Properties

 Base field $$\Q(\sqrt{77})$$ Label 2.2.77.1-28.1-a4 Conductor $$(2 a + 6)$$ Conductor norm $$28$$ CM no base-change yes: 1694.e3,98.a3 Q-curve yes Torsion order $$2$$ 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

$$y^2 + \left(a + 1\right) x y + y = x^{3} - a x^{2} + \left(-321 a - 1235\right) x + 4940 a + 19211$$
magma: E := ChangeRing(EllipticCurve([a + 1, -a, 1, -321*a - 1235, 4940*a + 19211]),K);

sage: E = EllipticCurve(K, [a + 1, -a, 1, -321*a - 1235, 4940*a + 19211])

gp (2.8): E = ellinit([a + 1, -a, 1, -321*a - 1235, 4940*a + 19211],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

 $$\mathfrak{N}$$ = $$(2 a + 6)$$ = $$\left(2\right) \cdot \left(a + 3\right)$$ magma: Conductor(E);  sage: E.conductor() $$N(\mathfrak{N})$$ = $$28$$ = $$4 \cdot 7$$ magma: Norm(Conductor(E));  sage: E.conductor().norm() $$\mathfrak{D}$$ = $$(941192)$$ = $$\left(2\right)^{3} \cdot \left(a + 3\right)^{12}$$ magma: Discriminant(E);  sage: E.discriminant()  gp (2.8): E.disc $$N(\mathfrak{D})$$ = $$885842380864$$ = $$4^{3} \cdot 7^{12}$$ magma: Norm(Discriminant(E));  sage: E.discriminant().norm()  gp (2.8): norm(E.disc) $$j$$ = $$\frac{4956477625}{941192}$$ 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: $$\Z/2\Z$$ 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] $\left(\frac{9}{4} a + 7 : -\frac{23}{4} a - \frac{203}{8} : 1\right)$ 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$$(j)_{-}$$
$$\left(a + 3\right)$$ $$7$$ $$2$$ $$I_{12}$$ Non-split multiplicative $$1$$ $$1$$ $$12$$ $$12$$
$$\left(2\right)$$ $$4$$ $$3$$ $$I_{3}$$ Split multiplicative $$-1$$ $$1$$ $$3$$ $$3$$

Galois Representations

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

prime Image of Galois Representation
$$2$$ 2B
$$3$$ 3Cs

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 3 and 6.
Its isogeny class 28.1-a consists of curves linked by isogenies of degrees dividing 18.

Base change

This curve is the base-change of elliptic curves 1694.e3, 98.a3, defined over $$\Q$$, so it is also a $$\Q$$-curve.