Properties

Label 231231cb
Number of curves $6$
Conductor $231231$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("231231.cb1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 231231cb

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
231231.cb5 231231cb1 [1, 1, 0, -142419, 29615832] [2] 2949120 \(\Gamma_0(N)\)-optimal
231231.cb4 231231cb2 [1, 1, 0, -2543664, 1560169395] [2, 2] 5898240  
231231.cb1 231231cb3 [1, 1, 0, -40696779, 99911269242] [2] 11796480  
231231.cb3 231231cb4 [1, 1, 0, -2810469, 1212522480] [2, 2] 11796480  
231231.cb6 231231cb5 [1, 1, 0, 7950666, 8273979267] [2] 23592960  
231231.cb2 231231cb6 [1, 1, 0, -17840484, -28093000767] [2] 23592960  

Rank

sage: E.rank()
 

The elliptic curves in class 231231cb have rank \(0\).

Modular form 231231.2.a.cb

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

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 Cremona numbering.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.