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Properties

Label 61710.o
Number of curves $8$
Conductor $61710$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("o1")
 
E.isogeny_class()
 

Elliptic curves in class 61710.o

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
61710.o1 61710n8 \([1, 1, 0, -13729872, 19575769734]\) \(161572377633716256481/914742821250\) \(1620522707156471250\) \([2]\) \(2621440\) \(2.6857\)  
61710.o2 61710n4 \([1, 1, 0, -2632962, -1645526076]\) \(1139466686381936641/4080\) \(7227968880\) \([2]\) \(655360\) \(1.9925\)  
61710.o3 61710n6 \([1, 1, 0, -873622, 293965984]\) \(41623544884956481/2962701562500\) \(5248606542764062500\) \([2, 2]\) \(1310720\) \(2.3391\)  
61710.o4 61710n3 \([1, 1, 0, -174242, -22573404]\) \(330240275458561/67652010000\) \(119849662487610000\) \([2, 2]\) \(655360\) \(1.9925\)  
61710.o5 61710n2 \([1, 1, 0, -164562, -25761996]\) \(278202094583041/16646400\) \(29490113030400\) \([2, 2]\) \(327680\) \(1.6460\)  
61710.o6 61710n1 \([1, 1, 0, -9682, -454604]\) \(-56667352321/16711680\) \(-29605760532480\) \([2]\) \(163840\) \(1.2994\) \(\Gamma_0(N)\)-optimal
61710.o7 61710n5 \([1, 1, 0, 370258, -134849304]\) \(3168685387909439/6278181696900\) \(-11122181845141860900\) \([2]\) \(1310720\) \(2.3391\)  
61710.o8 61710n7 \([1, 1, 0, 792548, 1284670666]\) \(31077313442863199/420227050781250\) \(-744457854309082031250\) \([2]\) \(2621440\) \(2.6857\)  

Rank

sage: E.rank()
 

The elliptic curves in class 61710.o have rank \(1\).

Complex multiplication

The elliptic curves in class 61710.o do not have complex multiplication.

Modular form 61710.2.a.o

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

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.