Base field \(\Q(\sqrt{-3}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x + 1 \); 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 |
|---|---|---|
| $\left(-\frac{34641}{1444} a + \frac{146253}{1444} : \frac{4310395}{6859} a - \frac{81228285}{54872} : 1\right)$ | $6.2007877639778075145698148472999494264$ | $\infty$ |
| $\left(-75 a - 19 : 47 a - 38 : 1\right)$ | $0$ | $2$ |
| $\left(-31 a + 13 : 9 a - 16 : 1\right)$ | $0$ | $2$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((297a-57)\) | = | \((-2a+1)^{2}\cdot(-3a+1)^{2}\cdot(4a-3)^{2}\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 74529 \) | = | \(3^{2}\cdot7^{2}\cdot13^{2}\) |
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| Discriminant: | $\Delta$ | = | $-44792940860361a+47113994774880$ | ||
| Discriminant ideal: | $\frak{D}_{\mathrm{min}} = (\Delta)$ | = | \((-44792940860361a+47113994774880)\) | = | \((-2a+1)^{8}\cdot(-3a+1)^{12}\cdot(4a-3)^{12}\) |
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| Discriminant norm: | $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ | = | \( 2115761672920660979069533041 \) | = | \(3^{8}\cdot7^{12}\cdot13^{12}\) |
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| j-invariant: | $j$ | = | \( \frac{1915717851108899}{1703607756123} a + \frac{2297367303009589}{1703607756123} \) | ||
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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: | $\mathrm{Reg}(E/K)$ | ≈ | \( 6.2007877639778075145698148472999494264 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 12.401575527955615029139629694599898853 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 0.1431735989906350290127031730394283626560 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 64 \) = \(2^{2}\cdot2^{2}\cdot2^{2}\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(4\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 4.1005222103936045259204964682604068643 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}4.100522210 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 0.143174 \cdot 12.401576 \cdot 64 } { {4^2 \cdot 1.732051} } \\ & \approx 4.100522210 \end{aligned}$$
Local data at primes of bad reduction
This elliptic curve is not semistable. There are 3 primes $\frak{p}$ of bad reduction.
| $\mathfrak{p}$ | $N(\mathfrak{p})$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((-2a+1)\) | \(3\) | \(4\) | \(I_{2}^{*}\) | Additive | \(-1\) | \(2\) | \(8\) | \(2\) |
| \((-3a+1)\) | \(7\) | \(4\) | \(I_{6}^{*}\) | Additive | \(-1\) | \(2\) | \(12\) | \(6\) |
| \((4a-3)\) | \(13\) | \(4\) | \(I_{6}^{*}\) | Additive | \(1\) | \(2\) | \(12\) | \(6\) |
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 |
| \(3\) | 3B[2] |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3 and 6.
Its isogeny class
74529.3-b
consists of curves linked by isogenies of
degrees dividing 12.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.