Properties

Label 348726r
Number of curves $6$
Conductor $348726$
CM no
Rank $2$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("348726.r1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 348726r

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
348726.r5 348726r1 [1, 0, 1, 45478, -7641844] [2] 3538944 \(\Gamma_0(N)\)-optimal
348726.r4 348726r2 [1, 0, 1, -416602, -88598260] [2, 2] 7077888  
348726.r3 348726r3 [1, 0, 1, -1831722, 868022860] [2, 2] 14155776  
348726.r2 348726r4 [1, 0, 1, -6394762, -6224581684] [2] 14155776  
348726.r1 348726r5 [1, 0, 1, -28567382, 58766768156] [2] 28311552  
348726.r6 348726r6 [1, 0, 1, 2262018, 4198689724] [2] 28311552  

Rank

sage: E.rank()
 

The elliptic curves in class 348726r have rank \(2\).

Modular form 348726.2.a.r

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

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 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.