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SageMath
E = EllipticCurve("cm1")
E.isogeny_class()
Elliptic curves in class 182070.cm
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
182070.cm1 | 182070br2 | \([1, -1, 1, -79538, -8613169]\) | \(15537040571177/1786050\) | \(6396875600850\) | \([2]\) | \(737280\) | \(1.4842\) | |
182070.cm2 | 182070br1 | \([1, -1, 1, -4568, -156553]\) | \(-2942649737/1296540\) | \(-4643657843580\) | \([2]\) | \(368640\) | \(1.1377\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 182070.cm have rank \(1\).
Complex multiplication
The elliptic curves in class 182070.cm do not have complex multiplication.Modular form 182070.2.a.cm
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.