2\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(4\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 3.32184869258351 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 4 \) (rounded) |
BSD formula
$$\begin{aligned}3.321848693 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 4 \cdot 64.276255 \cdot 2.480679 \cdot 4 } { {4^2 \cdot 48.000000} } \\ & \approx 3.321848693 \end{aligned}$$
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 | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((a^3-4a+1)\) | \(2\) | \(2\) | \(I_{2}^{*}\) | Additive | \(1\) | \(6\) | \(12\) | \(0\) |
| \((a^2-2)\) | \(9\) | \(2\) | \(I_{2}\) | Split multiplicative | \(-1\) | \(1\) | \(2\) | \(2\) |
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, 4 and 8.
Its isogeny class
576.1-d
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.a1 |
| \(\Q\) | 192.c1 |
| \(\Q\) | 288.b1 |
| \(\Q\) | 576.g1 |
| \(\Q(\sqrt{3}) \) | 2.2.12.1-192.1-b5 |
| \(\Q(\sqrt{3}) \) | 2.2.12.1-768.1-g5 |
| \(\Q(\sqrt{6}) \) | 2.2.24.1-96.1-b5 |
| \(\Q(\sqrt{6}) \) | 2.2.24.1-96.1-c5 |
| \(\Q(\sqrt{2}) \) | 2.2.8.1-288.1-c4 |
| \(\Q(\sqrt{2}) \) | 2.2.8.1-2592.1-a4 |