Properties

Label 198.e
Number of curves 4
Conductor 198
CM no
Rank 0
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("198.e1")
sage: E.isogeny_class()

Elliptic curves in class 198.e

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
198.e1 198b3 [1, -1, 1, -725, 7661] 6 96  
198.e2 198b4 [1, -1, 1, -365, 15005] 6 192  
198.e3 198b1 [1, -1, 1, -50, -115] 2 32 \(\Gamma_0(N)\)-optimal
198.e4 198b2 [1, -1, 1, 40, -547] 2 64  

Rank

sage: E.rank()

The elliptic curves in class 198.e have rank \(0\).

Modular form 198.2.a.e

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

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 LMFDB numbering.

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.