Properties

Label 201810cg
Number of curves 8
Conductor 201810
CM no
Rank 0
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("201810.p1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 201810cg

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
201810.p7 201810cg1 [1, 1, 0, -478117, -127398131] [2] 2903040 \(\Gamma_0(N)\)-optimal
201810.p6 201810cg2 [1, 1, 0, -554997, -83776419] [2, 2] 5806080  
201810.p5 201810cg3 [1, 1, 0, -1415092, 491827024] [2] 8709120  
201810.p4 201810cg4 [1, 1, 0, -4187577, 3238581249] [2] 11612160  
201810.p8 201810cg5 [1, 1, 0, 1847503, -612806919] [2] 11612160  
201810.p2 201810cg6 [1, 1, 0, -21096372, 37284011856] [2, 2] 17418240  
201810.p1 201810cg7 [1, 1, 0, -337534452, 2386710180624] [2] 34836480  
201810.p3 201810cg8 [1, 1, 0, -19558772, 42951297936] [2] 34836480  

Rank

sage: E.rank()
 

The elliptic curves in class 201810cg have rank \(0\).

Modular form 201810.2.a.p

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

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 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.