Properties

Label 222.b
Number of curves $4$
Conductor $222$
CM no
Rank $0$
Graph

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

Elliptic curves in class 222.b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
222.b1 222c3 \([1, 1, 0, -804, -9108]\) \(57588477431113/78653268\) \(78653268\) \([2]\) \(144\) \(0.41989\)  
222.b2 222c4 \([1, 1, 0, -604, 5428]\) \(24431916147913/202409388\) \(202409388\) \([4]\) \(144\) \(0.41989\)  
222.b3 222c2 \([1, 1, 0, -64, -80]\) \(29704593673/15968016\) \(15968016\) \([2, 2]\) \(72\) \(0.073321\)  
222.b4 222c1 \([1, 1, 0, 16, 0]\) \(410172407/255744\) \(-255744\) \([2]\) \(36\) \(-0.27325\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 222.b have rank \(0\).

Complex multiplication

The elliptic curves in class 222.b do not have complex multiplication.

Modular form 222.2.a.b

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.