Properties

Label 2.0.3.1-171.2-a1
Base field \(\Q(\sqrt{-3}) \)
Conductor \((-15a+6)\)
Conductor norm \( 171 \)
CM no
Base change no
Q-curve no
Torsion order \( 6 \)
Rank \( 0 \)

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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

\({y}^2+{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(4a-9\right){x}-10a+13\)
sage: E = EllipticCurve([K([1,0]),K([-1,0]),K([0,1]),K([-9,4]),K([13,-10])])
 
gp: E = ellinit([Pol(Vecrev([1,0])),Pol(Vecrev([-1,0])),Pol(Vecrev([0,1])),Pol(Vecrev([-9,4])),Pol(Vecrev([13,-10]))], K);
 
magma: E := EllipticCurve([K![1,0],K![-1,0],K![0,1],K![-9,4],K![13,-10]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-15a+6)\) = \((-2a+1)^{2}\cdot(-5a+2)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 171 \) = \(3^{2}\cdot19\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((8667a+4536)\) = \((-2a+1)^{9}\cdot(-5a+2)^{3}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 135005697 \) = \(3^{9}\cdot19^{3}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{29840721}{6859} a - \frac{5426511}{6859} \)
sage: E.j_invariant()
 
gp: E.j
 
magma: jInvariant(E);
 
Endomorphism ring: \(\Z\)
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
sage: E.has_cm(), E.cm_discriminant()
 
magma: HasComplexMultiplication(E);
 
Sato-Tate group: $\mathrm{SU}(2)$

Mordell-Weil group

Rank: \(0\)
Torsion structure: \(\Z/6\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(5 a - 1 : -15 a + 11 : 1\right)$
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: \( 2.71313752790744 \)
Tamagawa product: \( 6 \)  =  \(2\cdot3\)
Torsion order: \(6\)
Leading coefficient: \( 0.522143560695278 \)
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\((j)_{-}\)
\((-2a+1)\) \(3\) \(2\) \(III^{*}\) Additive \(1\) \(2\) \(9\) \(0\)
\((-5a+2)\) \(19\) \(3\) \(I_{3}\) 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
\(2\) 2B
\(3\) 3B.1.1[2]

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3 and 6.
Its isogeny class 171.2-a consists of curves linked by isogenies of degrees dividing 6.

Base change

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