# Properties

 Base field $$\Q(\zeta_{11})^+$$ Label 5.5.14641.1-43.5-a1 Conductor $$(43,-a^{4} + a^{3} + 4 a^{2} - 4 a - 2)$$ Conductor norm $$43$$ CM no base-change no Q-curve no Torsion order $$1$$ Rank not available

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## 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 $$1$$.

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);

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 3, 3, -4, -1, 1]);

## Weierstrass equation

$$y^2 + \left(a^{3} - 3 a + 1\right) x y + \left(a^{4} + a^{3} - 4 a^{2} - 2 a + 3\right) y = x^{3} + \left(-a^{3} + 2 a\right) x^{2} + \left(-66 a^{4} + 214 a^{3} - 135 a^{2} - 134 a + 75\right) x - 1428 a^{4} + 3862 a^{3} - 836 a^{2} - 3010 a + 777$$
sage: E = EllipticCurve(K, [a^3 - 3*a + 1, -a^3 + 2*a, a^4 + a^3 - 4*a^2 - 2*a + 3, -66*a^4 + 214*a^3 - 135*a^2 - 134*a + 75, -1428*a^4 + 3862*a^3 - 836*a^2 - 3010*a + 777])

gp: E = ellinit([a^3 - 3*a + 1, -a^3 + 2*a, a^4 + a^3 - 4*a^2 - 2*a + 3, -66*a^4 + 214*a^3 - 135*a^2 - 134*a + 75, -1428*a^4 + 3862*a^3 - 836*a^2 - 3010*a + 777],K)

magma: E := ChangeRing(EllipticCurve([a^3 - 3*a + 1, -a^3 + 2*a, a^4 + a^3 - 4*a^2 - 2*a + 3, -66*a^4 + 214*a^3 - 135*a^2 - 134*a + 75, -1428*a^4 + 3862*a^3 - 836*a^2 - 3010*a + 777]),K);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(43,-a^{4} + a^{3} + 4 a^{2} - 4 a - 2)$$ = $$\left(a^{4} + a^{3} - 4 a^{2} - 2 a + 3\right)$$ sage: E.conductor()  magma: Conductor(E); $$N(\mathfrak{N})$$ = $$43$$ = $$43$$ sage: E.conductor().norm()  magma: Norm(Conductor(E)); $$\mathfrak{D}$$ = $$(271818611107,a + 29077453113,a^{2} + 250768899963,a^{3} - 3 a + 269254326726,a^{4} - 4 a^{2} + 198383995888)$$ = $$\left(a^{4} + a^{3} - 4 a^{2} - 2 a + 3\right)^{7}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); $$N(\mathfrak{D})$$ = $$271818611107$$ = $$43^{7}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); $$j$$ = $$\frac{227781670175826902353974618817}{271818611107} a^{4} - \frac{417029511149911281170191819502}{271818611107} a^{3} - \frac{564646067884749778380802910776}{271818611107} a^{2} + \frac{1152469922914316585435506406013}{271818611107} a - \frac{274161584173414880187077710523}{271818611107}$$ sage: E.j_invariant()  gp: E.j  magma: jInvariant(E); $$\text{End} (E)$$ = $$\Z$$ (no Complex Multiplication ) sage: E.has_cm(), E.cm_discriminant()  magma: HasComplexMultiplication(E); $$\text{ST} (E)$$ = $\mathrm{SU}(2)$

## Mordell-Weil group

Rank not available.

sage: E.rank()

magma: Rank(E);

Regulator: not available

sage: gens = E.gens(); gens

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

sage: E.regulator_of_points(gens)

magma: Regulator(gens);

## Torsion subgroup

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

## Local data at primes of bad reduction

sage: E.local_data()

magma: LocalInformation(E);

prime Norm Tamagawa number Kodaira symbol Reduction type Root number ord($$\mathfrak{N}$$) ord($$\mathfrak{D}$$) ord$$(j)_{-}$$
$$\left(a^{4} + a^{3} - 4 a^{2} - 2 a + 3\right)$$ $$43$$ $$7$$ $$I_{7}$$ Split multiplicative $$-1$$ $$1$$ $$7$$ $$7$$

## Galois Representations

The mod $$p$$ Galois Representation has maximal image for all primes $$p < 1000$$ 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 43.5-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.