Properties

Label 57.c
Number of curves $4$
Conductor $57$
CM no
Rank $0$
Graph

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

Elliptic curves in class 57.c

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
57.c1 57b3 \([1, 0, 1, -102, 385]\) \(115714886617/1539\) \(1539\) \([4]\) \(6\) \(-0.24501\)  
57.c2 57b1 \([1, 0, 1, -7, 5]\) \(30664297/3249\) \(3249\) \([2, 2]\) \(3\) \(-0.59158\) \(\Gamma_0(N)\)-optimal
57.c3 57b2 \([1, 0, 1, -2, -1]\) \(389017/57\) \(57\) \([2]\) \(6\) \(-0.93816\)  
57.c4 57b4 \([1, 0, 1, 8, 29]\) \(67419143/390963\) \(-390963\) \([2]\) \(6\) \(-0.24501\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 57.2.a.c

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{3} - q^{4} - 2q^{5} + q^{6} - 3q^{8} + q^{9} - 2q^{10} - q^{12} + 6q^{13} - 2q^{15} - q^{16} - 6q^{17} + q^{18} - q^{19} + O(q^{20})\)  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 LMFDB numbering.

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.