Properties

Base field 5.5.65657.1
Label 5.5.65657.1-45.1-c1
Conductor \((45,-a^{3} + 4 a)\)
Conductor norm \( 45 \)
CM no
base-change no
Q-curve no
Torsion order \( 2 \)
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 \(1\).

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);
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 5, 2, -5, -1, 1]);
 

Weierstrass equation

\( y^2 + \left(-a^{4} + 2 a^{3} + 5 a^{2} - 6 a - 5\right) x y + \left(a^{3} - 3 a\right) y = x^{3} + \left(a^{4} - 5 a^{2} - 2 a + 2\right) x^{2} + \left(2 a^{4} - 4 a^{3} - 11 a^{2} + 13 a + 13\right) x + 2 a^{4} - 3 a^{3} - 9 a^{2} + 5 a + 6 \)
sage: E = EllipticCurve(K, [-a^4 + 2*a^3 + 5*a^2 - 6*a - 5, a^4 - 5*a^2 - 2*a + 2, a^3 - 3*a, 2*a^4 - 4*a^3 - 11*a^2 + 13*a + 13, 2*a^4 - 3*a^3 - 9*a^2 + 5*a + 6])
 
gp: E = ellinit([-a^4 + 2*a^3 + 5*a^2 - 6*a - 5, a^4 - 5*a^2 - 2*a + 2, a^3 - 3*a, 2*a^4 - 4*a^3 - 11*a^2 + 13*a + 13, 2*a^4 - 3*a^3 - 9*a^2 + 5*a + 6],K)
 
magma: E := ChangeRing(EllipticCurve([-a^4 + 2*a^3 + 5*a^2 - 6*a - 5, a^4 - 5*a^2 - 2*a + 2, a^3 - 3*a, 2*a^4 - 4*a^3 - 11*a^2 + 13*a + 13, 2*a^4 - 3*a^3 - 9*a^2 + 5*a + 6]),K);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((45,-a^{3} + 4 a)\) = \( \left(-a^{4} + a^{3} + 4 a^{2} - 2 a - 2\right)^{2} \cdot \left(-a^{2} + a + 2\right) \)
sage: E.conductor()
 
magma: Conductor(E);
 
\(N(\mathfrak{N}) \) = \( 45 \) = \( 3^{2} \cdot 5 \)
sage: E.conductor().norm()
 
magma: Norm(Conductor(E));
 
\(\mathfrak{D}\) = \((98415,a + 4718,a^{4} - a^{3} - 4 a^{2} + 2 a + 5279,-a^{4} + a^{3} + 5 a^{2} - 3 a + 70684,-a^{4} + 2 a^{3} + 4 a^{2} - 6 a + 90526)\) = \( \left(-a^{4} + a^{3} + 4 a^{2} - 2 a - 2\right)^{9} \cdot \left(-a^{2} + a + 2\right) \)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
\(N(\mathfrak{D})\) = \( 98415 \) = \( 3^{9} \cdot 5 \)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
\(j\) = \( \frac{34992}{5} a^{4} - \frac{81598}{5} a^{3} - \frac{54171}{5} a^{2} + \frac{89407}{5} a + \frac{46644}{5} \)
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: \(\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Generator: $\left(-a^{4} + a^{3} + 4 a^{2} - a - 2 : -a^{4} + 5 a^{2} + a - 3 : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(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\((j)_{-}\)
\( \left(-a^{4} + a^{3} + 4 a^{2} - 2 a - 2\right) \) \(3\) \(2\) \(III^*\) Additive \(1\) \(2\) \(9\) \(0\)
\( \left(-a^{2} + a + 2\right) \) \(5\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.

prime Image of Galois Representation
\(2\) 2B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2.
Its isogeny class 45.1-c consists of curves linked by isogenies of degree 2.

Base change

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