Properties

Label 172a
Number of curves 2
Conductor 172
CM no
Rank 1
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("172.a1")
sage: E.isogeny_class()

Elliptic curves in class 172a

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
172.a1 172a1 [0, 1, 0, -13, 15] 3 12 \(\Gamma_0(N)\)-optimal
172.a2 172a2 [0, 1, 0, 67, 79] 1 36  

Rank

sage: E.rank()

The elliptic curves in class 172a have rank \(1\).

Modular form 172.2.a.a

sage: E.q_eigenform(10)
\( q - 2q^{3} - 4q^{7} + q^{9} - 3q^{11} - q^{13} - 3q^{17} + 2q^{19} + O(q^{20}) \)

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 Cremona numbering.

\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.