Properties

Label 54.a
Number of curves $3$
Conductor $54$
CM no
Rank $0$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("a1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 54.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
54.a1 54a2 \([1, -1, 0, -123, -667]\) \(-1167051/512\) \(-90699264\) \([]\) \(18\) \(0.23410\)  
54.a2 54a3 \([1, -1, 0, -3, 3]\) \(-132651/2\) \(-54\) \([3]\) \(18\) \(-0.86451\)  
54.a3 54a1 \([1, -1, 0, 12, 8]\) \(9261/8\) \(-157464\) \([3]\) \(6\) \(-0.31520\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 54.a have rank \(0\).

Complex multiplication

The elliptic curves in class 54.a do not have complex multiplication.

Modular form 54.2.a.a

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{4} + 3 q^{5} - q^{7} - q^{8} - 3 q^{10} - 3 q^{11} - 4 q^{13} + q^{14} + q^{16} + 2 q^{19} + O(q^{20})\)  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.

\(\left(\begin{array}{rrr} 1 & 9 & 3 \\ 9 & 1 & 3 \\ 3 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.