Properties

Label 57222.b
Number of curves $2$
Conductor $57222$
CM no
Rank $1$
Graph

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

Elliptic curves in class 57222.b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
57222.b1 57222z1 \([1, -1, 0, -77483844, -260289446576]\) \(595099203230897/5780865024\) \(499759029976203487936512\) \([2]\) \(23396352\) \(3.3670\) \(\Gamma_0(N)\)-optimal
57222.b2 57222z2 \([1, -1, 0, -20886084, -632419718576]\) \(-11655394135217/1991891886336\) \(-172200172949886458856999168\) \([2]\) \(46792704\) \(3.7136\)  

Rank

sage: E.rank()
 

The elliptic curves in class 57222.b have rank \(1\).

Complex multiplication

The elliptic curves in class 57222.b do not have complex multiplication.

Modular form 57222.2.a.b

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.