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SageMath
E = EllipticCurve("bf1")
E.isogeny_class()
Elliptic curves in class 6720.bf
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
6720.bf1 | 6720m3 | \([0, -1, 0, -1296065, -567489663]\) | \(7347751505995469192/72930375\) | \(2389782528000\) | \([2]\) | \(61440\) | \(1.9515\) | |
6720.bf2 | 6720m4 | \([0, -1, 0, -116065, -425663]\) | \(5276930158229192/3050936350875\) | \(99973082345472000\) | \([2]\) | \(61440\) | \(1.9515\) | |
6720.bf3 | 6720m2 | \([0, -1, 0, -81065, -8832663]\) | \(14383655824793536/45209390625\) | \(185177664000000\) | \([2, 2]\) | \(30720\) | \(1.6049\) | |
6720.bf4 | 6720m1 | \([0, -1, 0, -2940, -254538]\) | \(-43927191786304/415283203125\) | \(-26578125000000\) | \([2]\) | \(15360\) | \(1.2583\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 6720.bf have rank \(1\).
Complex multiplication
The elliptic curves in class 6720.bf do not have complex multiplication.Modular form 6720.2.a.bf
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.