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The vertices of the line_graph are the edges of the graph, including edge labels by default. If you don't want the labels, you can do:

sage: G = Graph([(2,3),(2,4),(2,1),(1,5),(5,6),(1,0),(0,12),(0,13),(14,0),(15,0),(7,0),(7,10),(11,7),(7,9),(7,8)])
sage: h = G.line_graph(labels=False)

Then you can either use the method that has already been proposed here, or use the Javascript plotting (it's better to increase the link distance).