Properties

Base field 5.5.65657.1
Label 5.5.65657.1-37.1-b1
Conductor \((37,a^{3} - 2 a^{2} - 2 a + 2)\)
Conductor norm \( 37 \)
CM no
base-change no
Q-curve not determined
Torsion order \( 1 \)
Rank not available

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Base field 5.5.65657.1

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

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 5, 2, -5, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^5 - x^4 - 5*x^3 + 2*x^2 + 5*x + 1)
gp (2.8): K = nfinit(a^5 - a^4 - 5*a^3 + 2*a^2 + 5*a + 1);

Weierstrass equation

\( y^2 + \left(-2 a^{4} + 3 a^{3} + 9 a^{2} - 8 a - 7\right) y = x^{3} + \left(-a^{3} + 4 a + 2\right) x^{2} + \left(-20 a^{4} + 24 a^{3} + 92 a^{2} - 55 a - 81\right) x + 136 a^{4} - 170 a^{3} - 637 a^{2} + 426 a + 579 \)
magma: E := ChangeRing(EllipticCurve([0, -a^3 + 4*a + 2, -2*a^4 + 3*a^3 + 9*a^2 - 8*a - 7, -20*a^4 + 24*a^3 + 92*a^2 - 55*a - 81, 136*a^4 - 170*a^3 - 637*a^2 + 426*a + 579]),K);
sage: E = EllipticCurve(K, [0, -a^3 + 4*a + 2, -2*a^4 + 3*a^3 + 9*a^2 - 8*a - 7, -20*a^4 + 24*a^3 + 92*a^2 - 55*a - 81, 136*a^4 - 170*a^3 - 637*a^2 + 426*a + 579])
gp (2.8): E = ellinit([0, -a^3 + 4*a + 2, -2*a^4 + 3*a^3 + 9*a^2 - 8*a - 7, -20*a^4 + 24*a^3 + 92*a^2 - 55*a - 81, 136*a^4 - 170*a^3 - 637*a^2 + 426*a + 579],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

\(\mathfrak{N} \) = \((37,a^{3} - 2 a^{2} - 2 a + 2)\) = \( \left(a^{4} - a^{3} - 3 a^{2} + 3 a + 2\right) \)
magma: Conductor(E);
sage: E.conductor()
\(N(\mathfrak{N}) \) = \( 37 \) = \( 37 \)
magma: Norm(Conductor(E));
sage: E.conductor().norm()
\(\mathfrak{D}\) = \((50653,a + 9401,a^{4} - a^{3} - 4 a^{2} + 2 a + 46110,-a^{4} + a^{3} + 5 a^{2} - 3 a + 5826,-a^{4} + 2 a^{3} + 4 a^{2} - 6 a + 22207)\) = \( \left(a^{4} - a^{3} - 3 a^{2} + 3 a + 2\right)^{3} \)
magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
\(N(\mathfrak{D})\) = \( 50653 \) = \( 37^{3} \)
magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
\(j\) = \( -\frac{18254600376320}{50653} a^{4} + \frac{49539534499840}{50653} a^{3} + \frac{6263292940288}{50653} a^{2} - \frac{47056864038912}{50653} a - \frac{10663276101632}{50653} \)
magma: jInvariant(E);
sage: E.j_invariant()
gp (2.8): 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 rank and generators

Rank not available.
magma: Rank(E);
sage: E.rank()
magma: Generators(E); // includes torsion
sage: E.gens()

Regulator: not available

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

Torsion subgroup

Structure: Trivial
magma: TorsionSubgroup(E);
sage: E.torsion_subgroup().gens()
gp (2.8): elltors(E)[2]
magma: Order(TorsionSubgroup(E));
sage: E.torsion_order()
gp (2.8): elltors(E)[1]

Local data at primes of bad reduction

magma: LocalInformation(E);
sage: E.local_data()
prime Norm Tamagawa number Kodaira symbol Reduction type ord(\(\mathfrak{N}\)) ord(\(\mathfrak{D}\)) ord\((j)_{-}\)
\( \left(a^{4} - a^{3} - 3 a^{2} + 3 a + 2\right) \) 37 \(1\) \( I_{3} \) Non-split multiplicative 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
\(3\) 3B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3.
Its isogeny class 37.1-b consists of curves linked by isogenies of degree3.

Base change

This curve is not the base-change of an elliptic curve defined over \(\Q\). It has not yet been determined whether or not it is a \(\Q\)-curve.