Properties

Label 9702.bf
Number of curves $4$
Conductor $9702$
CM no
Rank $0$
Graph

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Show commands: 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)
 
\(q + q^{2} + q^{4} - 2 q^{5} + q^{8} - 2 q^{10} - q^{11} - 2 q^{13} + q^{16} - 6 q^{17} + 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

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.