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
\(\Z \oplus \Z \oplus \Z/{3}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(2 a + 4 : 7 a + 7 : 1\right)$ | $0.41066978745816666522070401078281713878$ | $\infty$ |
| $\left(4 a - 6 : -7 a - 7 : 1\right)$ | $0.41066978745816666522070401078281713878$ | $\infty$ |
| $\left(0 : -3 a + 1 : 1\right)$ | $0$ | $3$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((108a-288)\) | = | \((-2a+1)^{4}\cdot(2)^{2}\cdot(-3a+1)^{2}\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 63504 \) | = | \(3^{4}\cdot4^{2}\cdot7^{2}\) |
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| Discriminant: | $\Delta$ | = | $16848a-23760$ | ||
| Discriminant ideal: | $\frak{D}_{\mathrm{min}} = (\Delta)$ | = | \((16848a-23760)\) | = | \((-2a+1)^{6}\cdot(2)^{4}\cdot(-3a+1)^{4}\) |
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| Discriminant norm: | $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ | = | \( 448084224 \) | = | \(3^{6}\cdot4^{4}\cdot7^{4}\) |
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| j-invariant: | $j$ | = | \( 0 \) | ||
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| Endomorphism ring: | $\mathrm{End}(E)$ | = | \(\Z[(1+\sqrt{-3})/2]\) (complex multiplication) | ||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z[(1+\sqrt{-3})/2]\) | ||
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| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{U}(1)$ | ||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | \( 2 \) |
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| Mordell-Weil rank: | $r$ | = | \(2\) |
| Regulator: | $\mathrm{Reg}(E/K)$ | ≈ | \( 0.12648725574820183713246775598693332611 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 0.505949022992807348529871023947733304440 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 5.3406162896350754631819216273272823700 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 27 \) = \(3\cdot3\cdot3\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(3\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 4.6801391427651059127939919905648741520 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}4.680139143 \approx L^{(2)}(E/K,1)/2! & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 5.340616 \cdot 0.505949 \cdot 27 } { {3^2 \cdot 1.732051} } \\ & \approx 4.680139143 \end{aligned}$$
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 | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((-2a+1)\) | \(3\) | \(3\) | \(IV\) | Additive | \(-1\) | \(4\) | \(6\) | \(0\) |
| \((2)\) | \(4\) | \(3\) | \(IV\) | Additive | \(1\) | \(2\) | \(4\) | \(0\) |
| \((-3a+1)\) | \(7\) | \(3\) | \(IV\) | Additive | \(1\) | \(2\) | \(4\) | \(0\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(3\) | 3B.1.1[2] |
For all other primes \(p\), the image is a split Cartan subgroup if \(\left(\frac{ -3 }{p}\right)=+1\) or a nonsplit Cartan subgroup if \(\left(\frac{ -3 }{p}\right)=-1\).
Isogenies and isogeny class
This curve has no rational isogenies other than endomorphisms. Its isogeny class 63504.1-CMa consists of this curve only.
Base change
This elliptic curve is a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.