Properties

Label 42350bb
Number of curves 6
Conductor 42350
CM no
Rank 1
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("42350.bl1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 42350bb

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
42350.bl5 42350bb1 [1, 1, 0, -1575, -49375] [2] 69120 \(\Gamma_0(N)\)-optimal
42350.bl4 42350bb2 [1, 1, 0, -31825, -2197125] [2] 138240  
42350.bl6 42350bb3 [1, 1, 0, 13550, 1024500] [2] 207360  
42350.bl3 42350bb4 [1, 1, 0, -107450, 11067500] [2] 414720  
42350.bl2 42350bb5 [1, 1, 0, -515825, 142791125] [2] 622080  
42350.bl1 42350bb6 [1, 1, 0, -8259825, 9133575125] [2] 1244160  

Rank

sage: E.rank()
 

The elliptic curves in class 42350bb have rank \(1\).

Modular form 42350.2.a.bl

sage: E.q_eigenform(10)
 
\( q - q^{2} + 2q^{3} + q^{4} - 2q^{6} + q^{7} - q^{8} + q^{9} + 2q^{12} - 4q^{13} - q^{14} + q^{16} + 6q^{17} - q^{18} - 2q^{19} + O(q^{20}) \)

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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 6 & 9 & 18 \\ 2 & 1 & 6 & 3 & 18 & 9 \\ 3 & 6 & 1 & 2 & 3 & 6 \\ 6 & 3 & 2 & 1 & 6 & 3 \\ 9 & 18 & 3 & 6 & 1 & 2 \\ 18 & 9 & 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.