Properties

Label 210a
Number of curves 8
Conductor 210
CM no
Rank 0
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("210.d1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 210a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
210.d7 210a1 [1, 0, 0, -41, -39] [6] 48 \(\Gamma_0(N)\)-optimal
210.d5 210a2 [1, 0, 0, -361, 2585] [2, 6] 96  
210.d4 210a3 [1, 0, 0, -2681, -53655] [2] 144  
210.d2 210a4 [1, 0, 0, -5761, 167825] [6] 192  
210.d6 210a5 [1, 0, 0, -81, 6561] [6] 192  
210.d3 210a6 [1, 0, 0, -2701, -52819] [2, 2] 288  
210.d1 210a7 [1, 0, 0, -6451, 124931] [2] 576  
210.d8 210a8 [1, 0, 0, 729, -176985] [2] 576  

Rank

sage: E.rank()
 

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

Modular form 210.2.a.d

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

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.