Properties

Label 80b
Number of curves 4
Conductor 80
CM no
Rank 0
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("80.b1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 80b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
80.b4 80b1 [0, -1, 0, 4, -4] [2] 4 \(\Gamma_0(N)\)-optimal
80.b3 80b2 [0, -1, 0, -1, 0] [2] 8  
80.b2 80b3 [0, -1, 0, -36, 140] [2] 12  
80.b1 80b4 [0, -1, 0, -41, 116] [2] 24  

Rank

sage: E.rank()
 

The elliptic curves in class 80b have rank \(0\).

Modular form 80.2.a.b

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

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 Cremona numbering.

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.