Properties

Base field \(\Q(\zeta_{11})^+\)
Label 5.5.14641.1-43.2-a2
Conductor \((43,a^{4} - 2 a^{2} - a - 1)\)
Conductor norm \( 43 \)
CM no
base-change no
Q-curve no
Torsion order \( 1 \)
Rank not available

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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

\( y^2 + \left(a^{4} + a^{3} - 3 a^{2} - 3 a + 1\right) x y + \left(a^{4} + a^{3} - 3 a^{2} - 3 a + 1\right) y = x^{3} + \left(-a^{4} + 4 a^{2} + a - 3\right) x^{2} + \left(-30 a^{4} + 30 a^{3} + 25 a^{2} + 160 a - 215\right) x - 226 a^{4} + 563 a^{3} - 458 a^{2} + 913 a - 1012 \)
magma: E := ChangeRing(EllipticCurve([a^4 + a^3 - 3*a^2 - 3*a + 1, -a^4 + 4*a^2 + a - 3, a^4 + a^3 - 3*a^2 - 3*a + 1, -30*a^4 + 30*a^3 + 25*a^2 + 160*a - 215, -226*a^4 + 563*a^3 - 458*a^2 + 913*a - 1012]),K);
 
sage: E = EllipticCurve(K, [a^4 + a^3 - 3*a^2 - 3*a + 1, -a^4 + 4*a^2 + a - 3, a^4 + a^3 - 3*a^2 - 3*a + 1, -30*a^4 + 30*a^3 + 25*a^2 + 160*a - 215, -226*a^4 + 563*a^3 - 458*a^2 + 913*a - 1012])
 
gp: E = ellinit([a^4 + a^3 - 3*a^2 - 3*a + 1, -a^4 + 4*a^2 + a - 3, a^4 + a^3 - 3*a^2 - 3*a + 1, -30*a^4 + 30*a^3 + 25*a^2 + 160*a - 215, -226*a^4 + 563*a^3 - 458*a^2 + 913*a - 1012],K)
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((43,a^{4} - 2 a^{2} - a - 1)\) = \( \left(-a^{4} + 2 a^{2} + a + 1\right) \)
magma: Conductor(E);
 
sage: E.conductor()
 
\(N(\mathfrak{N}) \) = \( 43 \) = \( 43 \)
magma: Norm(Conductor(E));
 
sage: E.conductor().norm()
 
\(\mathfrak{D}\) = \((271818611107,a + 150821116011,a^{2} + 242741157992,a^{3} - 3 a + 73434615221,a^{4} - 4 a^{2} + 250768899967)\) = \( \left(-a^{4} + 2 a^{2} + a + 1\right)^{7} \)
magma: Discriminant(E);
 
sage: E.discriminant()
 
gp: E.disc
 
\(N(\mathfrak{D})\) = \( 271818611107 \) = \( 43^{7} \)
magma: Norm(Discriminant(E));
 
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
\(j\) = \( -\frac{70548898331353450135096255010}{271818611107} a^{4} + \frac{189247840974084378816217200685}{271818611107} a^{3} - \frac{842216448583260989645063883}{6321363049} a^{2} - \frac{150714011772341855278459782553}{271818611107} a + \frac{41930794528416841925025385323}{271818611107} \)
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\((j)_{-}\)
\( \left(-a^{4} + 2 a^{2} + a + 1\right) \) \(43\) \(7\) \(I_{7}\) Split multiplicative \(-1\) \(1\) \(7\) \(7\)

Galois Representations

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

prime Image of Galois Representation
\(7\) 7B.1.3

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 7.
Its isogeny class 43.2-a consists of curves linked by isogenies of degree 7.

Base change

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