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SageMath
E = EllipticCurve("cr1")
E.isogeny_class()
Elliptic curves in class 32448cr
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
32448.a2 | 32448cr1 | \([0, -1, 0, -3605, -73659]\) | \(1048576/117\) | \(578290332672\) | \([2]\) | \(64512\) | \(0.98995\) | \(\Gamma_0(N)\)-optimal |
32448.a1 | 32448cr2 | \([0, -1, 0, -13745, 544881]\) | \(3631696/507\) | \(40094796398592\) | \([2]\) | \(129024\) | \(1.3365\) |
Rank
sage: E.rank()
The elliptic curves in class 32448cr have rank \(0\).
Complex multiplication
The elliptic curves in class 32448cr do not have complex multiplication.Modular form 32448.2.a.cr
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.