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SageMath
E = EllipticCurve("bf1")
E.isogeny_class()
Elliptic curves in class 10080.bf
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
10080.bf1 | 10080by2 | \([0, 0, 0, -2916147, -1916735686]\) | \(7347751505995469192/72930375\) | \(27221116608000\) | \([2]\) | \(122880\) | \(2.1542\) | |
10080.bf2 | 10080by3 | \([0, 0, 0, -261147, -1567186]\) | \(5276930158229192/3050936350875\) | \(1138755891091392000\) | \([2]\) | \(122880\) | \(2.1542\) | |
10080.bf3 | 10080by1 | \([0, 0, 0, -182397, -29901436]\) | \(14383655824793536/45209390625\) | \(2109289329000000\) | \([2, 2]\) | \(61440\) | \(1.8076\) | \(\Gamma_0(N)\)-optimal |
10080.bf4 | 10080by4 | \([0, 0, 0, -105852, -55191904]\) | \(-43927191786304/415283203125\) | \(-1240029000000000000\) | \([4]\) | \(122880\) | \(2.1542\) |
Rank
sage: E.rank()
The elliptic curves in class 10080.bf have rank \(0\).
Complex multiplication
The elliptic curves in class 10080.bf do not have complex multiplication.Modular form 10080.2.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.