Properties

Label 177870.ff
Number of curves 8
Conductor 177870
CM no
Rank 0
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("177870.ff1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 177870.ff

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
177870.ff1 177870dy8 [1, 1, 1, -2082457616, -36578228819791] [2] 79626240  
177870.ff2 177870dy6 [1, 1, 1, -130156496, -571548803407] [2, 2] 39813120  
177870.ff3 177870dy7 [1, 1, 1, -120670096, -658383514447] [2] 79626240  
177870.ff4 177870dy5 [1, 1, 1, -25835741, -49667231041] [2] 26542080  
177870.ff5 177870dy3 [1, 1, 1, -8730576, -7549690191] [2] 19906560  
177870.ff6 177870dy2 [1, 1, 1, -3424121, 1278863543] [2, 2] 13271040  
177870.ff7 177870dy1 [1, 1, 1, -2949801, 1948034199] [2] 6635520 \(\Gamma_0(N)\)-optimal
177870.ff8 177870dy4 [1, 1, 1, 11398379, 9407522543] [2] 26542080  

Rank

sage: E.rank()
 

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

Modular form 177870.2.a.ff

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

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.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 3 & 4 & 6 & 12 & 12 \\ 2 & 1 & 2 & 6 & 2 & 3 & 6 & 6 \\ 4 & 2 & 1 & 12 & 4 & 6 & 12 & 3 \\ 3 & 6 & 12 & 1 & 12 & 2 & 4 & 4 \\ 4 & 2 & 4 & 12 & 1 & 6 & 3 & 12 \\ 6 & 3 & 6 & 2 & 6 & 1 & 2 & 2 \\ 12 & 6 & 12 & 4 & 3 & 2 & 1 & 4 \\ 12 & 6 & 3 & 4 & 12 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.