Properties

Label 177870hp
Number of curves $8$
Conductor $177870$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("177870.bs1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 177870hp

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
177870.bs7 177870hp1 [1, 1, 0, -243212, -16345776] [2] 3317760 \(\Gamma_0(N)\)-optimal
177870.bs5 177870hp2 [1, 1, 0, -2140492, 1192980496] [2, 2] 6635520  
177870.bs4 177870hp3 [1, 1, 0, -15895772, -24399903744] [2] 9953280  
177870.bs2 177870hp4 [1, 1, 0, -34157092, 76822593016] [2] 13271040  
177870.bs6 177870hp5 [1, 1, 0, -480372, 2998194984] [2] 13271040  
177870.bs3 177870hp6 [1, 1, 0, -16014352, -24017530676] [2, 2] 19906560  
177870.bs1 177870hp7 [1, 1, 0, -38248102, 57264612574] [2] 39813120  
177870.bs8 177870hp8 [1, 1, 0, 4322118, -80825425974] [2] 39813120  

Rank

sage: E.rank()
 

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

Modular form 177870.2.a.bs

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

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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.