Properties

Label 609.a
Number of curves $6$
Conductor $609$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("609.a1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 609.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
609.a1 609b4 [1, 1, 1, -204624, 35542050] [4] 1536  
609.a2 609b5 [1, 1, 1, -42469, -2756140] [2] 3072  
609.a3 609b3 [1, 1, 1, -13034, 528806] [2, 2] 1536  
609.a4 609b2 [1, 1, 1, -12789, 551346] [2, 4] 768  
609.a5 609b1 [1, 1, 1, -784, 8720] [4] 384 \(\Gamma_0(N)\)-optimal
609.a6 609b6 [1, 1, 1, 12481, 2376092] [2] 3072  

Rank

sage: E.rank()
 

The elliptic curves in class 609.a have rank \(1\).

Modular form 609.2.a.a

sage: E.q_eigenform(10)
 
\( q - q^{2} - q^{3} - q^{4} - 2q^{5} + q^{6} + q^{7} + 3q^{8} + q^{9} + 2q^{10} + 4q^{11} + q^{12} - 2q^{13} - q^{14} + 2q^{15} - q^{16} + 2q^{17} - q^{18} - 4q^{19} + O(q^{20}) \)

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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.