Properties

Base field 5.5.65657.1
Label 5.5.65657.1-45.1-b4
Conductor \((45,-a^{3} + 4 a)\)
Conductor norm \( 45 \)
CM no
base-change no
Q-curve no
Torsion order \( 3 \)
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

\( y^2 + \left(a^{2} - a - 1\right) x y + \left(-a^{4} + 2 a^{3} + 4 a^{2} - 6 a - 3\right) y = x^{3} + \left(-a^{2} + a + 2\right) x^{2} + \left(4 a^{4} - 5 a^{3} - 19 a^{2} + 13 a + 17\right) x + 8 a^{4} - 10 a^{3} - 37 a^{2} + 25 a + 32 \)
magma: E := ChangeRing(EllipticCurve([a^2 - a - 1, -a^2 + a + 2, -a^4 + 2*a^3 + 4*a^2 - 6*a - 3, 4*a^4 - 5*a^3 - 19*a^2 + 13*a + 17, 8*a^4 - 10*a^3 - 37*a^2 + 25*a + 32]),K);
sage: E = EllipticCurve(K, [a^2 - a - 1, -a^2 + a + 2, -a^4 + 2*a^3 + 4*a^2 - 6*a - 3, 4*a^4 - 5*a^3 - 19*a^2 + 13*a + 17, 8*a^4 - 10*a^3 - 37*a^2 + 25*a + 32])
gp (2.8): E = ellinit([a^2 - a - 1, -a^2 + a + 2, -a^4 + 2*a^3 + 4*a^2 - 6*a - 3, 4*a^4 - 5*a^3 - 19*a^2 + 13*a + 17, 8*a^4 - 10*a^3 - 37*a^2 + 25*a + 32],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

\(\mathfrak{N} \) = \((45,-a^{3} + 4 a)\) = \( \left(-a^{4} + a^{3} + 4 a^{2} - 2 a - 2\right)^{2} \cdot \left(-a^{2} + a + 2\right) \)
magma: Conductor(E);
sage: E.conductor()
\(N(\mathfrak{N}) \) = \( 45 \) = \( 3^{2} \cdot 5 \)
magma: Norm(Conductor(E));
sage: E.conductor().norm()
\(\mathfrak{D}\) = \((2278125,a + 1754318,a^{4} - a^{3} - 4 a^{2} + 2 a + 2177699,-a^{4} + a^{3} + 5 a^{2} - 3 a + 1947859,-a^{4} + 2 a^{3} + 4 a^{2} - 6 a + 2197336)\) = \( \left(-a^{4} + a^{3} + 4 a^{2} - 2 a - 2\right)^{6} \cdot \left(-a^{2} + a + 2\right)^{5} \)
magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
\(N(\mathfrak{D})\) = \( 2278125 \) = \( 3^{6} \cdot 5^{5} \)
magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
\(j\) = \( \frac{81829946}{3125} a^{4} - \frac{143808649}{3125} a^{3} - \frac{300612723}{3125} a^{2} + \frac{393132181}{3125} a + \frac{109010922}{3125} \)
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 group

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: \(\Z/3\Z\)
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]
Generator: $\left(0 : 2 a^{4} - 3 a^{3} - 9 a^{2} + 8 a + 8 : 1\right)$
magma: [f(P): P in Generators(T)] where T,f:=TorsionSubgroup(E);
sage: E.torsion_subgroup().gens()
gp (2.8): elltors(E)[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\((j)_{-}\)
\( \left(-a^{4} + a^{3} + 4 a^{2} - 2 a - 2\right) \) \(3\) \(1\) \(I_{0}^*\) Additive \(-1\) \(2\) \(6\) \(0\)
\( \left(-a^{2} + a + 2\right) \) \(5\) \(1\) \(I_{5}\) Non-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
\(3\) 3B.1.1
\(5\) 5B.4.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3, 5 and 15.
Its isogeny class 45.1-b consists of curves linked by isogenies of degrees dividing 15.

Base change

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