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Properties

Base field 5.5.24217.1
Label 5.5.24217.1-59.1-a1
Conductor \((59,-a^{4} + 4 a^{2} + 1)\)
Conductor norm \( 59 \)
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

\( y^2 + \left(2 a^{4} - 9 a^{2} + 3\right) x y + \left(3 a^{4} - a^{3} - 14 a^{2} + 3 a + 6\right) y = x^{3} + \left(4 a^{4} - 2 a^{3} - 19 a^{2} + 6 a + 7\right) x^{2} + \left(-4 a^{4} + 3 a^{3} + 18 a^{2} - 9 a - 5\right) x + 3 a^{4} - 2 a^{3} - 13 a^{2} + 7 a + 3 \)
magma: E := ChangeRing(EllipticCurve([2*a^4 - 9*a^2 + 3, 4*a^4 - 2*a^3 - 19*a^2 + 6*a + 7, 3*a^4 - a^3 - 14*a^2 + 3*a + 6, -4*a^4 + 3*a^3 + 18*a^2 - 9*a - 5, 3*a^4 - 2*a^3 - 13*a^2 + 7*a + 3]),K);
 
sage: E = EllipticCurve(K, [2*a^4 - 9*a^2 + 3, 4*a^4 - 2*a^3 - 19*a^2 + 6*a + 7, 3*a^4 - a^3 - 14*a^2 + 3*a + 6, -4*a^4 + 3*a^3 + 18*a^2 - 9*a - 5, 3*a^4 - 2*a^3 - 13*a^2 + 7*a + 3])
 
gp (2.8): E = ellinit([2*a^4 - 9*a^2 + 3, 4*a^4 - 2*a^3 - 19*a^2 + 6*a + 7, 3*a^4 - a^3 - 14*a^2 + 3*a + 6, -4*a^4 + 3*a^3 + 18*a^2 - 9*a - 5, 3*a^4 - 2*a^3 - 13*a^2 + 7*a + 3],K)
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((59,-a^{4} + 4 a^{2} + 1)\) = \( \left(a^{4} - 4 a^{2} - 1\right) \)
magma: Conductor(E);
 
sage: E.conductor()
 
\(N(\mathfrak{N}) \) = \( 59 \) = \( 59 \)
magma: Norm(Conductor(E));
 
sage: E.conductor().norm()
 
\(\mathfrak{D}\) = \((3481,a^{4} - 5 a^{2} + 1645,-a^{4} + 5 a^{2} + a + 2704,2 a^{4} - a^{3} - 9 a^{2} + 3 a + 842,-a^{4} + a^{3} + 5 a^{2} - 3 a + 2756)\) = \( \left(a^{4} - 4 a^{2} - 1\right)^{2} \)
magma: Discriminant(E);
 
sage: E.discriminant()
 
gp (2.8): E.disc
 
\(N(\mathfrak{D})\) = \( 3481 \) = \( 59^{2} \)
magma: Norm(Discriminant(E));
 
sage: E.discriminant().norm()
 
gp (2.8): norm(E.disc)
 
\(j\) = \( -\frac{4479742}{3481} a^{4} + \frac{1301710}{3481} a^{3} + \frac{19202738}{3481} a^{2} - \frac{677371}{3481} a - \frac{7584461}{3481} \)
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\((j)_{-}\)
\( \left(a^{4} - 4 a^{2} - 1\right) \) \(59\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)

Galois Representations

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

Isogenies and isogeny class

This curve has no rational isogenies. Its isogeny class 59.1-a consists of this curve only.

Base change

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