Properties

Label 57c
Number of curves 2
Conductor 57
CM no
Rank 0
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("57.b1")
sage: E.isogeny_class()

Elliptic curves in class 57c

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
57.b2 57c1 [0, 1, 1, 20, -32] 5 12 \(\Gamma_0(N)\)-optimal
57.b1 57c2 [0, 1, 1, -4390, -113432] 1 60  

Rank

sage: E.rank()

The elliptic curves in class 57c have rank \(0\).

Modular form 57.2.a.b

sage: E.q_eigenform(10)
\( q - 2q^{2} + q^{3} + 2q^{4} + q^{5} - 2q^{6} + 3q^{7} + q^{9} - 2q^{10} - 3q^{11} + 2q^{12} - 6q^{13} - 6q^{14} + q^{15} - 4q^{16} + 3q^{17} - 2q^{18} - q^{19} + O(q^{20}) \)

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.

\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.