Properties

Label 2.0.3.1-149769.1-CMb2
Base field \(\Q(\sqrt{-3}) \)
Conductor \((-432a+117)\)
Conductor norm \( 149769 \)
CM yes (\(-27\))
Base change no
Q-curve yes
Torsion order \( 1 \)
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(Pol(Vecrev([1, -1, 1])));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, 1]);
 

Weierstrass equation

\({y}^2+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-479a+130\right){x}-3378a+2952\)
sage: E = EllipticCurve([K([0,0]),K([-1,-1]),K([1,0]),K([130,-479]),K([2952,-3378])])
 
gp: E = ellinit([Pol(Vecrev([0,0])),Pol(Vecrev([-1,-1])),Pol(Vecrev([1,0])),Pol(Vecrev([130,-479])),Pol(Vecrev([2952,-3378]))], K);
 
magma: E := EllipticCurve([K![0,0],K![-1,-1],K![1,0],K![130,-479],K![2952,-3378]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-432a+117)\) = \((-2a+1)^{4}\cdot(-7a+1)^{2}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 149769 \) = \(3^{4}\cdot43^{2}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((589680a+206397)\) = \((-2a+1)^{4}\cdot(-7a+1)^{6}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 512030406969 \) = \(3^{4}\cdot43^{6}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -12288000 \)
sage: E.j_invariant()
 
gp: E.j
 
magma: jInvariant(E);
 
Endomorphism ring: \(\Z[(1+\sqrt{-27})/2]\) (complex multiplication)
Geometric endomorphism ring: \(\Z[(1+\sqrt{-27})/2]\)
sage: E.has_cm(), E.cm_discriminant()
 
magma: HasComplexMultiplication(E);
 
Sato-Tate group: $\mathrm{U}(1)$

Mordell-Weil group

Rank: \(2\)
Generators $\left(-6 a - 9 : 24 a - 7 : 1\right)$ $\left(23 a - 7 : -81 a + 137 : 1\right)$
Heights \(0.229084257992616\) \(1.60358980594831\)
Torsion structure: trivial
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 

BSD invariants

Analytic rank: \( 2 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(2\)
Regulator: \( 0.354237281505186 \)
Period: \( 0.713924910415157 \)
Tamagawa product: \( 4 \)  =  \(1\cdot2^{2}\)
Torsion order: \(1\)
Leading coefficient: \( 4.67235844785444 \)
Analytic order of Ш: \( 1 \) (rounded)

Local data at primes of bad reduction

sage: E.local_data()
 
magma: LocalInformation(E);
 
prime Norm Tamagawa number Kodaira symbol Reduction type Root number ord(\(\mathfrak{N}\)) ord(\(\mathfrak{D}\)) ord\((j)_{-}\)
\((-2a+1)\) \(3\) \(1\) \(II\) Additive \(1\) \(4\) \(4\) \(0\)
\((-7a+1)\) \(43\) \(4\) \(I_0^{*}\) Additive \(-1\) \(2\) \(6\) \(0\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.

prime Image of Galois Representation
\(43\) 43Cs.9.1

For all other primes \(p\), the image is a Borel subgroup if \(p=3\), 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 non-trivial cyclic isogenies (excluding endomorphisms) of degree \(d\) for \(d=\) 3.
Its isogeny class 149769.1-CMb consists of curves linked by isogenies of degree 3.

Base change

This curve is not the base change of an elliptic curve defined over \(\Q\). It is a \(\Q\)-curve.