Properties

Label 39675bc
Number of curves 8
Conductor 39675
CM no
Rank 1
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("39675.bk1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 39675bc

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
39675.bk7 39675bc1 [1, 0, 1, -276, -283427] [2] 76032 \(\Gamma_0(N)\)-optimal
39675.bk6 39675bc2 [1, 0, 1, -66401, -6499177] [2, 2] 152064  
39675.bk5 39675bc3 [1, 0, 1, -132526, 8577323] [2, 2] 304128  
39675.bk4 39675bc4 [1, 0, 1, -1058276, -419119177] [2] 304128  
39675.bk8 39675bc5 [1, 0, 1, 462599, 64519073] [2] 608256  
39675.bk2 39675bc6 [1, 0, 1, -1785651, 917796073] [2, 2] 608256  
39675.bk3 39675bc7 [1, 0, 1, -1455026, 1268258573] [2] 1216512  
39675.bk1 39675bc8 [1, 0, 1, -28566276, 58763946073] [2] 1216512  

Rank

sage: E.rank()
 

The elliptic curves in class 39675bc have rank \(1\).

Modular form 39675.2.a.bk

sage: E.q_eigenform(10)
 
\( q + q^{2} + q^{3} - q^{4} + q^{6} - 3q^{8} + q^{9} + 4q^{11} - q^{12} + 2q^{13} - q^{16} + 2q^{17} + q^{18} - 4q^{19} + O(q^{20}) \)

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.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 2 & 2 & 4 & 4 \\ 4 & 2 & 4 & 1 & 8 & 8 & 16 & 16 \\ 8 & 4 & 2 & 8 & 1 & 4 & 8 & 8 \\ 8 & 4 & 2 & 8 & 4 & 1 & 2 & 2 \\ 16 & 8 & 4 & 16 & 8 & 2 & 1 & 4 \\ 16 & 8 & 4 & 16 & 8 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.