Properties

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

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Show commands for: SageMath
sage: E = EllipticCurve("bd1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 57222.bd

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
57222.bd1 57222x1 [1, -1, 0, -509850, 140162548] [2] 884736 \(\Gamma_0(N)\)-optimal
57222.bd2 57222x2 [1, -1, 0, -405810, 198945148] [2] 1769472  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 57222.2.a.bd

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

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.