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SageMath
E = EllipticCurve("bf1")
E.isogeny_class()
Elliptic curves in class 9702.bf
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
9702.bf1 | 9702bx3 | \([1, -1, 1, -99896, -12122575]\) | \(1285429208617/614922\) | \(52739474657562\) | \([2]\) | \(49152\) | \(1.5879\) | |
9702.bf2 | 9702bx4 | \([1, -1, 1, -55796, 5002337]\) | \(223980311017/4278582\) | \(366957381520422\) | \([2]\) | \(49152\) | \(1.5879\) | |
9702.bf3 | 9702bx2 | \([1, -1, 1, -7286, -120319]\) | \(498677257/213444\) | \(18306263930724\) | \([2, 2]\) | \(24576\) | \(1.2413\) | |
9702.bf4 | 9702bx1 | \([1, -1, 1, 1534, -14479]\) | \(4657463/3696\) | \(-316991583216\) | \([4]\) | \(12288\) | \(0.89477\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 9702.bf have rank \(0\).
Complex multiplication
The elliptic curves in class 9702.bf do not have complex multiplication.Modular form 9702.2.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.