Properties

Label 57498b
Number of curves $6$
Conductor $57498$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("57498.f1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 57498b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
57498.f5 57498b1 [1, 1, 0, -5504, 344832] [2] 193536 \(\Gamma_0(N)\)-optimal
57498.f4 57498b2 [1, 1, 0, -115024, 14954800] [2, 2] 387072  
57498.f3 57498b3 [1, 1, 0, -142404, 7261020] [2, 2] 774144  
57498.f1 57498b4 [1, 1, 0, -1839964, 959876932] [2] 774144  
57498.f6 57498b5 [1, 1, 0, 528406, 56766798] [2] 1548288  
57498.f2 57498b6 [1, 1, 0, -1251294, -534099078] [2] 1548288  

Rank

sage: E.rank()
 

The elliptic curves in class 57498b have rank \(1\).

Modular form 57498.2.a.f

sage: E.q_eigenform(10)
 
\( q - q^{2} - q^{3} + q^{4} + 2q^{5} + q^{6} - q^{7} - q^{8} + q^{9} - 2q^{10} - 4q^{11} - q^{12} - 6q^{13} + q^{14} - 2q^{15} + q^{16} - 2q^{17} - q^{18} + 4q^{19} + O(q^{20}) \)

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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.