Properties

Label 6270l
Number of curves 8
Conductor 6270
CM no
Rank 0
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("6270.l1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 6270l

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
6270.l7 6270l1 [1, 0, 1, -86508, 8587306] [6] 64512 \(\Gamma_0(N)\)-optimal
6270.l5 6270l2 [1, 0, 1, -1336508, 594587306] [2, 6] 129024  
6270.l4 6270l3 [1, 0, 1, -1745883, -886935194] [2] 193536  
6270.l2 6270l4 [1, 0, 1, -21384008, 38059355306] [6] 258048  
6270.l6 6270l5 [1, 0, 1, -1289008, 638819306] [6] 258048  
6270.l3 6270l6 [1, 0, 1, -2257883, -324349594] [2, 2] 387072  
6270.l1 6270l7 [1, 0, 1, -21424283, 37908785126] [2] 774144  
6270.l8 6270l8 [1, 0, 1, 8716517, -2549957914] [2] 774144  

Rank

sage: E.rank()
 

The elliptic curves in class 6270l have rank \(0\).

Modular form 6270.2.a.l

sage: E.q_eigenform(10)
 
\( q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} - 4q^{7} - q^{8} + q^{9} - q^{10} - q^{11} + q^{12} + 2q^{13} + 4q^{14} + q^{15} + q^{16} + 6q^{17} - q^{18} + q^{19} + O(q^{20}) \)

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.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.