Properties

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

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Show commands for: SageMath
sage: E = EllipticCurve("177450.hb1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 177450.hb

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
177450.hb1 177450dr8 [1, 1, 1, -169804672463, -26932316475430969] [2] 445906944  
177450.hb2 177450dr6 [1, 1, 1, -10612798713, -420820204853469] [2, 2] 222953472  
177450.hb3 177450dr7 [1, 1, 1, -10482372963, -431667192777969] [2] 445906944  
177450.hb4 177450dr5 [1, 1, 1, -2097311213, -36909542328469] [2] 148635648  
177450.hb5 177450dr3 [1, 1, 1, -671458213, -6405484770469] [4] 111476736  
177450.hb6 177450dr2 [1, 1, 1, -174936213, -157577078469] [2, 2] 74317824  
177450.hb7 177450dr1 [1, 1, 1, -108688213, 433752569531] [4] 37158912 \(\Gamma_0(N)\)-optimal
177450.hb8 177450dr4 [1, 1, 1, 687470787, -1249384340469] [2] 148635648  

Rank

sage: E.rank()
 

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

Modular form 177450.2.a.hb

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.