Properties

Label 2.0.3.1-138384.2-b2
Base field \(\Q(\sqrt{-3}) \)
Conductor norm \( 138384 \)
CM no
Base change no
Q-curve yes
Torsion order \( 3 \)
Rank \( 2 \)

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Base field \(\Q(\sqrt{-3}) \)

Generator \(a\), with minimal polynomial \( x^{2} - x + 1 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -1, 1]))
 
gp: K = nfinit(Polrev([1, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, 1]);
 

Weierstrass equation

\({y}^2={x}^{3}+\left(-36a+3\right){x}+84a-47\)
sage: E = EllipticCurve([K([0,0]),K([0,0]),K([0,0]),K([3,-36]),K([-47,84])])
 
gp: E = ellinit([Polrev([0,0]),Polrev([0,0]),Polrev([0,0]),Polrev([3,-36]),Polrev([-47,84])], K);
 
magma: E := EllipticCurve([K![0,0],K![0,0],K![0,0],K![3,-36],K![-47,84]]);
 

This is a global minimal model.

sage: E.is_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

sage: E.local_data()
 
magma: LocalInformation(E);
 

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.