Properties

Label 77b
Number of curves 3
Conductor 77
CM no
Rank 0
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("77.b1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 77b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
77.b2 77b1 [0, 1, 1, -49, 600] [3] 20 \(\Gamma_0(N)\)-optimal
77.b3 77b2 [0, 1, 1, 441, -15815] [] 60  
77.b1 77b3 [0, 1, 1, -89, 295] [3] 60  

Rank

sage: E.rank()
 

The elliptic curves in class 77b have rank \(0\).

Modular form 77.2.a.b

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

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.

\(\left(\begin{array}{rrr} 1 & 3 & 3 \\ 3 & 1 & 9 \\ 3 & 9 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.