Properties

Base field 5.5.36497.1
Label 5.5.36497.1-25.1-a2
Conductor \((5,-a^{2} + 2 a + 2)\)
Conductor norm \( 25 \)
CM no
base-change no
Q-curve not determined
Torsion order \( 4 \)
Rank not available

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Base field 5.5.36497.1

Generator \(a\), with minimal polynomial \( x^{5} - 2 x^{4} - 3 x^{3} + 5 x^{2} + x - 1 \); class number \(1\).

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 1, 5, -3, -2, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^5 - 2*x^4 - 3*x^3 + 5*x^2 + x - 1)
gp (2.8): K = nfinit(a^5 - 2*a^4 - 3*a^3 + 5*a^2 + a - 1);

Weierstrass equation

\( y^2 + \left(a^{4} - 2 a^{3} - 3 a^{2} + 5 a + 2\right) x y + \left(a^{4} - 2 a^{3} - 3 a^{2} + 6 a + 2\right) y = x^{3} + \left(a^{4} - a^{3} - 5 a^{2} + 4 a + 4\right) x^{2} + \left(20 a^{4} - 18 a^{3} - 76 a^{2} + 10 a + 20\right) x + 69 a^{4} - 48 a^{3} - 268 a^{2} - 6 a + 54 \)
magma: E := ChangeRing(EllipticCurve([a^4 - 2*a^3 - 3*a^2 + 5*a + 2, a^4 - a^3 - 5*a^2 + 4*a + 4, a^4 - 2*a^3 - 3*a^2 + 6*a + 2, 20*a^4 - 18*a^3 - 76*a^2 + 10*a + 20, 69*a^4 - 48*a^3 - 268*a^2 - 6*a + 54]),K);
sage: E = EllipticCurve(K, [a^4 - 2*a^3 - 3*a^2 + 5*a + 2, a^4 - a^3 - 5*a^2 + 4*a + 4, a^4 - 2*a^3 - 3*a^2 + 6*a + 2, 20*a^4 - 18*a^3 - 76*a^2 + 10*a + 20, 69*a^4 - 48*a^3 - 268*a^2 - 6*a + 54])
gp (2.8): E = ellinit([a^4 - 2*a^3 - 3*a^2 + 5*a + 2, a^4 - a^3 - 5*a^2 + 4*a + 4, a^4 - 2*a^3 - 3*a^2 + 6*a + 2, 20*a^4 - 18*a^3 - 76*a^2 + 10*a + 20, 69*a^4 - 48*a^3 - 268*a^2 - 6*a + 54],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

\(\mathfrak{N} \) = \((5,-a^{2} + 2 a + 2)\) = \( \left(a^{4} - 2 a^{3} - 2 a^{2} + 5 a\right) \)
magma: Conductor(E);
sage: E.conductor()
\(N(\mathfrak{N}) \) = \( 25 \) = \( 25 \)
magma: Norm(Conductor(E));
sage: E.conductor().norm()
\(\mathfrak{D}\) = \((625,625 a,a^{4} - 2 a^{3} - 3 a^{2} + 32 a + 177,a^{4} - a^{3} - 4 a^{2} + 488 a + 145,a^{2} + 238 a + 463)\) = \( \left(a^{4} - 2 a^{3} - 2 a^{2} + 5 a\right)^{4} \)
magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
\(N(\mathfrak{D})\) = \( 390625 \) = \( 25^{4} \)
magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
\(j\) = \( \frac{1549868991089}{625} a^{4} - \frac{2463604541892}{625} a^{3} - \frac{5660845609864}{625} a^{2} + \frac{5425940335044}{625} a + \frac{3777390596864}{625} \)
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 rank and generators

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: \(\Z/2\Z\times\Z/2\Z\)
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]
Generators: $\left(a^{4} - a^{3} - 4 a^{2} : -a^{4} + a^{3} + 4 a^{2} - a - 1 : 1\right)$,$\left(\frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - 3 a - \frac{9}{4} : \frac{1}{2} a^{4} - \frac{3}{2} a^{3} - \frac{15}{8} a^{2} + \frac{17}{4} a + \frac{11}{4} : 1\right)$
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\((j)_{-}\)
\( \left(a^{4} - 2 a^{3} - 2 a^{2} + 5 a\right) \) 25 \(4\) \( I_{4} \) Split multiplicative 1 4 4

Galois Representations

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

prime Image of Galois Representation
\(2\) 2Cs

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2.
Its isogeny class 25.1-a consists of curves linked by isogenies of degrees dividing 4.

Base change

This curve is not the base-change of an elliptic curve defined over \(\Q\). It has not yet been determined whether or not it is a \(\Q\)-curve.