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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 574c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
574.e2 | 574c1 | \([1, 1, 0, -84, 80]\) | \(66775173193/32915456\) | \(32915456\) | \([2]\) | \(224\) | \(0.13561\) | \(\Gamma_0(N)\)-optimal |
574.e1 | 574c2 | \([1, 1, 0, -724, -7728]\) | \(42060685455433/516618368\) | \(516618368\) | \([2]\) | \(448\) | \(0.48218\) |
Rank
sage: E.rank()
The elliptic curves in class 574c have rank \(0\).
Complex multiplication
The elliptic curves in class 574c do not have complex multiplication.Modular form 574.2.a.c
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the Cremona numbering.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.