Properties

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

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

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

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^2 - x + 1)
 
gp: K = nfinit(a^2 - a + 1);
 

Weierstrass equation

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

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((19)\) = \( \left(-5 a + 3\right) \cdot \left(-5 a + 2\right) \)
magma: Conductor(E);
 
sage: E.conductor()
 
\(N(\mathfrak{N}) \) = \( 361 \) = \( 19^{2} \)
magma: Norm(Conductor(E));
 
sage: E.conductor().norm()
 
\(\mathfrak{D}\) = \((6859)\) = \( \left(-5 a + 3\right)^{3} \cdot \left(-5 a + 2\right)^{3} \)
magma: Discriminant(E);
 
sage: E.discriminant()
 
gp: E.disc
 
\(N(\mathfrak{D})\) = \( 47045881 \) = \( 19^{6} \)
magma: Norm(Discriminant(E));
 
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
\(j\) = \( -\frac{89915392}{6859} \)
magma: jInvariant(E);
 
sage: E.j_invariant()
 
gp: E.j
 
\( \text{End} (E) \) = \(\Z\)   (no Complex Multiplication )
magma: HasComplexMultiplication(E);
 
sage: E.has_cm(), E.cm_discriminant()
 
\( \text{ST} (E) \) = $\mathrm{SU}(2)$

Mordell-Weil group

Rank: \( 1 \)

magma: Rank(E);
 
sage: E.rank()
 

Generator: $\left(2 a : a - 1 : 1\right)$

magma: gens := [P:P in Generators(E)|Order(P) eq 0]; gens;
 
sage: gens = E.gens(); gens
 

Height: 0.6779006064770846

magma: [Height(P):P in gens];
 
sage: [P.height() for P in gens]
 

Regulator: 0.677900606477

magma: Regulator(gens);
 
sage: E.regulator_of_points(gens)
 

Torsion subgroup

Structure: \(\Z/3\Z\times\Z/3\Z\)
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
Generators: $\left(-5 a : -10 : 1\right)$,$\left(3 a - 1 : a - 3 : 1\right)$
magma: [piT(P) : P in Generators(T)];
 
sage: T.gens()
 
gp: T[3]
 

Local data at primes of bad reduction

magma: LocalInformation(E);
 
sage: E.local_data()
 
prime Norm Tamagawa number Kodaira symbol Reduction type Root number ord(\(\mathfrak{N}\)) ord(\(\mathfrak{D}\)) ord\((j)_{-}\)
\( \left(-5 a + 3\right) \) \(19\) \(3\) \(I_{3}\) Split multiplicative \(-1\) \(1\) \(3\) \(3\)
\( \left(-5 a + 2\right) \) \(19\) \(3\) \(I_{3}\) Split multiplicative \(-1\) \(1\) \(3\) \(3\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p \) 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.