Properties

Label 350658.dw
Number of curves $6$
Conductor $350658$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("350658.dw1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 350658.dw

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
350658.dw1 350658dw6 [1, -1, 1, -86176949, -307893151695] [2] 41943040  
350658.dw2 350658dw3 [1, -1, 1, -19290569, 32615063385] [2] 20971520  
350658.dw3 350658dw4 [1, -1, 1, -5525609, -4547331687] [2, 2] 20971520  
350658.dw4 350658dw2 [1, -1, 1, -1256729, 464333433] [2, 2] 10485760  
350658.dw5 350658dw1 [1, -1, 1, 137191, 40024185] [2] 5242880 \(\Gamma_0(N)\)-optimal
350658.dw6 350658dw5 [1, -1, 1, 6823651, -21999305919] [2] 41943040  

Rank

sage: E.rank()
 

The elliptic curves in class 350658.dw have rank \(1\).

Modular form 350658.2.a.dw

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

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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.