Properties

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

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Show commands for: SageMath

sage: E = EllipticCurve("210.e1")
sage: E.isogeny_class()

Elliptic curves in class 210e

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
210.e7 210e1 [1, 0, 0, 210, 900] 8 128 \(\Gamma_0(N)\)-optimal
210.e6 210e2 [1, 0, 0, -1070, 7812] 16 256  
210.e5 210e3 [1, 0, 0, -7550, -247500] 8 512  
210.e4 210e4 [1, 0, 0, -15070, 710612] 8 512  
210.e2 210e5 [1, 0, 0, -120050, -16020000] 4 1024  
210.e8 210e6 [1, 0, 0, 1270, -789048] 4 1024  
210.e1 210e7 [1, 0, 0, -1920800, -1024800150] 2 2048  
210.e3 210e8 [1, 0, 0, -119300, -16229850] 2 2048  

Rank

sage: E.rank()

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

Modular form 210.2.a.e

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.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 2 & 2 & 4 & 4 \\ 4 & 2 & 4 & 1 & 8 & 8 & 16 & 16 \\ 8 & 4 & 2 & 8 & 1 & 4 & 2 & 2 \\ 8 & 4 & 2 & 8 & 4 & 1 & 8 & 8 \\ 16 & 8 & 4 & 16 & 2 & 8 & 1 & 4 \\ 16 & 8 & 4 & 16 & 2 & 8 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.