Properties

Label 80223.j
Number of curves $6$
Conductor $80223$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("80223.j1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 80223.j

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
80223.j1 80223c6 [1, 1, 0, -2441056, 1466719159] [2] 1310720  
80223.j2 80223c4 [1, 1, 0, -168071, 17918520] [2, 2] 655360  
80223.j3 80223c2 [1, 1, 0, -65826, -6313545] [2, 2] 327680  
80223.j4 80223c1 [1, 1, 0, -65221, -6438296] [2] 163840 \(\Gamma_0(N)\)-optimal
80223.j5 80223c3 [1, 1, 0, 26739, -22549446] [2] 655360  
80223.j6 80223c5 [1, 1, 0, 468994, 122014941] [2] 1310720  

Rank

sage: E.rank()
 

The elliptic curves in class 80223.j have rank \(0\).

Modular form 80223.2.a.j

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

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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 8 & 4 \\ 2 & 1 & 2 & 4 & 4 & 2 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 8 & 4 & 2 & 1 & 4 & 8 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.