Properties

Label 67270.bd
Number of curves $4$
Conductor $67270$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("67270.bd1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 67270.bd

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
67270.bd1 67270y4 [1, -1, 1, -257248, 50281697] [2] 460800  
67270.bd2 67270y3 [1, -1, 1, -84268, -8777519] [2] 460800  
67270.bd3 67270y2 [1, -1, 1, -16998, 694097] [2, 2] 230400  
67270.bd4 67270y1 [1, -1, 1, 2222, 63681] [2] 115200 \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 67270.bd have rank \(1\).

Modular form 67270.2.a.bd

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

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.