Properties

Label 84270.bb
Number of curves 8
Conductor 84270
CM no
Rank 1
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("84270.bb1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 84270.bb

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
84270.bb1 84270bg7 [1, 1, 1, -14981860, -22326372163] [2] 3594240  
84270.bb2 84270bg8 [1, 1, 1, -1273940, -75856195] [2] 3594240  
84270.bb3 84270bg6 [1, 1, 1, -936860, -348756163] [2, 2] 1797120  
84270.bb4 84270bg5 [1, 1, 1, -810455, 280487927] [2] 1198080  
84270.bb5 84270bg4 [1, 1, 1, -192475, -28075105] [2] 1198080  
84270.bb6 84270bg2 [1, 1, 1, -52025, 4116035] [2, 2] 599040  
84270.bb7 84270bg3 [1, 1, 1, -37980, -9339075] [2] 898560  
84270.bb8 84270bg1 [1, 1, 1, 4155, 318267] [2] 299520 \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 84270.bb have rank \(1\).

Modular form 84270.2.a.bb

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

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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.