# Properties

 Base field $$\Q(\sqrt{77})$$ Label 2.2.77.1-11.1-a2 Conductor $$(a - 6)$$ Conductor norm $$11$$ CM no base-change yes: 121.d2,539.a2 Q-curve yes Torsion order $$1$$ Rank not available

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## Base field $$\Q(\sqrt{77})$$

Generator $$a$$, with minimal polynomial $$x^{2} - x - 19$$; class number $$1$$.

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-19, -1, 1]);

sage: x = polygen(QQ); K.<a> = NumberField(x^2 - x - 19)

gp (2.8): K = nfinit(a^2 - a - 19);

## Weierstrass equation

$$y^2 + y = x^{3} + \left(a + 1\right) x^{2} + \left(-92 a - 355\right) x + 1671 a + 6495$$
magma: E := ChangeRing(EllipticCurve([0, a + 1, 1, -92*a - 355, 1671*a + 6495]),K);

sage: E = EllipticCurve(K, [0, a + 1, 1, -92*a - 355, 1671*a + 6495])

gp (2.8): E = ellinit([0, a + 1, 1, -92*a - 355, 1671*a + 6495],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(a - 6)$$ = $$\left(a - 6\right)$$ magma: Conductor(E);  sage: E.conductor() $$N(\mathfrak{N})$$ = $$11$$ = $$11$$ magma: Norm(Conductor(E));  sage: E.conductor().norm() $$\mathfrak{D}$$ = $$(161051)$$ = $$\left(a - 6\right)^{10}$$ magma: Discriminant(E);  sage: E.discriminant()  gp (2.8): E.disc $$N(\mathfrak{D})$$ = $$25937424601$$ = $$11^{10}$$ magma: Norm(Discriminant(E));  sage: E.discriminant().norm()  gp (2.8): norm(E.disc) $$j$$ = $$-\frac{122023936}{161051}$$ 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: Trivial 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]

## 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$$(j)_{-}$$
$$\left(a - 6\right)$$ $$11$$ $$10$$ $$I_{10}$$ 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-a consists of curves linked by isogenies of degrees dividing 25.

## Base change

This curve is the base-change of elliptic curves 121.d2, 539.a2, defined over $$\Q$$, so it is also a $$\Q$$-curve.