Properties

Label 177870.k
Number of curves 8
Conductor 177870
CM no
Rank 1
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("177870.k1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 177870.k

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
177870.k1 177870il8 [1, 1, 0, -11388423323, -467786756340273] [2] 125829120  
177870.k2 177870il6 [1, 1, 0, -711776573, -7309388000823] [2, 2] 62914560  
177870.k3 177870il7 [1, 1, 0, -707329823, -7405218131373] [2] 125829120  
177870.k4 177870il3 [1, 1, 0, -89350153, 324953928853] [2] 31457280  
177870.k5 177870il4 [1, 1, 0, -44764073, -112723333323] [2, 2] 31457280  
177870.k6 177870il2 [1, 1, 0, -6344153, 3604500453] [2, 2] 15728640  
177870.k7 177870il1 [1, 1, 0, 1244967, 403409637] [2] 7864320 \(\Gamma_0(N)\)-optimal
177870.k8 177870il5 [1, 1, 0, 7529707, -360271629087] [2] 62914560  

Rank

sage: E.rank()
 

The elliptic curves in class 177870.k have rank \(1\).

Modular form 177870.2.a.k

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 LMFDB numbering.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 16 & 4 & 8 & 16 & 8 \\ 2 & 1 & 2 & 8 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 16 & 4 & 8 & 16 & 8 \\ 16 & 8 & 16 & 1 & 4 & 2 & 4 & 8 \\ 4 & 2 & 4 & 4 & 1 & 2 & 4 & 2 \\ 8 & 4 & 8 & 2 & 2 & 1 & 2 & 4 \\ 16 & 8 & 16 & 4 & 4 & 2 & 1 & 8 \\ 8 & 4 & 8 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.