Properties

Base field 5.5.65657.1
Label 5.5.65657.1-45.1-c2
Conductor \((45,-a^{3} + 4 a)\)
Conductor norm \( 45 \)
CM no
base-change no
Q-curve yes
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 \(1\).

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

\( y^2 + \left(-a^{4} + 2 a^{3} + 5 a^{2} - 7 a - 5\right) x y + \left(-a^{4} + 2 a^{3} + 5 a^{2} - 6 a - 4\right) y = x^{3} + \left(-3 a^{4} + 4 a^{3} + 13 a^{2} - 11 a - 9\right) x^{2} + \left(24 a^{4} - 31 a^{3} - 110 a^{2} + 79 a + 89\right) x + 93 a^{4} - 125 a^{3} - 415 a^{2} + 318 a + 327 \)
magma: E := ChangeRing(EllipticCurve([-a^4 + 2*a^3 + 5*a^2 - 7*a - 5, -3*a^4 + 4*a^3 + 13*a^2 - 11*a - 9, -a^4 + 2*a^3 + 5*a^2 - 6*a - 4, 24*a^4 - 31*a^3 - 110*a^2 + 79*a + 89, 93*a^4 - 125*a^3 - 415*a^2 + 318*a + 327]),K);
sage: E = EllipticCurve(K, [-a^4 + 2*a^3 + 5*a^2 - 7*a - 5, -3*a^4 + 4*a^3 + 13*a^2 - 11*a - 9, -a^4 + 2*a^3 + 5*a^2 - 6*a - 4, 24*a^4 - 31*a^3 - 110*a^2 + 79*a + 89, 93*a^4 - 125*a^3 - 415*a^2 + 318*a + 327])
gp (2.8): E = ellinit([-a^4 + 2*a^3 + 5*a^2 - 7*a - 5, -3*a^4 + 4*a^3 + 13*a^2 - 11*a - 9, -a^4 + 2*a^3 + 5*a^2 - 6*a - 4, 24*a^4 - 31*a^3 - 110*a^2 + 79*a + 89, 93*a^4 - 125*a^3 - 415*a^2 + 318*a + 327],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()
\(\mathfrak{D}\) = \((492075,a + 4718,a^{4} - a^{3} - 4 a^{2} + 2 a + 300524,-a^{4} + a^{3} + 5 a^{2} - 3 a + 70684,-a^{4} + 2 a^{3} + 4 a^{2} - 6 a + 484186)\) = \( \left(-a^{4} + a^{3} + 4 a^{2} - 2 a - 2\right)^{9} \cdot \left(-a^{2} + a + 2\right)^{2} \)
magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
\(N(\mathfrak{D})\) = \( 492075 \) = \( 3^{9} \cdot 5^{2} \)
magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
\(j\) = \( \frac{11314724189}{25} a^{4} - \frac{30736776391}{25} a^{3} - \frac{3562664832}{25} a^{2} + \frac{29192288229}{25} a + \frac{6597394298}{25} \)
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: \(\Z/2\Z\)
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: $\left(\frac{3}{4} a^{4} - \frac{5}{4} a^{3} - 2 a^{2} + \frac{11}{4} a - 2 : -\frac{7}{4} a^{4} + \frac{13}{4} a^{3} + \frac{53}{8} a^{2} - \frac{19}{2} a - \frac{45}{8} : 1\right)$
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 ord(\(\mathfrak{N}\)) ord(\(\mathfrak{D}\)) ord\((j)_{-}\)
\( \left(-a^{4} + a^{3} + 4 a^{2} - 2 a - 2\right) \) \(3\) \(2\) \( III^* \) Additive \(2\) \(9\) \(0\)
\( \left(-a^{2} + a + 2\right) \) \(5\) \(2\) \( I_{2} \) Split multiplicative \(1\) \(2\) \(2\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p \) 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 degree2.

Base change

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