Properties

Label 127050fh
Number of curves 8
Conductor 127050
CM no
Rank 1
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("127050.fh1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 127050fh

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
127050.fh7 127050fh1 [1, 1, 1, 635187, -146561469] [2] 3932160 \(\Gamma_0(N)\)-optimal
127050.fh6 127050fh2 [1, 1, 1, -3236813, -1315905469] [2, 2] 7864320  
127050.fh5 127050fh3 [1, 1, 1, -22838813, 41063618531] [2, 2] 15728640  
127050.fh4 127050fh4 [1, 1, 1, -45586813, -118456005469] [2] 15728640  
127050.fh8 127050fh5 [1, 1, 1, 3841687, 131297069531] [2] 31457280  
127050.fh2 127050fh6 [1, 1, 1, -363151313, 2663511743531] [2, 2] 31457280  
127050.fh3 127050fh7 [1, 1, 1, -360882563, 2698436881031] [2] 62914560  
127050.fh1 127050fh8 [1, 1, 1, -5810420063, 170472072856031] [2] 62914560  

Rank

sage: E.rank()
 

The elliptic curves in class 127050fh have rank \(1\).

Modular form 127050.2.a.fh

sage: E.q_eigenform(10)
 
\( q + q^{2} - q^{3} + q^{4} - q^{6} - q^{7} + q^{8} + q^{9} - q^{12} - 2q^{13} - q^{14} + q^{16} + 2q^{17} + q^{18} - 4q^{19} + O(q^{20}) \)

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 Cremona numbering.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.