Properties

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

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Show commands for: SageMath

sage: E = EllipticCurve("198.d1")
sage: E.isogeny_class()

Elliptic curves in class 198.d

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
198.d1 198c4 [1, -1, 1, -1325, 4969] 2 192  
198.d2 198c2 [1, -1, 1, -1025, 12881] 6 64  
198.d3 198c3 [1, -1, 1, -785, -8207] 2 96  
198.d4 198c1 [1, -1, 1, -65, 209] 6 32 \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()

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

Modular form 198.2.a.d

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

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.