A276182 Numbers N such that the modular curve X_0(N) is hyperelliptic.

22, 23, 26, 28, 29, 30, 31, 33, 35, 37, 39, 40, 41, 46, 47, 48, 50, 59, 71

Offset: 1

Author

Gheorghe Coserea, Oct 17 2016

Keywords

• nonn
• fini
• full

Comments

"The only case where the hyperelliptic involution is not defined by an element of SL(2, R) is N=37."

"For N = 40, 48 the hyperelliptic involution v is not of Atkin-Lehner type. The remaining sixteen values are listed in the table below, together with their genera and hyperelliptic involutions v." (see Ogg link)

n N g v

1 22 2 11

2 23 2 23

3 26 2 26

4 28 2 7

5 29 2 29

6 30 3 15

7 31 2 31

8 33 3 11

9 35 3 35

10 39 3 39

11 41 3 41

12 46 5 23

13 47 4 47

14 50 2 50

15 59 5 59

16 71 6 71

References

• J. S. Balakrishnan, B. Mazur, and N. Dogra, Ogg's torsion conjecture: fifty years later, Bull. Amer. Math. Soc., 62:2 (2025), 235-268.

Links

• Andrew P. Ogg, Hyperelliptic modular curves, Bulletin de la S. M. F., 102 (1974), p. 449-462.

Crossrefs

Cf. A001617, A260990.