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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 4641.c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
4641.c1 | 4641a3 | \([1, 1, 1, -24752, -1509184]\) | \(1677087406638588673/4641\) | \(4641\) | \([2]\) | \(5120\) | \(0.82194\) | |
4641.c2 | 4641a2 | \([1, 1, 1, -1547, -24064]\) | \(409460675852593/21538881\) | \(21538881\) | \([2, 2]\) | \(2560\) | \(0.47537\) | |
4641.c3 | 4641a4 | \([1, 1, 1, -1462, -26716]\) | \(-345608484635233/94427721297\) | \(-94427721297\) | \([2]\) | \(5120\) | \(0.82194\) | |
4641.c4 | 4641a1 | \([1, 1, 1, -102, -366]\) | \(117433042273/22801233\) | \(22801233\) | \([4]\) | \(1280\) | \(0.12880\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 4641.c have rank \(1\).
Complex multiplication
The elliptic curves in class 4641.c do not have complex multiplication.Modular form 4641.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.