Properties

Label 106470.fp
Number of curves 8
Conductor 106470
CM no
Rank 1
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("106470.fp1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 106470.fp

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
106470.fp1 106470gc8 [1, -1, 1, -2921536832, 60781355487389] [2] 37748736  
106470.fp2 106470gc6 [1, -1, 1, -182596082, 949742591789] [2, 2] 18874368  
106470.fp3 106470gc7 [1, -1, 1, -181455332, 962194106189] [2] 37748736  
106470.fp4 106470gc4 [1, -1, 1, -22921502, -42221557699] [2] 9437184  
106470.fp5 106470gc3 [1, -1, 1, -11483582, 14647001789] [2, 2] 9437184  
106470.fp6 106470gc2 [1, -1, 1, -1627502, -468282499] [2, 2] 4718592  
106470.fp7 106470gc1 [1, -1, 1, 319378, -52428931] [2] 2359296 \(\Gamma_0(N)\)-optimal
106470.fp8 106470gc5 [1, -1, 1, 1931638, 46811333261] [2] 18874368  

Rank

sage: E.rank()
 

The elliptic curves in class 106470.fp have rank \(1\).

Modular form 106470.2.a.fp

sage: E.q_eigenform(10)
 
\( q + q^{2} + q^{4} + q^{5} + q^{7} + q^{8} + q^{10} - 4q^{11} + q^{14} + q^{16} - 2q^{17} - 4q^{19} + O(q^{20}) \)

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.