Properties

Label 690.k
Number of curves 6
Conductor 690
CM no
Rank 0
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("690.k1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 690.k

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
690.k1 690k4 [1, 0, 0, -110400, 14109732] [4] 1536  
690.k2 690k5 [1, 0, 0, -25830, -1370850] [2] 3072  
690.k3 690k3 [1, 0, 0, -7080, 207900] [2, 2] 1536  
690.k4 690k2 [1, 0, 0, -6900, 220032] [2, 4] 768  
690.k5 690k1 [1, 0, 0, -420, 3600] [8] 384 \(\Gamma_0(N)\)-optimal
690.k6 690k6 [1, 0, 0, 8790, 1010922] [2] 3072  

Rank

sage: E.rank()
 

The elliptic curves in class 690.k have rank \(0\).

Modular form 690.2.a.k

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

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 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.