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Properties

Label 66654.x
Number of curves $2$
Conductor $66654$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("x1")
 
E.isogeny_class()
 

Elliptic curves in class 66654.x

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
66654.x1 66654e2 \([1, -1, 0, -1622013, -727539643]\) \(306177219/28672\) \(44194880974853984256\) \([]\) \(3179520\) \(2.5080\)  
66654.x2 66654e1 \([1, -1, 0, -344478, 77733252]\) \(2138072571/5488\) \(11603808554997456\) \([3]\) \(1059840\) \(1.9587\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 66654.x have rank \(1\).

Complex multiplication

The elliptic curves in class 66654.x do not have complex multiplication.

Modular form 66654.2.a.x

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{4} + 3 q^{5} + q^{7} - q^{8} - 3 q^{10} - 6 q^{11} - 4 q^{13} - q^{14} + q^{16} + 3 q^{17} - 4 q^{19} + O(q^{20})\) Copy content 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.