Properties

Label 177450.hh
Number of curves $8$
Conductor $177450$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("177450.hh1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 177450.hh

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
177450.hh1 177450dv7 [1, 1, 1, -27255563, 34445453531] [2] 31850496  
177450.hh2 177450dv4 [1, 1, 1, -24340313, 46210642031] [2] 10616832  
177450.hh3 177450dv6 [1, 1, 1, -11411813, -14448358969] [2, 2] 15925248  
177450.hh4 177450dv3 [1, 1, 1, -11327313, -14678367969] [2] 7962624  
177450.hh5 177450dv2 [1, 1, 1, -1525313, 717532031] [2, 2] 5308416  
177450.hh6 177450dv5 [1, 1, 1, -342313, 1803526031] [2] 10616832  
177450.hh7 177450dv1 [1, 1, 1, -173313, -9843969] [2] 2654208 \(\Gamma_0(N)\)-optimal
177450.hh8 177450dv8 [1, 1, 1, 3079937, -48619905469] [2] 31850496  

Rank

sage: E.rank()
 

The elliptic curves in class 177450.hh have rank \(1\).

Modular form 177450.2.a.hh

sage: E.q_eigenform(10)
 
\( q + q^{2} - q^{3} + q^{4} - q^{6} + q^{7} + q^{8} + q^{9} - q^{12} + q^{14} + q^{16} + 6q^{17} + q^{18} + 4q^{19} + O(q^{20}) \)

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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.