Properties

Label 693.d
Number of curves 6
Conductor 693
CM no
Rank 0
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("693.d1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 693.d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
693.d1 693d5 [1, -1, 0, -40671, 3167194] [2] 1280  
693.d2 693d3 [1, -1, 0, -2556, 49387] [2, 2] 640  
693.d3 693d2 [1, -1, 0, -351, -1328] [2, 2] 320  
693.d4 693d1 [1, -1, 0, -306, -1985] [2] 160 \(\Gamma_0(N)\)-optimal
693.d5 693d6 [1, -1, 0, 279, 150880] [2] 1280  
693.d6 693d4 [1, -1, 0, 1134, -10535] [2] 640  

Rank

sage: E.rank()
 

The elliptic curves in class 693.d have rank \(0\).

Modular form 693.2.a.d

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

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.