# Properties

 Label 4.4.1125.1-45.1-b6 Base field $$\Q(\zeta_{15})^+$$ Conductor $$(a^3-5a+1)$$ Conductor norm $$45$$ CM no Base change yes: 15.a4,75.b4 Q-curve yes Torsion order $$16$$ Rank $$0$$

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Base field$$\Q(\zeta_{15})^+$$

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 4, -4, -1, 1]))

gp: K = nfinit(Pol(Vecrev([1, 4, -4, -1, 1])));

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

## Weierstrass equation

$${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}-80{x}+242$$
sage: E = EllipticCurve([K([1,0,0,0]),K([1,0,0,0]),K([1,0,0,0]),K([-80,0,0,0]),K([242,0,0,0])])

gp: E = ellinit([Pol(Vecrev([1,0,0,0])),Pol(Vecrev([1,0,0,0])),Pol(Vecrev([1,0,0,0])),Pol(Vecrev([-80,0,0,0])),Pol(Vecrev([242,0,0,0]))], K);

magma: E := EllipticCurve([K![1,0,0,0],K![1,0,0,0],K![1,0,0,0],K![-80,0,0,0],K![242,0,0,0]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(a^3-5a+1)$$ = $$(-a-1)\cdot(-a^3+a^2+3a-2)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$45$$ = $$5\cdot9$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(15)$$ = $$(-a-1)^{4}\cdot(-a^3+a^2+3a-2)^{2}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$50625$$ = $$5^{4}\cdot9^{2}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{56667352321}{15}$$ sage: E.j_invariant()  gp: E.j  magma: jInvariant(E); Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) sage: E.has_cm(), E.cm_discriminant()  magma: HasComplexMultiplication(E); Sato-Tate group: $\mathrm{SU}(2)$

## Mordell-Weil group

 Rank: $$0$$ Torsion structure: $$\Z/16\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generator: $\left(7 a^{3} + 6 a^{2} - 17 a + 1 : 47 a^{3} + 38 a^{2} - 119 a - 29 : 1\right)$ sage: T.gens()  gp: T[3]  magma: [piT(P) : P in Generators(T)];

## BSD invariants

 Analytic rank: $$0$$ sage: E.rank()  magma: Rank(E); Mordell-Weil rank: $$0$$ Regulator: $$1$$ Period: $$985.145157251357$$ Tamagawa product: $$8$$  =  $$2^{2}\cdot2$$ Torsion order: $$16$$ Leading coefficient: $$0.917854808049480$$ Analytic order of Ш: $$1$$ (rounded)

## 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$$(j)_{-}$$
$$(-a-1)$$ $$5$$ $$4$$ $$I_{4}$$ Split multiplicative $$-1$$ $$1$$ $$4$$ $$4$$
$$(-a^3+a^2+3a-2)$$ $$9$$ $$2$$ $$I_{2}$$ Split multiplicative $$-1$$ $$1$$ $$2$$ $$2$$

## Galois Representations

The mod $$p$$ Galois Representation has maximal image for all primes $$p < 1000$$ except those listed.

prime Image of Galois Representation
$$2$$ 2B

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 4, 8, 16 and 32.
Its isogeny class 45.1-b consists of curves linked by isogenies of degrees dividing 32.

## Base change

This curve is the base change of elliptic curves 15.a4, 75.b4, defined over $$\Q$$, so it is also a $$\Q$$-curve.