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Properties

Base field 5.5.24217.1
Label 5.5.24217.1-23.1-a1
Conductor \((23,a^{3} - 3 a)\)
Conductor norm \( 23 \)
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

\( y^2 + \left(-a^{4} + a^{3} + 5 a^{2} - 2 a - 3\right) x y = x^{3} + \left(-3 a^{4} + a^{3} + 15 a^{2} - 2 a - 8\right) x^{2} + \left(-2 a^{4} + 2 a^{3} + 10 a^{2} - 2 a - 6\right) x + 17 a^{4} - 6 a^{3} - 81 a^{2} + 13 a + 45 \)
magma: E := ChangeRing(EllipticCurve([-a^4 + a^3 + 5*a^2 - 2*a - 3, -3*a^4 + a^3 + 15*a^2 - 2*a - 8, 0, -2*a^4 + 2*a^3 + 10*a^2 - 2*a - 6, 17*a^4 - 6*a^3 - 81*a^2 + 13*a + 45]),K);
 
sage: E = EllipticCurve(K, [-a^4 + a^3 + 5*a^2 - 2*a - 3, -3*a^4 + a^3 + 15*a^2 - 2*a - 8, 0, -2*a^4 + 2*a^3 + 10*a^2 - 2*a - 6, 17*a^4 - 6*a^3 - 81*a^2 + 13*a + 45])
 
gp (2.8): E = ellinit([-a^4 + a^3 + 5*a^2 - 2*a - 3, -3*a^4 + a^3 + 15*a^2 - 2*a - 8, 0, -2*a^4 + 2*a^3 + 10*a^2 - 2*a - 6, 17*a^4 - 6*a^3 - 81*a^2 + 13*a + 45],K)
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((23,a^{3} - 3 a)\) = \( \left(a^{3} - 3 a\right) \)
magma: Conductor(E);
 
sage: E.conductor()
 
\(N(\mathfrak{N}) \) = \( 23 \) = \( 23 \)
magma: Norm(Conductor(E));
 
sage: E.conductor().norm()
 
\(\mathfrak{D}\) = \((529,a^{4} - 5 a^{2} + 167,-a^{4} + 5 a^{2} + a + 332,2 a^{4} - a^{3} - 9 a^{2} + 3 a + 423,-a^{4} + a^{3} + 5 a^{2} - 3 a + 431)\) = \( \left(a^{3} - 3 a\right)^{2} \)
magma: Discriminant(E);
 
sage: E.discriminant()
 
gp (2.8): E.disc
 
\(N(\mathfrak{D})\) = \( 529 \) = \( 23^{2} \)
magma: Norm(Discriminant(E));
 
sage: E.discriminant().norm()
 
gp (2.8): norm(E.disc)
 
\(j\) = \( \frac{2370155}{529} a^{4} + \frac{9772390}{529} a^{3} - \frac{30902810}{529} a^{2} - \frac{40834965}{529} a - \frac{14158252}{529} \)
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\((j)_{-}\)
\( \left(a^{3} - 3 a\right) \) \(23\) \(2\) \(I_{2}\) 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 23.1-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.