# Properties

 Base field $$\Q(\sqrt{209})$$ Label 2.2.209.1-11.1-b2 Conductor $$(-70 a - 471)$$ Conductor norm $$11$$ CM no base-change yes: 3971.b2,121.d2 Q-curve yes Torsion order $$1$$ Rank not available

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## Base field $$\Q(\sqrt{209})$$

Generator $$a$$, with minimal polynomial $$x^{2} - x - 52$$; class number $$1$$.

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-52, -1, 1]);

sage: x = polygen(QQ); K.<a> = NumberField(x^2 - x - 52)

gp: K = nfinit(a^2 - a - 52);

## Weierstrass equation

$$y^2 + y = x^{3} + \left(-a - 1\right) x^{2} + \left(-6195627759 a - 41686761846\right) x + 1292706345403712 a + 8697866248261405$$
magma: E := ChangeRing(EllipticCurve([0, -a - 1, 1, -6195627759*a - 41686761846, 1292706345403712*a + 8697866248261405]),K);

sage: E = EllipticCurve(K, [0, -a - 1, 1, -6195627759*a - 41686761846, 1292706345403712*a + 8697866248261405])

gp: E = ellinit([0, -a - 1, 1, -6195627759*a - 41686761846, 1292706345403712*a + 8697866248261405],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(-70 a - 471)$$ = $$\left(-70 a - 471\right)$$ magma: Conductor(E);  sage: E.conductor() $$N(\mathfrak{N})$$ = $$11$$ = $$11$$ magma: Norm(Conductor(E));  sage: E.conductor().norm() $$\mathfrak{D}$$ = $$(161051)$$ = $$\left(-70 a - 471\right)^{10}$$ magma: Discriminant(E);  sage: E.discriminant()  gp: E.disc $$N(\mathfrak{D})$$ = $$25937424601$$ = $$11^{10}$$ magma: Norm(Discriminant(E));  sage: E.discriminant().norm()  gp: norm(E.disc) $$j$$ = $$-\frac{122023936}{161051}$$ 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$$(j)_{-}$$
$$\left(-70 a - 471\right)$$ $$11$$ $$10$$ $$I_{10}$$ Split multiplicative $$-1$$ $$1$$ $$10$$ $$10$$

## Galois Representations

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

prime Image of Galois Representation
$$5$$ 5Cs.4.1

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 5.
Its isogeny class 11.1-b consists of curves linked by isogenies of degrees dividing 25.

## Base change

This curve is the base-change of elliptic curves 3971.b2, 121.d2, defined over $$\Q$$, so it is also a $$\Q$$-curve.