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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 221a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
221.a2 | 221a1 | \([1, -1, 1, -733, 7804]\) | \(43499078731809/82055753\) | \(82055753\) | \([2]\) | \(120\) | \(0.40860\) | \(\Gamma_0(N)\)-optimal |
221.a1 | 221a2 | \([1, -1, 1, -11718, 491144]\) | \(177930109857804849/634933\) | \(634933\) | \([2]\) | \(240\) | \(0.75517\) |
Rank
sage: E.rank()
The elliptic curves in class 221a have rank \(0\).
Complex multiplication
The elliptic curves in class 221a do not have complex multiplication.Modular form 221.2.a.a
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.