Properties

Label 6041.a
Number of curves $2$
Conductor $6041$
CM no
Rank $0$
Graph

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

Elliptic curves in class 6041.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
6041.a1 6041a2 \([1, 1, 0, -4602, 118265]\) \(10782289820289961/42287\) \(42287\) \([2]\) \(6180\) \(0.52334\)  
6041.a2 6041a1 \([1, 1, 0, -287, 1760]\) \(-2628643361401/5213383\) \(-5213383\) \([2]\) \(3090\) \(0.17677\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 6041.a have rank \(0\).

Complex multiplication

The elliptic curves in class 6041.a do not have complex multiplication.

Modular form 6041.2.a.a

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