Properties

Label 356928fa
Number of curves $6$
Conductor $356928$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("356928.fa1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 356928fa

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
356928.fa5 356928fa1 [0, 1, 0, -259809, -73204065] [2] 5505024 \(\Gamma_0(N)\)-optimal
356928.fa4 356928fa2 [0, 1, 0, -4640289, -3848301729] [2, 2] 11010048  
356928.fa3 356928fa3 [0, 1, 0, -5127009, -2992161249] [2, 2] 22020096  
356928.fa1 356928fa4 [0, 1, 0, -74241249, -246240605025] [2] 22020096  
356928.fa2 356928fa5 [0, 1, 0, -32545569, 69189939807] [2] 44040192  
356928.fa6 356928fa6 [0, 1, 0, 14504031, -20373484065] [2] 44040192  

Rank

sage: E.rank()
 

The elliptic curves in class 356928fa have rank \(0\).

Modular form 356928.2.a.fa

sage: E.q_eigenform(10)
 
\( q + q^{3} - 2q^{5} + q^{9} + q^{11} - 2q^{15} - 6q^{17} + 4q^{19} + O(q^{20}) \)

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.