Properties

Label 57222bh
Number of curves 4
Conductor 57222
CM no
Rank 1
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("57222.bm1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 57222bh

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
57222.bm3 57222bh1 [1, -1, 1, -14360, -621129] [2] 147456 \(\Gamma_0(N)\)-optimal
57222.bm4 57222bh2 [1, -1, 1, 11650, -2639505] [2] 294912  
57222.bm1 57222bh3 [1, -1, 1, -209435, 36802059] [2] 442368  
57222.bm2 57222bh4 [1, -1, 1, -105395, 73299291] [2] 884736  

Rank

sage: E.rank()
 

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

Modular form 57222.2.a.bm

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

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 Cremona numbering.

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.