Properties

Label 356928t
Number of curves $6$
Conductor $356928$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("356928.t1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 356928t

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
356928.t5 356928t1 [0, -1, 0, -259809, 73204065] [2] 5505024 \(\Gamma_0(N)\)-optimal
356928.t4 356928t2 [0, -1, 0, -4640289, 3848301729] [2, 2] 11010048  
356928.t1 356928t3 [0, -1, 0, -74241249, 246240605025] [2] 22020096  
356928.t3 356928t4 [0, -1, 0, -5127009, 2992161249] [2, 2] 22020096  
356928.t6 356928t5 [0, -1, 0, 14504031, 20373484065] [2] 44040192  
356928.t2 356928t6 [0, -1, 0, -32545569, -69189939807] [2] 44040192  

Rank

sage: E.rank()
 

The elliptic curves in class 356928t have rank \(1\).

Modular form 356928.2.a.t

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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.