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 |
|---|---|---|
| $\left(4 a + 1 : 6 a - 5 : 1\right)$ | $1.1868364784358381697913212721369349892$ | $\infty$ |
| $\left(-a + 1 : -9 a : 1\right)$ | $1.2510842975811834387540865657497474871$ | $\infty$ |
| $\left(3 : 6 a - 5 : 1\right)$ | $0$ | $3$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((372)\) | = | \((-2a+1)^{2}\cdot(2)^{2}\cdot(-6a+1)\cdot(6a-5)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||||
| Conductor norm: | $N(\frak{N})$ | = | \( 138384 \) | = | \(3^{2}\cdot4^{2}\cdot31\cdot31\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||||
| Discriminant: | $\Delta$ | = | $-321408a-147312$ | ||
| Discriminant ideal: | $\frak{D}_{\mathrm{min}} = (\Delta)$ | = | \((-321408a-147312)\) | = | \((-2a+1)^{6}\cdot(2)^{4}\cdot(-6a+1)\cdot(6a-5)^{3}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||||
| Discriminant norm: | $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ | = | \( 172351183104 \) | = | \(3^{6}\cdot4^{4}\cdot31\cdot31^{3}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||||
| j-invariant: | $j$ | = | \( -\frac{185048064}{29791} a + \frac{379461888}{29791} \) | ||
sage: E.j_invariant()
gp: E.j
magma: jInvariant(E);
| |||||
| Endomorphism ring: | $\mathrm{End}(E)$ | = | \(\Z\) | ||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) | ||
sage: E.has_cm(), E.cm_discriminant()
magma: HasComplexMultiplication(E);
| |||||
| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | ||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | \( 2 \) |
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | $r$ | = | \(2\) |
| Regulator: | $\mathrm{Reg}(E/K)$ | ≈ | \( 0.46383067915812151697601387049061572289 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 1.85532271663248606790405548196246289156 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 2.8798055469335009862035990878990204366 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 18 \) = \(2\cdot3\cdot1\cdot3\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(3\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 6.1695287775182601686499565680088645650 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$\displaystyle 6.169528778 \approx L^{(2)}(E/K,1)/2! \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \approx \frac{ 1 \cdot 2.879806 \cdot 1.855323 \cdot 18 } { {3^2 \cdot 1.732051} } \approx 6.169528778$
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 | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((-2a+1)\) | \(3\) | \(2\) | \(I_0^{*}\) | Additive | \(-1\) | \(2\) | \(6\) | \(0\) |
| \((2)\) | \(4\) | \(3\) | \(IV\) | Additive | \(1\) | \(2\) | \(4\) | \(0\) |
| \((-6a+1)\) | \(31\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
| \((6a-5)\) | \(31\) | \(3\) | \(I_{3}\) | Split multiplicative | \(-1\) | \(1\) | \(3\) | \(3\) |
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] |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3.
Its isogeny class
138384.2-b
consists of curves linked by isogenies of
degree 3.
Base change
This elliptic curve is a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.