Properties

Label 6930o
Number of curves 8
Conductor 6930
CM no
Rank 1
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("6930.q1")
sage: E.isogeny_class()

Elliptic curves in class 6930o

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
6930.q8 6930o1 [1, -1, 0, 14301, -85995] 2 27648 \(\Gamma_0(N)\)-optimal
6930.q7 6930o2 [1, -1, 0, -57699, -647595] 4 55296  
6930.q6 6930o3 [1, -1, 0, -182259, 32927013] 6 82944  
6930.q4 6930o4 [1, -1, 0, -675099, -212909715] 2 110592  
6930.q5 6930o5 [1, -1, 0, -592299, 174808125] 2 110592  
6930.q3 6930o6 [1, -1, 0, -2994759, 1995489513] 12 165888  
6930.q2 6930o7 [1, -1, 0, -3073509, 1885066263] 6 331776  
6930.q1 6930o8 [1, -1, 0, -47916009, 127676162763] 6 331776  

Rank

sage: E.rank()

The elliptic curves in class 6930o have rank \(1\).

Modular form 6930.2.a.q

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

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 Cremona numbering.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.