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Properties

Base field 5.5.24217.1
Label 5.5.24217.1-61.2-a1
Conductor \((61,3 a^{4} - a^{3} - 15 a^{2} + 2 a + 7)\)
Conductor norm \( 61 \)
CM no
base-change no
Q-curve no
Torsion order \( 5 \)
Rank not available

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Base field 5.5.24217.1

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

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 3, -1, -5, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^5 - 5*x^3 - x^2 + 3*x + 1)
 
gp (2.8): K = nfinit(a^5 - 5*a^3 - a^2 + 3*a + 1);
 

Weierstrass equation

\( y^2 + \left(2 a^{4} - a^{3} - 9 a^{2} + 3 a + 3\right) x y + \left(a^{2} + a - 2\right) y = x^{3} + \left(2 a^{4} - 10 a^{2} - a + 4\right) x^{2} + \left(4 a^{4} - 2 a^{3} - 20 a^{2} + 6 a + 9\right) x + 2 a^{4} - 4 a^{3} - 7 a^{2} + 9 a + 4 \)
magma: E := ChangeRing(EllipticCurve([2*a^4 - a^3 - 9*a^2 + 3*a + 3, 2*a^4 - 10*a^2 - a + 4, a^2 + a - 2, 4*a^4 - 2*a^3 - 20*a^2 + 6*a + 9, 2*a^4 - 4*a^3 - 7*a^2 + 9*a + 4]),K);
 
sage: E = EllipticCurve(K, [2*a^4 - a^3 - 9*a^2 + 3*a + 3, 2*a^4 - 10*a^2 - a + 4, a^2 + a - 2, 4*a^4 - 2*a^3 - 20*a^2 + 6*a + 9, 2*a^4 - 4*a^3 - 7*a^2 + 9*a + 4])
 
gp (2.8): E = ellinit([2*a^4 - a^3 - 9*a^2 + 3*a + 3, 2*a^4 - 10*a^2 - a + 4, a^2 + a - 2, 4*a^4 - 2*a^3 - 20*a^2 + 6*a + 9, 2*a^4 - 4*a^3 - 7*a^2 + 9*a + 4],K)
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((61,3 a^{4} - a^{3} - 15 a^{2} + 2 a + 7)\) = \( \left(a^{3} + a^{2} - 3 a\right) \)
magma: Conductor(E);
 
sage: E.conductor()
 
\(N(\mathfrak{N}) \) = \( 61 \) = \( 61 \)
magma: Norm(Conductor(E));
 
sage: E.conductor().norm()
 
\(\mathfrak{D}\) = \((61,a^{4} - 5 a^{2} + 49,-a^{4} + 5 a^{2} + a + 50,2 a^{4} - a^{3} - 9 a^{2} + 3 a + 16,-a^{4} + a^{3} + 5 a^{2} - 3 a + 53)\) = \( \left(a^{3} + a^{2} - 3 a\right) \)
magma: Discriminant(E);
 
sage: E.discriminant()
 
gp (2.8): E.disc
 
\(N(\mathfrak{D})\) = \( 61 \) = \( 61 \)
magma: Norm(Discriminant(E));
 
sage: E.discriminant().norm()
 
gp (2.8): norm(E.disc)
 
\(j\) = \( -\frac{115152428}{61} a^{4} - \frac{248328657}{61} a^{3} + \frac{38281223}{61} a^{2} + \frac{189872228}{61} a + \frac{54566560}{61} \)
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: \(\Z/5\Z\)
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: $\left(-a^{4} + 5 a^{2} - 1 : -a^{2} + 2 : 1\right)$
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\((j)_{-}\)
\( \left(a^{3} + a^{2} - 3 a\right) \) \(61\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)

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 61.2-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.