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Properties

Base field 5.5.36497.1
Label 5.5.36497.1-27.1-b1
Conductor \((27,-a^{3} + 2 a^{2} + 3 a - 3)\)
Conductor norm \( 27 \)
CM no
base-change no
Q-curve no
Torsion order \( 1 \)
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

\( y^2 + \left(2 a^{4} - 3 a^{3} - 6 a^{2} + 6 a + 1\right) x y + \left(2 a^{4} - 3 a^{3} - 6 a^{2} + 7 a + 2\right) y = x^{3} + \left(-a^{4} + a^{3} + 3 a^{2} - a + 1\right) x^{2} + \left(-5 a^{4} + 6 a^{3} + 16 a^{2} - 9 a - 5\right) x - 3 a^{4} + 3 a^{3} + 10 a^{2} - 5 a - 3 \)
magma: E := ChangeRing(EllipticCurve([2*a^4 - 3*a^3 - 6*a^2 + 6*a + 1, -a^4 + a^3 + 3*a^2 - a + 1, 2*a^4 - 3*a^3 - 6*a^2 + 7*a + 2, -5*a^4 + 6*a^3 + 16*a^2 - 9*a - 5, -3*a^4 + 3*a^3 + 10*a^2 - 5*a - 3]),K);
 
sage: E = EllipticCurve(K, [2*a^4 - 3*a^3 - 6*a^2 + 6*a + 1, -a^4 + a^3 + 3*a^2 - a + 1, 2*a^4 - 3*a^3 - 6*a^2 + 7*a + 2, -5*a^4 + 6*a^3 + 16*a^2 - 9*a - 5, -3*a^4 + 3*a^3 + 10*a^2 - 5*a - 3])
 
gp (2.8): E = ellinit([2*a^4 - 3*a^3 - 6*a^2 + 6*a + 1, -a^4 + a^3 + 3*a^2 - a + 1, 2*a^4 - 3*a^3 - 6*a^2 + 7*a + 2, -5*a^4 + 6*a^3 + 16*a^2 - 9*a - 5, -3*a^4 + 3*a^3 + 10*a^2 - 5*a - 3],K)
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((27,-a^{3} + 2 a^{2} + 3 a - 3)\) = \( \left(a^{2} - 1\right)^{3} \)
magma: Conductor(E);
 
sage: E.conductor()
 
\(N(\mathfrak{N}) \) = \( 27 \) = \( 3^{3} \)
magma: Norm(Conductor(E));
 
sage: E.conductor().norm()
 
\(\mathfrak{D}\) = \((243,a + 31,a^{4} - 2 a^{3} - 3 a^{2} + 5 a + 197,a^{4} - a^{3} - 4 a^{2} + 2 a + 238,a^{2} - a + 223)\) = \( \left(a^{2} - 1\right)^{5} \)
magma: Discriminant(E);
 
sage: E.discriminant()
 
gp (2.8): E.disc
 
\(N(\mathfrak{D})\) = \( 243 \) = \( 3^{5} \)
magma: Norm(Discriminant(E));
 
sage: E.discriminant().norm()
 
gp (2.8): norm(E.disc)
 
\(j\) = \( -180015 a^{4} + 286223 a^{3} + 657933 a^{2} - 631336 a - 439430 \)
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\((j)_{-}\)
\( \left(a^{2} - 1\right) \) \(3\) \(3\) \(IV\) Additive \(1\) \(3\) \(5\) \(0\)

Galois Representations

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

prime Image of Galois Representation
\(3\) 3B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3.
Its isogeny class 27.1-b consists of curves linked by isogenies of degree 3.

Base change

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