Properties

Label 2.0.3.1-361.2-a4
Base field \(\Q(\sqrt{-3}) \)
Conductor \((19)\)
Conductor norm \( 361 \)
CM no
Base change yes: 19.a2,171.b2
Q-curve yes
Torsion order \( 9 \)
Rank \( 1 \)

Related objects

Downloads

Learn more about

Show commands for: 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

\({y}^2+{y}={x}^{3}-a{x}^{2}+\left(-9a+9\right){x}-15\)
sage: E = EllipticCurve([K([0,0]),K([0,-1]),K([1,0]),K([9,-9]),K([-15,0])])
 
gp: E = ellinit([Pol(Vecrev([0,0])),Pol(Vecrev([0,-1])),Pol(Vecrev([1,0])),Pol(Vecrev([9,-9])),Pol(Vecrev([-15,0]))], K);
 
magma: E := EllipticCurve([K![0,0],K![0,-1],K![1,0],K![9,-9],K![-15,0]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((19)\) = \((-5a+3)\cdot(-5a+2)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 361 \) = \(19\cdot19\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-6859)\) = \((-5a+3)^{3}\cdot(-5a+2)^{3}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 47045881 \) = \(19^{3}\cdot19^{3}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{89915392}{6859} \)
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: \(1\)
Generator $\left(2 a : a - 1 : 1\right)$
Height \(0.677900606477085\)
Torsion structure: \(\Z/3\Z\times\Z/3\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generators: $\left(-5 a : -10 : 1\right)$ $\left(3 a - 1 : a - 3 : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 1 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(1\)
Regulator: \( 0.677900606477085 \)
Period: \( 2.80592702532153 \)
Tamagawa product: \( 9 \)  =  \(3\cdot3\)
Torsion order: \(9\)
Leading coefficient: \( 0.488089257193140 \)
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\((j)_{-}\)
\((-5a+3)\) \(19\) \(3\) \(I_{3}\) Split multiplicative \(-1\) \(1\) \(3\) \(3\)
\((-5a+2)\) \(19\) \(3\) \(I_{3}\) 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
\(3\) 3Cs.1.1[2]

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3.
Its isogeny class 361.2-a consists of curves linked by isogenies of degrees dividing 9.

Base change

This curve is the base change of elliptic curves 19.a2, 171.b2, defined over \(\Q\), so it is also a \(\Q\)-curve.