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Properties

Label 4043.a
Number of curves $2$
Conductor $4043$
CM no
Rank $1$
Graph

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

Elliptic curves in class 4043.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
4043.a1 4043a1 \([0, -1, 1, -1190, 16212]\) \(-186521757331456/115472123\) \(-115472123\) \([5]\) \(4920\) \(0.48942\) \(\Gamma_0(N)\)-optimal
4043.a2 4043a2 \([0, -1, 1, 8040, -105468]\) \(57469555823243264/37822070293163\) \(-37822070293163\) \([]\) \(24600\) \(1.2941\)  

Rank

sage: E.rank()
 

The elliptic curves in class 4043.a have rank \(1\).

Complex multiplication

The elliptic curves in class 4043.a do not have complex multiplication.

Modular form 4043.2.a.a

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

\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.