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SageMath
E = EllipticCurve("cf1")
E.isogeny_class()
Elliptic curves in class 129472cf
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 129472.r2 | 129472cf1 | \([0, 1, 0, -333313, 73626111]\) | \(647214625/3332\) | \(21083292934602752\) | \([2]\) | \(884736\) | \(1.9782\) | \(\Gamma_0(N)\)-optimal |
| 129472.r1 | 129472cf2 | \([0, 1, 0, -518273, -17411201]\) | \(2433138625/1387778\) | \(8781191507262046208\) | \([2]\) | \(1769472\) | \(2.3247\) |
Rank
sage: E.rank()
The elliptic curves in class 129472cf have rank \(0\).
Complex multiplication
The elliptic curves in class 129472cf do not have complex multiplication.Modular form 129472.2.a.cf
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.
