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Properties

Label 6006.b
Number of curves $2$
Conductor $6006$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("b1")
 
E.isogeny_class()
 

Elliptic curves in class 6006.b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
6006.b1 6006e2 \([1, 1, 0, -1512, -1890]\) \(382672988497801/220768590042\) \(220768590042\) \([2]\) \(16128\) \(0.86624\)  
6006.b2 6006e1 \([1, 1, 0, 378, 0]\) \(5948434379159/3453077628\) \(-3453077628\) \([2]\) \(8064\) \(0.51966\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 6006.b have rank \(1\).

Complex multiplication

The elliptic curves in class 6006.b do not have complex multiplication.

Modular form 6006.2.a.b

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{3} + q^{4} - 4 q^{5} + q^{6} - q^{7} - q^{8} + q^{9} + 4 q^{10} + q^{11} - q^{12} + q^{13} + q^{14} + 4 q^{15} + q^{16} - 6 q^{17} - q^{18} + 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.