Properties

Label 210c
Number of curves 6
Conductor 210
CM no
Rank 0
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("210.c1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 210c

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
210.c6 210c1 [1, 1, 1, 10, -13] [4] 32 \(\Gamma_0(N)\)-optimal
210.c5 210c2 [1, 1, 1, -70, -205] [2, 4] 64  
210.c2 210c3 [1, 1, 1, -1050, -13533] [2, 2] 128  
210.c4 210c4 [1, 1, 1, -370, 2435] [4] 128  
210.c1 210c5 [1, 1, 1, -16800, -845133] [2] 256  
210.c3 210c6 [1, 1, 1, -980, -15325] [2] 256  

Rank

sage: E.rank()
 

The elliptic curves in class 210c have rank \(0\).

Modular form 210.2.a.c

sage: E.q_eigenform(10)
 
\( q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} + q^{7} + q^{8} + q^{9} + q^{10} + 4q^{11} - q^{12} - 2q^{13} + q^{14} - q^{15} + q^{16} + 2q^{17} + q^{18} - 4q^{19} + O(q^{20}) \)

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.