Properties

Label 177870ge
Number of curves $6$
Conductor $177870$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("177870.cp1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 177870ge

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
177870.cp6 177870ge1 [1, 0, 1, 59166, -7747148] [2] 1966080 \(\Gamma_0(N)\)-optimal
177870.cp5 177870ge2 [1, 0, 1, -415154, -80223244] [2, 2] 3932160  
177870.cp4 177870ge3 [1, 0, 1, -2193854, 1181942276] [2] 7864320  
177870.cp2 177870ge4 [1, 0, 1, -6225574, -5978961628] [2, 2] 7864320  
177870.cp3 177870ge5 [1, 0, 1, -5810544, -6810349724] [2] 15728640  
177870.cp1 177870ge6 [1, 0, 1, -99607324, -382643588428] [2] 15728640  

Rank

sage: E.rank()
 

The elliptic curves in class 177870ge have rank \(0\).

Modular form 177870.2.a.cp

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

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.