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Properties

Base field 5.5.24217.1
Label 5.5.24217.1-61.2-a2
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 \( 1 \)
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

\( y^2 + \left(a^{4} - 4 a^{2} + a\right) x y + \left(a^{4} - 5 a^{2} + 3\right) y = x^{3} + \left(a^{4} - 2 a^{3} - 4 a^{2} + 7 a + 2\right) x^{2} + \left(-33 a^{4} + 86 a^{3} + 225 a^{2} - 86 a - 137\right) x - 317 a^{4} + 1250 a^{3} + 1079 a^{2} - 939 a - 684 \)
magma: E := ChangeRing(EllipticCurve([a^4 - 4*a^2 + a, a^4 - 2*a^3 - 4*a^2 + 7*a + 2, a^4 - 5*a^2 + 3, -33*a^4 + 86*a^3 + 225*a^2 - 86*a - 137, -317*a^4 + 1250*a^3 + 1079*a^2 - 939*a - 684]),K);
 
sage: E = EllipticCurve(K, [a^4 - 4*a^2 + a, a^4 - 2*a^3 - 4*a^2 + 7*a + 2, a^4 - 5*a^2 + 3, -33*a^4 + 86*a^3 + 225*a^2 - 86*a - 137, -317*a^4 + 1250*a^3 + 1079*a^2 - 939*a - 684])
 
gp (2.8): E = ellinit([a^4 - 4*a^2 + a, a^4 - 2*a^3 - 4*a^2 + 7*a + 2, a^4 - 5*a^2 + 3, -33*a^4 + 86*a^3 + 225*a^2 - 86*a - 137, -317*a^4 + 1250*a^3 + 1079*a^2 - 939*a - 684],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()
 
\(\mathfrak{D}\) = \((226981,3721 a^{4} - 18605 a^{2} + 182329,503 a^{4} - 2515 a^{2} + a + 146441,2062 a^{4} - a^{3} - 10309 a^{2} + 3 a + 110289,214 a^{4} + a^{3} - 1070 a^{2} - 3 a + 164735)\) = \( \left(a^{3} + a^{2} - 3 a\right)^{5} \)
magma: Discriminant(E);
 
sage: E.discriminant()
 
gp (2.8): E.disc
 
\(N(\mathfrak{D})\) = \( 844596301 \) = \( 61^{5} \)
magma: Norm(Discriminant(E));
 
sage: E.discriminant().norm()
 
gp (2.8): norm(E.disc)
 
\(j\) = \( \frac{947582806658950190622182}{226981} a^{4} - \frac{684924891007902588272712}{226981} a^{3} - \frac{4242841621501708043901758}{226981} a^{2} + \frac{2119197020585266574178655}{226981} a + \frac{1310965894310847398348250}{226981} \)
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: Trivial
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]
 

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\((j)_{-}\)
\( \left(a^{3} + a^{2} - 3 a\right) \) \(61\) \(5\) \(I_{5}\) 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.1.2

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.