Properties

Base field \(\Q(\zeta_{11})^+\)
Label 5.5.14641.1-43.4-b1
Conductor \((43,2 a^{4} - a^{3} - 7 a^{2} + 3 a + 3)\)
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

\( y^2 + \left(a^{4} + a^{3} - 4 a^{2} - 2 a + 2\right) x y + \left(a^{2} - 2\right) y = x^{3} + \left(a^{2} - 3\right) x^{2} + \left(124 a^{4} + 151 a^{3} - 668 a^{2} - 260 a + 194\right) x + 581 a^{4} + 2293 a^{3} - 4817 a^{2} - 3846 a + 946 \)
magma: E := ChangeRing(EllipticCurve([a^4 + a^3 - 4*a^2 - 2*a + 2, a^2 - 3, a^2 - 2, 124*a^4 + 151*a^3 - 668*a^2 - 260*a + 194, 581*a^4 + 2293*a^3 - 4817*a^2 - 3846*a + 946]),K);
 
sage: E = EllipticCurve(K, [a^4 + a^3 - 4*a^2 - 2*a + 2, a^2 - 3, a^2 - 2, 124*a^4 + 151*a^3 - 668*a^2 - 260*a + 194, 581*a^4 + 2293*a^3 - 4817*a^2 - 3846*a + 946])
 
gp: E = ellinit([a^4 + a^3 - 4*a^2 - 2*a + 2, a^2 - 3, a^2 - 2, 124*a^4 + 151*a^3 - 668*a^2 - 260*a + 194, 581*a^4 + 2293*a^3 - 4817*a^2 - 3846*a + 946],K)
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((43,2 a^{4} - a^{3} - 7 a^{2} + 3 a + 3)\) = \( \left(a^{4} + a^{3} - 3 a^{2} - 3 a + 2\right) \)
magma: Conductor(E);
 
sage: E.conductor()
 
\(N(\mathfrak{N}) \) = \( 43 \) = \( 43 \)
magma: Norm(Conductor(E));
 
sage: E.conductor().norm()
 
\(\mathfrak{D}\) = \((43,a + 22,a^{2} + 32,a^{3} - 3 a + 4,a^{4} - 4 a^{2} + 9)\) = \( \left(a^{4} + a^{3} - 3 a^{2} - 3 a + 2\right) \)
magma: Discriminant(E);
 
sage: E.discriminant()
 
gp: E.disc
 
\(N(\mathfrak{D})\) = \( 43 \) = \( 43 \)
magma: Norm(Discriminant(E));
 
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
\(j\) = \( \frac{3443260198697718303446625}{43} a^{4} - \frac{6303449114592619950934258}{43} a^{3} - \frac{8535767899365460376030330}{43} a^{2} + \frac{17419485949931837709740318}{43} a - \frac{4143833314037896993105954}{43} \)
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\((j)_{-}\)
\( \left(a^{4} + a^{3} - 3 a^{2} - 3 a + 2\right) \) \(43\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)

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.
Its isogeny class 43.4-b consists of curves linked by isogenies of degree 5.

Base change

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