Properties

Base field 5.5.65657.1
Label 5.5.65657.1-225.1-a4
Conductor \((225,-a^{4} + a^{3} + 6 a^{2} - 4 a - 8)\)
Conductor norm \( 225 \)
CM no
base-change no
Q-curve no
Torsion order \( 1 \)
Rank not available

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Base field 5.5.65657.1

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

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

Weierstrass equation

\( y^2 + \left(-a^{4} + 2 a^{3} + 5 a^{2} - 7 a - 4\right) x y + \left(-2 a^{4} + 3 a^{3} + 9 a^{2} - 9 a - 6\right) y = x^{3} + \left(a^{3} - a^{2} - 4 a\right) x^{2} + \left(3 a^{4} - 5 a^{3} - 12 a^{2} + 16 a + 2\right) x + 9 a^{4} - 20 a^{3} - 32 a^{2} + 65 a + 14 \)
magma: E := ChangeRing(EllipticCurve([-a^4 + 2*a^3 + 5*a^2 - 7*a - 4, a^3 - a^2 - 4*a, -2*a^4 + 3*a^3 + 9*a^2 - 9*a - 6, 3*a^4 - 5*a^3 - 12*a^2 + 16*a + 2, 9*a^4 - 20*a^3 - 32*a^2 + 65*a + 14]),K);
 
sage: E = EllipticCurve(K, [-a^4 + 2*a^3 + 5*a^2 - 7*a - 4, a^3 - a^2 - 4*a, -2*a^4 + 3*a^3 + 9*a^2 - 9*a - 6, 3*a^4 - 5*a^3 - 12*a^2 + 16*a + 2, 9*a^4 - 20*a^3 - 32*a^2 + 65*a + 14])
 
gp (2.8): E = ellinit([-a^4 + 2*a^3 + 5*a^2 - 7*a - 4, a^3 - a^2 - 4*a, -2*a^4 + 3*a^3 + 9*a^2 - 9*a - 6, 3*a^4 - 5*a^3 - 12*a^2 + 16*a + 2, 9*a^4 - 20*a^3 - 32*a^2 + 65*a + 14],K)
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((225,-a^{4} + a^{3} + 6 a^{2} - 4 a - 8)\) = \( \left(-a^{4} + a^{3} + 4 a^{2} - 2 a - 2\right)^{2} \cdot \left(-a^{2} + a + 2\right)^{2} \)
magma: Conductor(E);
 
sage: E.conductor()
 
\(N(\mathfrak{N}) \) = \( 225 \) = \( 3^{2} \cdot 5^{2} \)
magma: Norm(Conductor(E));
 
sage: E.conductor().norm()
 
\(\mathfrak{D}\) = \((35595703125,a + 30731382443,a^{4} - a^{3} - 4 a^{2} + 2 a + 25189127699,-a^{4} + a^{3} + 5 a^{2} - 3 a + 26482872859,-a^{4} + 2 a^{3} + 4 a^{2} - 6 a + 15846556711)\) = \( \left(-a^{4} + a^{3} + 4 a^{2} - 2 a - 2\right)^{6} \cdot \left(-a^{2} + a + 2\right)^{11} \)
magma: Discriminant(E);
 
sage: E.discriminant()
 
gp (2.8): E.disc
 
\(N(\mathfrak{D})\) = \( 35595703125 \) = \( 3^{6} \cdot 5^{11} \)
magma: Norm(Discriminant(E));
 
sage: E.discriminant().norm()
 
gp (2.8): norm(E.disc)
 
\(j\) = \( \frac{81829946}{3125} a^{4} - \frac{143808649}{3125} a^{3} - \frac{300612723}{3125} a^{2} + \frac{393132181}{3125} a + \frac{109010922}{3125} \)
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\((j)_{-}\)
\( \left(-a^{4} + a^{3} + 4 a^{2} - 2 a - 2\right) \) \(3\) \(1\) \(I_{0}^*\) Additive \(-1\) \(2\) \(6\) \(0\)
\( \left(-a^{2} + a + 2\right) \) \(5\) \(2\) \(I_{5}^*\) Additive \(1\) \(2\) \(11\) \(5\)

Galois Representations

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

prime Image of Galois Representation
\(3\) 3B
\(5\) 5B.4.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3, 5 and 15.
Its isogeny class 225.1-a consists of curves linked by isogenies of degrees dividing 15.

Base change

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