# Properties

 Base field $$\Q(\sqrt{165})$$ Label 2.2.165.1-15.1-c7 Conductor $$(a - 8)$$ Conductor norm $$15$$ CM no base-change yes: 15.a4,27225.bp4 Q-curve yes Torsion order $$4$$ Rank not available

# Learn more about

Show commands for: Magma / SageMath / Pari/GP

## Base field $$\Q(\sqrt{165})$$

Generator $$a$$, with minimal polynomial $$x^{2} - x - 41$$; class number $$2$$.

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-41, -1, 1]);

sage: x = polygen(QQ); K.<a> = NumberField(x^2 - x - 41)

gp (2.8): K = nfinit(a^2 - a - 41);

## Weierstrass equation

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

sage: E = EllipticCurve(K, [1, 1, 1, -80, 242])

gp (2.8): E = ellinit([1, 1, 1, -80, 242],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(a - 8)$$ = $$\left(3, a + 1\right) \cdot \left(5, a + 2\right)$$ magma: Conductor(E);  sage: E.conductor() $$N(\mathfrak{N})$$ = $$15$$ = $$3 \cdot 5$$ magma: Norm(Conductor(E));  sage: E.conductor().norm() $$\mathfrak{D}$$ = $$(15)$$ = $$\left(3, a + 1\right)^{2} \cdot \left(5, a + 2\right)^{2}$$ magma: Discriminant(E);  sage: E.discriminant()  gp (2.8): E.disc $$N(\mathfrak{D})$$ = $$225$$ = $$3^{2} \cdot 5^{2}$$ magma: Norm(Discriminant(E));  sage: E.discriminant().norm()  gp (2.8): norm(E.disc) $$j$$ = $$\frac{56667352321}{15}$$ magma: jInvariant(E);  sage: E.j_invariant()  gp (2.8): 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()

magma: Generators(E); // includes torsion

sage: E.gens()

Regulator: not available

magma: Regulator(Generators(E));

sage: E.regulator_of_points(E.gens())

## Torsion subgroup

Structure: $$\Z/4\Z$$ magma: TorsionSubgroup(E);  sage: E.torsion_subgroup().gens()  gp (2.8): elltors(E)[2] magma: Order(TorsionSubgroup(E));  sage: E.torsion_order()  gp (2.8): elltors(E)[1] $\left(5 : -4 : 1\right)$ magma: [f(P): P in Generators(T)] where T,f:=TorsionSubgroup(E);  sage: E.torsion_subgroup().gens()  gp (2.8): elltors(E)[3]

## 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$$(j)_{-}$$
$$\left(3, a + 1\right)$$ $$3$$ $$2$$ $$I_{2}$$ Non-split multiplicative $$1$$ $$1$$ $$2$$ $$2$$
$$\left(5, a + 2\right)$$ $$5$$ $$2$$ $$I_{2}$$ Split multiplicative $$-1$$ $$1$$ $$2$$ $$2$$

## Galois Representations

The mod $$p$$ Galois Representation has maximal image for all primes $$p$$ 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 and 16.
Its isogeny class 15.1-c consists of curves linked by isogenies of degrees dividing 16.

## Base change

This curve is the base-change of elliptic curves 15.a4, 27225.bp4, defined over $$\Q$$, so it is also a $$\Q$$-curve.