Properties

Label 221a
Number of curves $2$
Conductor $221$
CM no
Rank $0$
Graph

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

Elliptic curves in class 221a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
221.a2 221a1 \([1, -1, 1, -733, 7804]\) \(43499078731809/82055753\) \(82055753\) \([2]\) \(120\) \(0.40860\) \(\Gamma_0(N)\)-optimal
221.a1 221a2 \([1, -1, 1, -11718, 491144]\) \(177930109857804849/634933\) \(634933\) \([2]\) \(240\) \(0.75517\)  

Rank

sage: E.rank()
 

The elliptic curves in class 221a have rank \(0\).

Complex multiplication

The elliptic curves in class 221a do not have complex multiplication.

Modular form 221.2.a.a

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