# SageMath code for working with elliptic curve isogeny class 2.2.13.1-81.4-c
# (Note that not all these functions may be available, and some may take a long time to execute.)


# Define the base number field: 
R.<x> = PolynomialRing(QQ);  K.<a> = NumberField(R([-3, -1, 1]))

# Define the isogeny class: 
E = EllipticCurve([K([0,0]),K([-1,1]),K([0,1]),K([-9,3]),K([19,-10])])
E.isogeny_class()

# Compute the Mordell-Weil rank: 
E.rank()

# Isogeny matrix: 
E.isogeny_class().matrix()

# Isogeny graph: 
E.isogeny_class().graph().plot(edge_labels=True)

# List of curves in the isogeny class: 
E.isogeny_class().curves