Properties

Label 60690.d
Number of curves 8
Conductor 60690
CM no
Rank 1
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("60690.d1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 60690.d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
60690.d1 60690a8 [1, 1, 0, -101506198, -393671178092] [2] 5308416  
60690.d2 60690a6 [1, 1, 0, -6344278, -6152807468] [2, 2] 2654208  
60690.d3 60690a7 [1, 1, 0, -5881878, -7087132908] [2] 5308416  
60690.d4 60690a5 [1, 1, 0, -1259323, -534904967] [2] 1769472  
60690.d5 60690a3 [1, 1, 0, -425558, -81384492] [2] 1327104  
60690.d6 60690a2 [1, 1, 0, -166903, 13708357] [2, 2] 884736  
60690.d7 60690a1 [1, 1, 0, -143783, 20917173] [2] 442368 \(\Gamma_0(N)\)-optimal
60690.d8 60690a4 [1, 1, 0, 555597, 101419857] [2] 1769472  

Rank

sage: E.rank()
 

The elliptic curves in class 60690.d have rank \(1\).

Modular form 60690.2.a.d

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

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.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 3 & 4 & 6 & 12 & 12 \\ 2 & 1 & 2 & 6 & 2 & 3 & 6 & 6 \\ 4 & 2 & 1 & 12 & 4 & 6 & 12 & 3 \\ 3 & 6 & 12 & 1 & 12 & 2 & 4 & 4 \\ 4 & 2 & 4 & 12 & 1 & 6 & 3 & 12 \\ 6 & 3 & 6 & 2 & 6 & 1 & 2 & 2 \\ 12 & 6 & 12 & 4 & 3 & 2 & 1 & 4 \\ 12 & 6 & 3 & 4 & 12 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.