Properties

Label 335730.v
Number of curves $6$
Conductor $335730$
CM no
Rank $2$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("335730.v1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 335730.v

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
335730.v1 335730v5 [1, 1, 0, -111014727, 450167758089] [2] 28311552  
335730.v2 335730v3 [1, 1, 0, -6938427, 7031687949] [2, 2] 14155776  
335730.v3 335730v6 [1, 1, 0, -6830127, 7261955409] [2] 28311552  
335730.v4 335730v4 [1, 1, 0, -1335707, -463815699] [2] 14155776  
335730.v5 335730v2 [1, 1, 0, -440427, 106119549] [2, 2] 7077888  
335730.v6 335730v1 [1, 1, 0, 21653, 6957181] [2] 3538944 \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 335730.v have rank \(2\).

Modular form 335730.2.a.v

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

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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.