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Properties

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

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

Elliptic curves in class 6006.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
6006.a1 6006c2 \([1, 1, 0, -2977, -32195]\) \(2919363263895961/1275513751512\) \(1275513751512\) \([2]\) \(13824\) \(1.0189\)  
6006.a2 6006c1 \([1, 1, 0, -2537, -50235]\) \(1806976738085401/932323392\) \(932323392\) \([2]\) \(6912\) \(0.67229\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 6006.2.a.a

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