Properties

Label 73689c
Number of curves $6$
Conductor $73689$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("73689.s1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 73689c

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
73689.s5 73689c1 [1, 1, 0, -94866, -12080889] [2] 491520 \(\Gamma_0(N)\)-optimal
73689.s4 73689c2 [1, 1, 0, -1547471, -741579120] [2, 2] 983040  
73689.s3 73689c3 [1, 1, 0, -1577116, -711726605] [2, 2] 1966080  
73689.s1 73689c4 [1, 1, 0, -24759506, -47430266319] [2] 1966080  
73689.s6 73689c5 [1, 1, 0, 1510199, -3155027696] [2] 3932160  
73689.s2 73689c6 [1, 1, 0, -5138751, 3642728346] [2] 3932160  

Rank

sage: E.rank()
 

The elliptic curves in class 73689c have rank \(0\).

Modular form 73689.2.a.s

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

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 Cremona numbering.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.