Properties

Label 3549c
Number of curves 6
Conductor 3549
CM no
Rank 1
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("3549.c1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 3549c

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
3549.c6 3549c1 [1, 0, 1, 165, -167] [2] 1152 \(\Gamma_0(N)\)-optimal
3549.c5 3549c2 [1, 0, 1, -680, -1519] [2, 2] 2304  
3549.c2 3549c3 [1, 0, 1, -8285, -290509] [2, 2] 4608  
3549.c3 3549c4 [1, 0, 1, -6595, 204323] [2] 4608  
3549.c1 3549c5 [1, 0, 1, -132500, -18574957] [2] 9216  
3549.c4 3549c6 [1, 0, 1, -5750, -471001] [2] 9216  

Rank

sage: E.rank()
 

The elliptic curves in class 3549c have rank \(1\).

Modular form 3549.2.a.c

sage: E.q_eigenform(10)
 
\( q + q^{2} + q^{3} - q^{4} + 2q^{5} + q^{6} + q^{7} - 3q^{8} + q^{9} + 2q^{10} - 4q^{11} - q^{12} + q^{14} + 2q^{15} - q^{16} - 6q^{17} + q^{18} - 4q^{19} + O(q^{20}) \)

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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)

Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with Cremona labels.