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
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(a - 1 : -15 a - 16 : 1\right)$ | $0.25758557850010162501961870653743174345$ | $\infty$ |
| $\left(-11 a + 15 : 5 a - 8 : 1\right)$ | $0$ | $2$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((-438a+272)\) | = | \((2)\cdot(-3a+1)\cdot(-4a+1)^{2}\cdot(6a-5)\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 146692 \) | = | \(4\cdot7\cdot13^{2}\cdot31\) |
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| Discriminant: | $\Delta$ | = | $-199820288a+254410752$ | ||
| Discriminant ideal: | $\frak{D}_{\mathrm{min}} = (\Delta)$ | = | \((-199820288a+254410752)\) | = | \((2)^{10}\cdot(-3a+1)^{3}\cdot(-4a+1)^{6}\cdot(6a-5)\) |
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| Discriminant norm: | $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ | = | \( 53816548494671872 \) | = | \(4^{10}\cdot7^{3}\cdot13^{6}\cdot31\) |
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| j-invariant: | $j$ | = | \( \frac{10621452329}{10888192} a - \frac{7274546105}{10888192} \) | ||
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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: | $\mathrm{Reg}(E/K)$ | ≈ | \( 0.25758557850010162501961870653743174345 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 0.515171157000203250039237413074863486900 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 1.10360860336339489200937253579300699244 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 40 \) = \(( 2 \cdot 5 )\cdot1\cdot2^{2}\cdot1\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(2\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 3.2825094886686198407059674421265588291 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}3.282509489 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 1.103609 \cdot 0.515171 \cdot 40 } { {2^2 \cdot 1.732051} } \\ & \approx 3.282509489 \end{aligned}$$
Local data at primes of bad reduction
This elliptic curve is not semistable. There are 4 primes $\frak{p}$ of bad reduction.
| $\mathfrak{p}$ | $N(\mathfrak{p})$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((2)\) | \(4\) | \(10\) | \(I_{10}\) | Split multiplicative | \(-1\) | \(1\) | \(10\) | \(10\) |
| \((-3a+1)\) | \(7\) | \(1\) | \(I_{3}\) | Non-split multiplicative | \(1\) | \(1\) | \(3\) | \(3\) |
| \((-4a+1)\) | \(13\) | \(4\) | \(I_0^{*}\) | Additive | \(1\) | \(2\) | \(6\) | \(0\) |
| \((6a-5)\) | \(31\) | \(1\) | \(I_{1}\) | 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.
Its isogeny class
146692.2-a
consists of curves linked by isogenies of
degree 2.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.