Properties

Label 333795w
Number of curves $6$
Conductor $333795$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("333795.w1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 333795w

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
333795.w4 333795w1 [1, 0, 0, -76591, 8152160] [2] 1179648 \(\Gamma_0(N)\)-optimal
333795.w3 333795w2 [1, 0, 0, -78036, 7828191] [2, 2] 2359296  
333795.w5 333795w3 [1, 0, 0, 73689, 34622826] [2] 4718592  
333795.w2 333795w4 [1, 0, 0, -252881, -39694680] [2, 2] 4718592  
333795.w6 333795w5 [1, 0, 0, 525974, -235810369] [2] 9437184  
333795.w1 333795w6 [1, 0, 0, -3829256, -2884343355] [2] 9437184  

Rank

sage: E.rank()
 

The elliptic curves in class 333795w have rank \(1\).

Modular form 333795.2.a.w

sage: E.q_eigenform(10)
 
\( q - q^{2} + q^{3} - q^{4} - q^{5} - q^{6} + q^{7} + 3q^{8} + q^{9} + q^{10} - q^{11} - q^{12} - 2q^{13} - q^{14} - q^{15} - q^{16} - q^{18} - 4q^{19} + O(q^{20}) \)

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.