Properties

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

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Show commands: SageMath
sage: E = EllipticCurve("c1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 57c

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
57.b2 57c1 \([0, 1, 1, 20, -32]\) \(841232384/1121931\) \(-1121931\) \([5]\) \(12\) \(-0.15331\) \(\Gamma_0(N)\)-optimal
57.b1 57c2 \([0, 1, 1, -4390, -113432]\) \(-9358714467168256/22284891\) \(-22284891\) \([]\) \(60\) \(0.65141\)  

Rank

sage: E.rank()
 

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

Complex multiplication

The elliptic curves in class 57c do not have complex multiplication.

Modular form 57.2.a.c

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

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.