Properties

Label 97682e
Number of curves 4
Conductor 97682
CM no
Rank 1
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("97682.h1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 97682e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
97682.h4 97682e1 [1, 1, 0, -147540, 13443344] [2] 1327104 \(\Gamma_0(N)\)-optimal
97682.h3 97682e2 [1, 1, 0, -2101180, 1171170408] [2] 2654208  
97682.h2 97682e3 [1, 1, 0, -5031640, -4345713588] [2] 3981312  
97682.h1 97682e4 [1, 1, 0, -5520050, -3451825606] [2] 7962624  

Rank

sage: E.rank()
 

The elliptic curves in class 97682e have rank \(1\).

Modular form 97682.2.a.h

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

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 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.