Properties

Base field 6.6.1397493.1
Label 6.6.1397493.1-37.1-a2
Conductor \((37,a^{4} - 2 a^{3} - 3 a^{2} + 3 a + 2)\)
Conductor norm \( 37 \)
CM no
base-change no
Q-curve no
Torsion order \( 1 \)
Rank not available

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Base field 6.6.1397493.1

Generator \(a\), with minimal polynomial \( x^{6} - 3 x^{5} - 3 x^{4} + 10 x^{3} + 3 x^{2} - 6 x + 1 \); class number \(1\).

sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 3*x^5 - 3*x^4 + 10*x^3 + 3*x^2 - 6*x + 1)
 
gp: K = nfinit(a^6 - 3*a^5 - 3*a^4 + 10*a^3 + 3*a^2 - 6*a + 1);
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -6, 3, 10, -3, -3, 1]);
 

Weierstrass equation

\( y^2 + \left(a^{4} - 2 a^{3} - 3 a^{2} + 4 a + 2\right) x y + \left(a^{5} - 2 a^{4} - 5 a^{3} + 6 a^{2} + 9 a - 2\right) y = x^{3} + \left(a^{4} - 4 a^{3} + 9 a - 1\right) x^{2} + \left(-10 a^{5} + 22 a^{4} + 51 a^{3} - 73 a^{2} - 89 a + 22\right) x - 38 a^{5} + 99 a^{4} + 142 a^{3} - 286 a^{2} - 220 a + 54 \)
sage: E = EllipticCurve(K, [a^4 - 2*a^3 - 3*a^2 + 4*a + 2, a^4 - 4*a^3 + 9*a - 1, a^5 - 2*a^4 - 5*a^3 + 6*a^2 + 9*a - 2, -10*a^5 + 22*a^4 + 51*a^3 - 73*a^2 - 89*a + 22, -38*a^5 + 99*a^4 + 142*a^3 - 286*a^2 - 220*a + 54])
 
gp: E = ellinit([a^4 - 2*a^3 - 3*a^2 + 4*a + 2, a^4 - 4*a^3 + 9*a - 1, a^5 - 2*a^4 - 5*a^3 + 6*a^2 + 9*a - 2, -10*a^5 + 22*a^4 + 51*a^3 - 73*a^2 - 89*a + 22, -38*a^5 + 99*a^4 + 142*a^3 - 286*a^2 - 220*a + 54],K)
 
magma: E := ChangeRing(EllipticCurve([a^4 - 2*a^3 - 3*a^2 + 4*a + 2, a^4 - 4*a^3 + 9*a - 1, a^5 - 2*a^4 - 5*a^3 + 6*a^2 + 9*a - 2, -10*a^5 + 22*a^4 + 51*a^3 - 73*a^2 - 89*a + 22, -38*a^5 + 99*a^4 + 142*a^3 - 286*a^2 - 220*a + 54]),K);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((37,a^{4} - 2 a^{3} - 3 a^{2} + 3 a + 2)\) = \( \left(37, a + 11\right) \)
sage: E.conductor()
 
magma: Conductor(E);
 
\(N(\mathfrak{N}) \) = \( 37 \) = \( 37 \)
sage: E.conductor().norm()
 
magma: Norm(Conductor(E));
 
\(\mathfrak{D}\) = \((50653,a^{3} - 2 a^{2} - 2 a + 44434,a^{5} - 3 a^{4} - 3 a^{3} + 9 a^{2} + 4 a + 8165,a + 10741,a^{4} - 3 a^{3} - a^{2} + 6 a + 17702,a^{2} - a + 7712)\) = \( \left(37, a + 11\right)^{3} \)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
\(N(\mathfrak{D})\) = \( 50653 \) = \( 37^{3} \)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
\(j\) = \( -\frac{13002440326011}{50653} a^{5} + \frac{36425460538638}{50653} a^{4} + \frac{46237058672607}{50653} a^{3} - \frac{120843085936113}{50653} a^{2} - \frac{62990925595713}{50653} a + \frac{65510248079025}{50653} \)
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\((j)_{-}\)
\( \left(37, a + 11\right) \) \(37\) \(3\) \(I_{3}\) 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
\(3\) 3B.1.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3.
Its isogeny class 37.1-a 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.