Properties

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

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Show commands for: SageMath

sage: E = EllipticCurve("210.b1")
sage: E.isogeny_class()

Elliptic curves in class 210.b

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
210.b1 210b7 [1, 0, 1, -351233, -80149132] 2 1152  
210.b2 210b6 [1, 0, 1, -21953, -1253644] 4 576  
210.b3 210b8 [1, 0, 1, -20353, -1443724] 4 1152  
210.b4 210b4 [1, 0, 1, -4358, -109132] 6 384  
210.b5 210b3 [1, 0, 1, -1473, -16652] 2 288  
210.b6 210b2 [1, 0, 1, -578, 2756] 12 192  
210.b7 210b1 [1, 0, 1, -498, 4228] 6 96 \(\Gamma_0(N)\)-optimal
210.b8 210b5 [1, 0, 1, 1922, 20756] 12 384  

Rank

sage: E.rank()

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

Modular form 210.2.a.b

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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.