Properties

Base field \(\Q(\zeta_{11})^+\)
Label 5.5.14641.1-11.1-a1
Conductor \((11,a^{4} + a^{3} - 4 a^{2} - 3 a + 2)\)
Conductor norm \( 11 \)
CM no
base-change yes: 11.a1
Q-curve yes
Torsion order \( 1 \)
Rank not available

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

Base field \(\Q(\zeta_{11})^+\)

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

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

Weierstrass equation

\( y^2 + y = x^{3} - x^{2} - 7820 x - 263580 \)
magma: E := ChangeRing(EllipticCurve([0, -1, 1, -7820, -263580]),K);
 
sage: E = EllipticCurve(K, [0, -1, 1, -7820, -263580])
 
gp: E = ellinit([0, -1, 1, -7820, -263580],K)
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((11,a^{4} + a^{3} - 4 a^{2} - 3 a + 2)\) = \( \left(a^{2} + a - 2\right) \)
magma: Conductor(E);
 
sage: E.conductor()
 
\(N(\mathfrak{N}) \) = \( 11 \) = \( 11 \)
magma: Norm(Conductor(E));
 
sage: E.conductor().norm()
 
\(\mathfrak{D}\) = \((11,11 a,11 a^{2} - 22,11 a^{3} - 33 a,11 a^{4} - 44 a^{2} + 22)\) = \( \left(a^{2} + a - 2\right)^{5} \)
magma: Discriminant(E);
 
sage: E.discriminant()
 
gp: E.disc
 
\(N(\mathfrak{D})\) = \( 161051 \) = \( 11^{5} \)
magma: Norm(Discriminant(E));
 
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
\(j\) = \( -\frac{52893159101157376}{11} \)
magma: jInvariant(E);
 
sage: E.j_invariant()
 
gp: 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()
 

Regulator: not available

magma: gens := [P:P in Generators(E)|Order(P) eq 0]; gens;
 
sage: gens = E.gens(); gens
 
magma: Regulator(gens);
 
sage: E.regulator_of_points(gens)
 

Torsion subgroup

Structure: Trivial
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 

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\((j)_{-}\)
\( \left(a^{2} + a - 2\right) \) \(11\) \(5\) \(I_{5}\) Split multiplicative \(-1\) \(1\) \(5\) \(5\)

Galois Representations

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

prime Image of Galois Representation
\(5\) 5B.1.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 5 and 25.
Its isogeny class 11.1-a consists of curves linked by isogenies of degrees dividing 25.

Base change

This curve is the base-change of elliptic curves 11.a1, defined over \(\Q\), so it is also a \(\Q\)-curve.