# Properties

 Label 2.2.77.1-44.1-e2 Base field $$\Q(\sqrt{77})$$ Conductor norm $$44$$ CM no Base change no Q-curve yes Torsion order $$2$$ Rank $$0$$

# Learn more

Show commands: Magma / PariGP / SageMath

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

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-19, -1, 1]))

gp: K = nfinit(Polrev([-19, -1, 1]));

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

## Weierstrass equation

$${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(-8a-37\right){x}-24a-97$$
sage: E = EllipticCurve([K([0,1]),K([-1,0]),K([0,1]),K([-37,-8]),K([-97,-24])])

gp: E = ellinit([Polrev([0,1]),Polrev([-1,0]),Polrev([0,1]),Polrev([-37,-8]),Polrev([-97,-24])], K);

magma: E := EllipticCurve([K![0,1],K![-1,0],K![0,1],K![-37,-8],K![-97,-24]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(2a-12)$$ = $$(2)\cdot(a-6)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$44$$ = $$4\cdot11$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(29282)$$ = $$(2)\cdot(a-6)^{8}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$857435524$$ = $$4\cdot11^{8}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{59776471}{29282}$$ sage: E.j_invariant()  gp: E.j  magma: jInvariant(E); Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) sage: E.has_cm(), E.cm_discriminant()  magma: HasComplexMultiplication(E); Sato-Tate group: $\mathrm{SU}(2)$

## Mordell-Weil group

 Rank: $$0$$ Torsion structure: $$\Z/2\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generator: $\left(-\frac{1}{4} a - \frac{7}{4} : \frac{1}{2} a + \frac{19}{8} : 1\right)$ sage: T.gens()  gp: T[3]  magma: [piT(P) : P in Generators(T)];

## BSD invariants

 Analytic rank: $$0$$ sage: E.rank()  magma: Rank(E); Mordell-Weil rank: $$0$$ Regulator: $$1$$ Period: $$9.8313794013912849757676597475880577502$$ Tamagawa product: $$8$$  =  $$1\cdot2^{3}$$ Torsion order: $$2$$ Leading coefficient: $$2.2407793279519221990140220113418871614$$ Analytic order of Ш: $$1$$ (rounded)

## Local data at primes of bad reduction

sage: E.local_data()

magma: LocalInformation(E);

prime Norm Tamagawa number Kodaira symbol Reduction type Root number ord($$\mathfrak{N}$$) ord($$\mathfrak{D}$$) ord$$(j)_{-}$$
$$(2)$$ $$4$$ $$1$$ $$I_{1}$$ Split multiplicative $$-1$$ $$1$$ $$1$$ $$1$$
$$(a-6)$$ $$11$$ $$8$$ $$I_{8}$$ Split multiplicative $$-1$$ $$1$$ $$8$$ $$8$$

## Galois Representations

The mod $$p$$ Galois Representation has maximal image for all primes $$p < 1000$$ 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.
Its isogeny class 44.1-e consists of curves linked by isogenies of degree 2.

## Base change

This elliptic curve is a $$\Q$$-curve.

It is not the base change of an elliptic curve defined over any subfield.