Properties

Label 257754.bb
Number of curves $6$
Conductor $257754$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("257754.bb1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 257754.bb

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
257754.bb1 257754bb5 [1, 0, 1, -4952416052, 134144366143754] [2] 212336640  
257754.bb2 257754bb3 [1, 0, 1, -310114892, 2087610425930] [2, 2] 106168320  
257754.bb3 257754bb6 [1, 0, 1, -105630052, 4799570167946] [2] 212336640  
257754.bb4 257754bb2 [1, 0, 1, -32751372, -18133417910] [2, 2] 53084160  
257754.bb5 257754bb1 [1, 0, 1, -25358092, -49090559926] [2] 26542080 \(\Gamma_0(N)\)-optimal
257754.bb6 257754bb4 [1, 0, 1, 126319668, -142590599606] [2] 106168320  

Rank

sage: E.rank()
 

The elliptic curves in class 257754.bb have rank \(1\).

Modular form 257754.2.a.bb

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

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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.