# Properties

 Base field $$\Q(\sqrt{77})$$ Label 2.2.77.1-17.1-a1 Conductor $$(a + 1)$$ Conductor norm $$17$$ CM no base-change no Q-curve no 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

$$y^2 + y = x^{3} + \left(-2 a - 5\right) x + a - 27$$
magma: E := ChangeRing(EllipticCurve([0, 0, 1, -2*a - 5, a - 27]),K);

sage: E = EllipticCurve(K, [0, 0, 1, -2*a - 5, a - 27])

gp (2.8): E = ellinit([0, 0, 1, -2*a - 5, a - 27],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(a + 1)$$ = $$\left(a + 1\right)$$ magma: Conductor(E);  sage: E.conductor() $$N(\mathfrak{N})$$ = $$17$$ = $$17$$ magma: Norm(Conductor(E));  sage: E.conductor().norm() $$\mathfrak{D}$$ = $$(-53 a + 4945)$$ = $$\left(a + 1\right)^{6}$$ magma: Discriminant(E);  sage: E.discriminant()  gp (2.8): E.disc $$N(\mathfrak{D})$$ = $$24137569$$ = $$17^{6}$$ magma: Norm(Discriminant(E));  sage: E.discriminant().norm()  gp (2.8): norm(E.disc) $$j$$ = $$-\frac{16539036917760}{24137569} a - \frac{64295071322112}{24137569}$$ 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$$(j)_{-}$$
$$\left(a + 1\right)$$ $$17$$ $$2$$ $$I_{6}$$ Non-split multiplicative $$1$$ $$1$$ $$6$$ $$6$$

## Galois Representations

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

prime Image of Galois Representation
$$3$$ 3Nn

## Isogenies and isogeny class

This curve has no rational isogenies. Its isogeny class 17.1-a consists of this curve only.

## Base change

This curve is not the base-change of an elliptic curve defined over $$\Q$$. It is not a $$\Q$$-curve.