# Properties

 Base field $$\Q(\zeta_{11})^+$$ Label 5.5.14641.1-43.3-a1 Conductor $$(43,-a^{3} - a^{2} + 4 a + 2)$$ Conductor norm $$43$$ CM no base-change no Q-curve no Torsion order $$1$$ Rank not available

# Related objects

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

$$y^2 + \left(a^{3} + a^{2} - 3 a - 1\right) x y + \left(a^{3} + a^{2} - 3 a - 1\right) y = x^{3} + \left(a^{4} - 5 a^{2} + a + 4\right) x^{2} + \left(201 a^{4} - 446 a^{3} - 237 a^{2} + 764 a - 199\right) x - 340 a^{4} + 1924 a^{3} - 3845 a^{2} + 2952 a - 615$$
magma: E := ChangeRing(EllipticCurve([a^3 + a^2 - 3*a - 1, a^4 - 5*a^2 + a + 4, a^3 + a^2 - 3*a - 1, 201*a^4 - 446*a^3 - 237*a^2 + 764*a - 199, -340*a^4 + 1924*a^3 - 3845*a^2 + 2952*a - 615]),K);

sage: E = EllipticCurve(K, [a^3 + a^2 - 3*a - 1, a^4 - 5*a^2 + a + 4, a^3 + a^2 - 3*a - 1, 201*a^4 - 446*a^3 - 237*a^2 + 764*a - 199, -340*a^4 + 1924*a^3 - 3845*a^2 + 2952*a - 615])

gp: E = ellinit([a^3 + a^2 - 3*a - 1, a^4 - 5*a^2 + a + 4, a^3 + a^2 - 3*a - 1, 201*a^4 - 446*a^3 - 237*a^2 + 764*a - 199, -340*a^4 + 1924*a^3 - 3845*a^2 + 2952*a - 615],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(43,-a^{3} - a^{2} + 4 a + 2)$$ = $$\left(a^{3} + a^{2} - 4 a - 2\right)$$ magma: Conductor(E);  sage: E.conductor() $$N(\mathfrak{N})$$ = $$43$$ = $$43$$ magma: Norm(Conductor(E));  sage: E.conductor().norm() $$\mathfrak{D}$$ = $$(271818611107,a + 73434615221,a^{2} + 2564284379,a^{3} - 3 a + 29077453113,a^{4} - 4 a^{2} + 120997495098)$$ = $$\left(a^{3} + a^{2} - 4 a - 2\right)^{7}$$ 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) $$j$$ = $$-\frac{346480612818557831035095564492}{271818611107} a^{4} + \frac{247862002283140572960026511999}{271818611107} a^{3} + \frac{1456471349605584774275478512978}{271818611107} a^{2} - \frac{14532257307132343958115316054}{6321363049} a - \frac{1217303334006333416200315022669}{271818611107}$$ 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$$(j)_{-}$$
$$\left(a^{3} + a^{2} - 4 a - 2\right)$$ $$43$$ $$7$$ $$I_{7}$$ 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 43.3-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.