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SageMath
E = EllipticCurve("cp1")
E.isogeny_class()
Elliptic curves in class 72450cp
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
72450.dt1 | 72450cp1 | \([1, -1, 1, -279155, 56839347]\) | \(-5702623460245179/252448\) | \(-106501500000\) | \([]\) | \(570240\) | \(1.5980\) | \(\Gamma_0(N)\)-optimal |
72450.dt2 | 72450cp2 | \([1, -1, 1, -255530, 66837097]\) | \(-5999796014211/2790817792\) | \(-858307290624000000\) | \([]\) | \(1710720\) | \(2.1474\) |
Rank
sage: E.rank()
The elliptic curves in class 72450cp have rank \(1\).
Complex multiplication
The elliptic curves in class 72450cp do not have complex multiplication.Modular form 72450.2.a.cp
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.