Properties

Base field 3.3.1129.1
Label 3.3.1129.1-19.1-b2
Conductor \((19,-a^{2} - a + 4)\)
Conductor norm \( 19 \)
CM no
base-change no
Q-curve no
Torsion order \( 1 \)
Rank not available

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Base field 3.3.1129.1

Generator \(a\), with minimal polynomial \( x^{3} - 7 x - 3 \); class number \(1\).

sage: x = polygen(QQ); K.<a> = NumberField(x^3 - 7*x - 3)
 
gp: K = nfinit(a^3 - 7*a - 3);
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3, -7, 0, 1]);
 

Weierstrass equation

\( y^2 + \left(a + 1\right) y = x^{3} + \left(a^{2} - a - 4\right) x^{2} + \left(-2 a^{2} + 4 a + 6\right) x + a^{2} - 3 a - 3 \)
sage: E = EllipticCurve(K, [0, a^2 - a - 4, a + 1, -2*a^2 + 4*a + 6, a^2 - 3*a - 3])
 
gp: E = ellinit([0, a^2 - a - 4, a + 1, -2*a^2 + 4*a + 6, a^2 - 3*a - 3],K)
 
magma: E := ChangeRing(EllipticCurve([0, a^2 - a - 4, a + 1, -2*a^2 + 4*a + 6, a^2 - 3*a - 3]),K);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((19,-a^{2} - a + 4)\) = \( \left(-a^{2} - a + 4\right) \)
sage: E.conductor()
 
magma: Conductor(E);
 
\(N(\mathfrak{N}) \) = \( 19 \) = \( 19 \)
sage: E.conductor().norm()
 
magma: Norm(Conductor(E));
 
\(\mathfrak{D}\) = \((6859,a + 2008,a^{2} - a + 5879)\) = \( \left(-a^{2} - a + 4\right)^{3} \)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
\(N(\mathfrak{D})\) = \( 6859 \) = \( 19^{3} \)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
\(j\) = \( -\frac{11923591168}{6859} a^{2} + \frac{28591403008}{6859} a + \frac{14920290304}{6859} \)
sage: E.j_invariant()
 
gp: E.j
 
magma: jInvariant(E);
 
\( \text{End} (E) \) = \(\Z\)   (no Complex Multiplication )
sage: E.has_cm(), E.cm_discriminant()
 
magma: HasComplexMultiplication(E);
 
\( \text{ST} (E) \) = $\mathrm{SU}(2)$

Mordell-Weil group

Rank not available.

sage: E.rank()
 
magma: Rank(E);
 

Regulator: not available

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

Torsion subgroup

Structure: Trivial
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 

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\((j)_{-}\)
\( \left(-a^{2} - a + 4\right) \) \(19\) \(1\) \(I_{3}\) Non-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
\(3\) 3B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3.
Its isogeny class 19.1-b 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 not a \(\Q\)-curve.