Properties

Label 169338b
Number of curves 2
Conductor 169338
CM no
Rank 1
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("169338.w1")
sage: E.isogeny_class()

Elliptic curves in class 169338b

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
169338.w2 169338b1 [1, 0, 0, -764, -33840] 2 338688 \(\Gamma_0(N)\)-optimal
169338.w1 169338b2 [1, 0, 0, -21044, -1173576] 2 677376  

Rank

sage: E.rank()

The elliptic curves in class 169338b have rank \(1\).

Modular form 169338.2.a.w

sage: E.q_eigenform(10)
\( q + q^{2} + q^{3} + q^{4} - 2q^{5} + q^{6} + 4q^{7} + q^{8} + q^{9} - 2q^{10} + 4q^{11} + q^{12} + 4q^{14} - 2q^{15} + q^{16} - 4q^{17} + q^{18} + 4q^{19} + O(q^{20}) \)

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.

\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.