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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 221067.c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
221067.c1 | 221067c2 | \([1, -1, 1, -18041, -864894]\) | \(669233048723/50759541\) | \(49251931872759\) | \([2]\) | \(442368\) | \(1.3724\) | |
221067.c2 | 221067c1 | \([1, -1, 1, -3686, 71052]\) | \(5706550403/1112643\) | \(1079596390257\) | \([2]\) | \(221184\) | \(1.0258\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 221067.c have rank \(2\).
Complex multiplication
The elliptic curves in class 221067.c do not have complex multiplication.Modular form 221067.2.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 LMFDB 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 LMFDB labels.