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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)
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.