Properties

Label 22743n
Number of curves $6$
Conductor $22743$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("n1")
 
E.isogeny_class()
 

Elliptic curves in class 22743n

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
22743.f6 22743n1 \([1, -1, 1, 3181, -16014]\) \(103823/63\) \(-2160676176687\) \([2]\) \(27648\) \(1.0560\) \(\Gamma_0(N)\)-optimal
22743.f5 22743n2 \([1, -1, 1, -13064, -119982]\) \(7189057/3969\) \(136122599131281\) \([2, 2]\) \(55296\) \(1.4026\)  
22743.f3 22743n3 \([1, -1, 1, -126779, 17301156]\) \(6570725617/45927\) \(1575132932804823\) \([2]\) \(110592\) \(1.7492\)  
22743.f2 22743n4 \([1, -1, 1, -159269, -24390012]\) \(13027640977/21609\) \(741111928603641\) \([2, 2]\) \(110592\) \(1.7492\)  
22743.f4 22743n5 \([1, -1, 1, -110534, -39634320]\) \(-4354703137/17294403\) \(-593136580192447347\) \([2]\) \(221184\) \(2.0957\)  
22743.f1 22743n6 \([1, -1, 1, -2547284, -1564182084]\) \(53297461115137/147\) \(5041577745603\) \([2]\) \(221184\) \(2.0957\)  

Rank

sage: E.rank()
 

The elliptic curves in class 22743n have rank \(1\).

Complex multiplication

The elliptic curves in class 22743n do not have complex multiplication.

Modular form 22743.2.a.n

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

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.