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Properties

Label 66654.f
Number of curves $2$
Conductor $66654$
CM no
Rank $2$
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Show commands: SageMath
E = EllipticCurve("f1")
 
E.isogeny_class()
 

Elliptic curves in class 66654.f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
66654.f1 66654c2 \([1, -1, 0, -3066, 60596]\) \(306177219/28672\) \(298541666304\) \([]\) \(138240\) \(0.94023\)  
66654.f2 66654c1 \([1, -1, 0, -651, -6219]\) \(2138072571/5488\) \(78385104\) \([]\) \(46080\) \(0.39092\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 66654.f have rank \(2\).

Complex multiplication

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

Modular form 66654.2.a.f

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.