Properties

Label 2.0.3.1-21609.3-d1
Base field \(\Q(\sqrt{-3}) \)
Conductor \((147)\)
Conductor norm \( 21609 \)
CM yes (\(-7\))
Base change no
Q-curve yes
Torsion order \( 2 \)
Rank \( 0 \)

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Show commands: Magma / Pari/GP / SageMath

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

\({y}^2+{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(-53a+33\right){x}-79a+162\)
sage: E = EllipticCurve([K([1,0]),K([-1,0]),K([0,1]),K([33,-53]),K([162,-79])])
 
gp: E = ellinit([Pol(Vecrev([1,0])),Pol(Vecrev([-1,0])),Pol(Vecrev([0,1])),Pol(Vecrev([33,-53])),Pol(Vecrev([162,-79]))], K);
 
magma: E := EllipticCurve([K![1,0],K![-1,0],K![0,1],K![33,-53],K![162,-79]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((147)\) = \((-2a+1)^{2}\cdot(-3a+1)^{2}\cdot(3a-2)^{2}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 21609 \) = \(3^{2}\cdot7^{2}\cdot7^{2}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((3333960a-2991303)\) = \((-2a+1)^{6}\cdot(-3a+1)^{9}\cdot(3a-2)^{3}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 10090298369529 \) = \(3^{6}\cdot7^{9}\cdot7^{3}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -3375 \)
sage: E.j_invariant()
 
gp: E.j
 
magma: jInvariant(E);
 
Endomorphism ring: \(\Z\)
Geometric endomorphism ring: \(\Z[(1+\sqrt{-7})/2]\) (potential complex multiplication)
sage: E.has_cm(), E.cm_discriminant()
 
magma: HasComplexMultiplication(E);
 
Sato-Tate group: $N(\mathrm{U}(1))$

Mordell-Weil group

Rank: \(0\)
Torsion structure: \(\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(7 a + 2 : -4 a - 1 : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 0 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(0\)
Regulator: \( 1 \)
Period: \( 1.07897936213582 \)
Tamagawa product: \( 8 \)  =  \(2\cdot2\cdot2\)
Torsion order: \(2\)
Leading coefficient: \( 2.49179610071666 \)
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\((j)_{-}\)
\((-2a+1)\) \(3\) \(2\) \(I_0^{*}\) Additive \(-1\) \(2\) \(6\) \(0\)
\((-3a+1)\) \(7\) \(2\) \(III^{*}\) Additive \(-1\) \(2\) \(9\) \(0\)
\((3a-2)\) \(7\) \(2\) \(III\) Additive \(-1\) \(2\) \(3\) \(0\)

Galois Representations

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

prime Image of Galois Representation
\(7\) 7B.6.2

For all other primes \(p\), the image is the normalizer of a split Cartan subgroup if \(\left(\frac{ -7 }{p}\right)=+1\) or the normalizer of a nonsplit Cartan subgroup if \(\left(\frac{ -7 }{p}\right)=-1\).

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 7 and 14.
Its isogeny class 21609.3-d consists of curves linked by isogenies of degrees dividing 14.

Base change

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