Properties

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

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Show commands for: SageMath

sage: E = EllipticCurve("6930.z1")
sage: E.isogeny_class()

Elliptic curves in class 6930ba

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
6930.z7 6930ba1 [1, -1, 1, -10553, -187063] 2 18432 \(\Gamma_0(N)\)-optimal
6930.z5 6930ba2 [1, -1, 1, -141773, -20499919] 4 36864  
6930.z4 6930ba3 [1, -1, 1, -429728, 108528707] 6 55296  
6930.z2 6930ba4 [1, -1, 1, -2268023, -1314110419] 2 73728  
6930.z6 6930ba5 [1, -1, 1, -115043, -28486843] 2 73728  
6930.z3 6930ba6 [1, -1, 1, -455648, 94718531] 12 110592  
6930.z1 6930ba7 [1, -1, 1, -2307848, -1265537149] 6 221184  
6930.z8 6930ba8 [1, -1, 1, 981832, 570811907] 6 221184  

Rank

sage: E.rank()

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

Modular form 6930.2.a.z

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.