Properties

Label 239343f
Number of curves $6$
Conductor $239343$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("239343.f1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 239343f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
239343.f4 239343f1 [1, 0, 1, -194587, -33054439] [2] 884736 \(\Gamma_0(N)\)-optimal
239343.f3 239343f2 [1, 0, 1, -196392, -32410415] [2, 2] 1769472  
239343.f2 239343f3 [1, 0, 1, -501437, 92658035] [2, 2] 3538944  
239343.f5 239343f4 [1, 0, 1, 79773, -116254109] [2] 3538944  
239343.f1 239343f5 [1, 0, 1, -7282822, 7563031751] [2] 7077888  
239343.f6 239343f6 [1, 0, 1, 1399228, 627885299] [2] 7077888  

Rank

sage: E.rank()
 

The elliptic curves in class 239343f have rank \(0\).

Modular form 239343.2.a.f

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

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.