Properties

Base field 5.5.65657.1
Label 5.5.65657.1-37.1-a2
Conductor \((37,a^{3} - 2 a^{2} - 2 a + 2)\)
Conductor norm \( 37 \)
CM no
base-change no
Q-curve no
Torsion order \( 5 \)
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: K = nfinit(a^5 - a^4 - 5*a^3 + 2*a^2 + 5*a + 1);
 

Weierstrass equation

\( y^2 + \left(-a^{4} + a^{3} + 5 a^{2} - 3 a - 4\right) y = x^{3} + \left(a^{4} - 2 a^{3} - 4 a^{2} + 7 a + 3\right) x^{2} + \left(49 a^{4} - 88 a^{3} - 178 a^{2} + 239 a + 64\right) x + 162 a^{4} - 287 a^{3} - 588 a^{2} + 779 a + 208 \)
magma: E := ChangeRing(EllipticCurve([0, a^4 - 2*a^3 - 4*a^2 + 7*a + 3, -a^4 + a^3 + 5*a^2 - 3*a - 4, 49*a^4 - 88*a^3 - 178*a^2 + 239*a + 64, 162*a^4 - 287*a^3 - 588*a^2 + 779*a + 208]),K);
 
sage: E = EllipticCurve(K, [0, a^4 - 2*a^3 - 4*a^2 + 7*a + 3, -a^4 + a^3 + 5*a^2 - 3*a - 4, 49*a^4 - 88*a^3 - 178*a^2 + 239*a + 64, 162*a^4 - 287*a^3 - 588*a^2 + 779*a + 208])
 
gp: E = ellinit([0, a^4 - 2*a^3 - 4*a^2 + 7*a + 3, -a^4 + a^3 + 5*a^2 - 3*a - 4, 49*a^4 - 88*a^3 - 178*a^2 + 239*a + 64, 162*a^4 - 287*a^3 - 588*a^2 + 779*a + 208],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()
 
\(\mathfrak{D}\) = \((69343957,a + 55930313,a^{4} - a^{3} - 4 a^{2} + 2 a + 6732306,-a^{4} + a^{3} + 5 a^{2} - 3 a + 25079061,-a^{4} + 2 a^{3} + 4 a^{2} - 6 a + 42266809)\) = \( \left(a^{4} - a^{3} - 3 a^{2} + 3 a + 2\right)^{5} \)
magma: Discriminant(E);
 
sage: E.discriminant()
 
gp: E.disc
 
\(N(\mathfrak{D})\) = \( 69343957 \) = \( 37^{5} \)
magma: Norm(Discriminant(E));
 
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
\(j\) = \( -\frac{2507214671425536}{69343957} a^{4} + \frac{3099587870568448}{69343957} a^{3} + \frac{11795529019346944}{69343957} a^{2} - \frac{7777196659007488}{69343957} a - \frac{10702583291781120}{69343957} \)
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 not available.

magma: Rank(E);
 
sage: E.rank()
 

Regulator: not available

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

Torsion subgroup

Structure: \(\Z/5\Z\)
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
Generator: $\left(-5 a^{4} + 9 a^{3} + 19 a^{2} - 26 a - 7 : 11 a^{4} - 19 a^{3} - 42 a^{2} + 54 a + 16 : 1\right)$
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\((j)_{-}\)
\( \left(a^{4} - a^{3} - 3 a^{2} + 3 a + 2\right) \) \(37\) \(5\) \(I_{5}\) Split multiplicative \(-1\) \(1\) \(5\) \(5\)

Galois Representations

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

prime Image of Galois Representation
\(5\) 5B.1.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 5.
Its isogeny class 37.1-a consists of curves linked by isogenies of degree 5.

Base change

This curve is not the base-change of an elliptic curve defined over \(\Q\). It is not a \(\Q\)-curve.