Properties

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

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Show commands for: SageMath
sage: E = EllipticCurve("b1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 221.b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
221.b1 221b1 \([1, 1, 0, -59, 152]\) \(23320116793/2873\) \(2873\) \([2]\) \(24\) \(-0.31302\) \(\Gamma_0(N)\)-optimal
221.b2 221b2 \([1, 1, 0, -54, 185]\) \(-17923019113/8254129\) \(-8254129\) \([2]\) \(48\) \(0.033555\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 221.2.a.b

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