Table 1 Weierstrass points on X 0 ( N )...

Table 1

Weierstrass points on X₀(N) for N ≤ 50 by W_Q

N	g₀(N)	W_Q	$g_{0}^{Q} (N)$	v(Q)	Weierstrass points
1 − 21	≤1	None
22	2	W₂, W₂₂	1	2	None
		W₁₁	0	6	$- \frac{1}{2} + \frac{\sqrt{- 11}}{22}, \pm \frac{1}{6} + \frac{\sqrt{- 11}}{66}, \pm \frac{1}{4} + \frac{\sqrt{- 11}}{44}, - \frac{1}{12} + \frac{\sqrt{- 11}}{132}$
23	2	W₂₃	0	6	$\frac{\sqrt{- 23}}{23}, \frac{1}{2} + \frac{\sqrt{- 23}}{46}, \pm \frac{1}{3} + \frac{\sqrt{- 23}}{69}, \pm \frac{1}{4} + \frac{\sqrt{- 23}}{92}$
24 − 25	≤1	None
26	2	W₂, W₁₃	1	2	None
		W₂₆	0	6	$\frac{\sqrt{- 26}}{26}, \pm \frac{1}{3} + \frac{\sqrt{- 26}}{78}, \pm \frac{2}{5} + \frac{\sqrt{- 26}}{130}, - \frac{5}{31} + \frac{\sqrt{- 26}}{806}$
27	1	none
28	2	W₄, W₂₈	1	2	None
		W₇	0	6	$\pm \frac{1}{4} + \frac{\sqrt{- 7}}{28}, \pm \frac{1}{8} + \frac{\sqrt{- 7}}{56}, \pm \frac{3}{8} + \frac{\sqrt{- 7}}{56}$
29	2	W₂₉	0	6	$\frac{\sqrt{- 29}}{29}, \frac{1}{2} + \frac{\sqrt{- 29}}{58}, \pm \frac{1}{3} + \frac{\sqrt{- 29}}{87}, \pm \frac{1}{5} + \frac{\sqrt{- 29}}{145}$
30	3	W₅, W₆, W₃₀	1	4	None
		W₂, W₃, W₁₀	2	0	None
		W₁₅	0	8	$- \frac{1}{2} + \frac{\sqrt{- 15}}{30}, \pm \frac{1}{4} + \frac{\sqrt{- 15}}{60}, \pm \frac{1}{6} + \frac{\sqrt{- 15}}{120}, - \frac{3}{8} + \frac{\sqrt{- 15}}{120}, - \frac{1}{16} + \frac{\sqrt{- 15}}{240}, - \frac{3}{34} + \frac{\sqrt{- 15}}{510}$
31	2	W₃₁	0	6	$\frac{\sqrt{- 31}}{31}, \frac{1}{2} + \frac{\sqrt{- 31}}{62}, \pm \frac{1}{4} + \frac{\sqrt{- 31}}{124}, \pm \frac{2}{5} + \frac{\sqrt{- 31}}{155}$
32	1	None
33	3	W₃	2	0	None
		W₁₁	0	8	$\pm \frac{1}{3} + \frac{\sqrt{- 11}}{33}, \pm \frac{1}{6} + \frac{\sqrt{- 11}}{66}, \pm \frac{2}{9} + \frac{\sqrt{- 11}}{99}, \pm \frac{1}{12} + \frac{\sqrt{- 11}}{132}$
		W₃₃	1	4	None
34	3	W₂, W₁₇	1	4	None
		W₃₄
35	3	W₅	1	4	None
		W₇	2	0	None
		W₃₅	0	8	$\frac{\sqrt{- 35}}{35}, \pm \frac{1}{4} + \frac{\sqrt{- 35}}{140}, - \frac{1}{6} + \frac{\sqrt{- 35}}{210}, \pm \frac{1}{12} + \frac{\sqrt{- 35}}{420}, - \frac{5}{12} + \frac{\sqrt{- 35}}{420}, - \frac{1}{18} + \frac{\sqrt{- 35}}{630}$
36	1	none
37	2	W₃₇	1	2	none
38	4	W₂	2	2	none
		W₁₉	1	6	$\pm \frac{1}{2} + \frac{\sqrt{- 76}}{38}, \pm \frac{1}{4} + \frac{\sqrt{- 76}}{152}, \pm \frac{1}{10} + \frac{\sqrt{- 76}}{380}$
		W₃₈	1	6	$\frac{\sqrt{- 38}}{38}, \frac{2}{39} + \frac{\sqrt{- 38}}{741}, \frac{2}{51} + \frac{\sqrt{- 38}}{1938}, \frac{20}{273} + \frac{\sqrt{- 38}}{546}, \frac{128}{2737} + \frac{\sqrt{- 38}}{2737}, \frac{88}{1297} + \frac{\sqrt{- 38}}{1297}$
39	3	W₃	1	4	none
		W₁₃	2	0	none
		W₃₉	0	8	$\frac{\sqrt{- 39}}{39}, \pm \frac{1}{4} + \frac{\sqrt{- 39}}{156}, \pm \frac{1}{8} + \frac{\sqrt{- 39}}{312}, - \frac{3}{8} + \frac{\sqrt{- 39}}{312}, - \frac{5}{16} + \frac{\sqrt{- 39}}{624}, - \frac{1}{20} + \frac{\sqrt{- 39}}{780}$
40	3	W₅, W₈	2	0	none
		W₄₀	1	4	none
		$(\begin{matrix} - 10 & 1 \\ - 120 & 10 \end{matrix})$	0	8	$\pm \frac{1}{4} + \frac{\sqrt{- 5}}{20}, \pm \frac{1}{12} + \frac{\sqrt{- 5}}{60}, \pm \frac{5}{12} + \frac{\sqrt{- 5}}{60}, \pm \frac{5}{36} + \frac{\sqrt{- 5}}{180}$
41	3	W₄₁	0	8	$\frac{\sqrt{- 41}}{41}, \frac{1}{2} + \frac{\sqrt{- 41}}{82}, \pm \frac{1}{3} + \frac{\sqrt{- 41}}{123}, \pm \frac{2}{5} + \frac{\sqrt{- 41}}{205}, \pm \frac{1}{6} + \frac{\sqrt{- 41}}{246}$
42	5	W₂, W₇	3	0	none
		W₃, W₆	2	4	none
		W₂₁, W₄₂
		W₁₄	1	8	$\pm \frac{1}{3} + \frac{\sqrt{- 14}}{42}, \pm \frac{1}{15} + \frac{\sqrt{- 14}}{210}, \frac{2}{57} + \frac{\sqrt{- 14}}{798}, - \frac{4}{75} + \frac{\sqrt{- 14}}{1050}, - \frac{4}{225} + \frac{\sqrt{- 14}}{3150}, - \frac{2}{3} + \frac{\sqrt{- 14}}{42}$
43	3	W₄₃	1	4	none
44	4	W₄	2	2	none
		W₁₁	1	6	$\pm \frac{1}{12} + \frac{\sqrt{- 11}}{132}, \pm \frac{1}{4} + \frac{\sqrt{- 11}}{44}, - \frac{3}{4} + \frac{\sqrt{- 11}}{44}, - \frac{3}{100} + \frac{\sqrt{- 11}}{1100}$
		W₄₄	1	6	$\frac{\sqrt{- 44}}{44}, \frac{52}{687} + \frac{\sqrt{- 44}}{687}, \frac{2}{59} + \frac{\sqrt{- 44}}{2596}, \frac{1}{15} + \frac{\sqrt{- 44}}{660}, \frac{4}{141} + \frac{\sqrt{- 44}}{6204}, \frac{1}{9} + \frac{\sqrt{- 44}}{396}$
45	3	W₅, W₉, W₄₅	1	4	none
46	5	W₂	3	0	none
		W₂₃	0	12	$- \frac{1}{2} + \frac{\sqrt{- 23}}{46}, \pm \frac{1}{4} + \frac{\sqrt{- 23}}{92}, \pm \frac{1}{6} + \frac{\sqrt{- 23}}{138}, \pm \frac{1}{8} + \frac{\sqrt{- 23}}{184}, \pm \frac{3}{8} + \frac{\sqrt{- 23}}{184}, \pm \frac{1}{12} + \frac{\sqrt{- 23}}{276}, - \frac{1}{24} + \frac{\sqrt{- 23}}{552}$
		W₄₆	2	4	none
47	4	W₄₇	0	10	$\frac{\sqrt{- 47}}{47}, \frac{1}{2} + \frac{\sqrt{- 47}}{94}, \pm \frac{1}{3} + \frac{\sqrt{- 47}}{141}, \pm \frac{1}{4} + \frac{\sqrt{- 47}}{188}, \pm \frac{1}{6} + \frac{\sqrt{- 47}}{282}, \pm \frac{2}{7} + \frac{\sqrt{- 47}}{329}$
48	3	W₃, W₁₆	2	0	none
		W₄₈	1	4	none
		$(\begin{matrix} - 6 & - 1 \\ - 48 & 6 \end{matrix})$	0	8	$\pm \frac{1}{8} + \frac{\sqrt{- 3}}{24}, \pm \frac{3}{8} + \frac{\sqrt{- 3}}{24}, \pm \frac{3}{56} + \frac{\sqrt{- 3}}{168}, \pm \frac{17}{56} + \frac{\sqrt{- 3}}{168}$
49	1	none
50	2	W₂, W₂₅	1	2	none
		W₅₀	0	6	$\frac{\sqrt{- 2}}{10}, \pm \frac{3}{11} + \frac{\sqrt{- 2}}{110}, \pm \frac{1}{17} + \frac{\sqrt{- 2}}{170}, - \frac{7}{43} + \frac{\sqrt{- 2}}{430}$

N	g₀(N)	W_Q	$g_{0}^{Q} (N)$	v(Q)	Weierstrass points
1 − 21	≤1	None
22	2	W₂, W₂₂	1	2	None
		W₁₁	0	6	$- \frac{1}{2} + \frac{\sqrt{- 11}}{22}, \pm \frac{1}{6} + \frac{\sqrt{- 11}}{66}, \pm \frac{1}{4} + \frac{\sqrt{- 11}}{44}, - \frac{1}{12} + \frac{\sqrt{- 11}}{132}$
23	2	W₂₃	0	6	$\frac{\sqrt{- 23}}{23}, \frac{1}{2} + \frac{\sqrt{- 23}}{46}, \pm \frac{1}{3} + \frac{\sqrt{- 23}}{69}, \pm \frac{1}{4} + \frac{\sqrt{- 23}}{92}$
24 − 25	≤1	None
26	2	W₂, W₁₃	1	2	None
		W₂₆	0	6	$\frac{\sqrt{- 26}}{26}, \pm \frac{1}{3} + \frac{\sqrt{- 26}}{78}, \pm \frac{2}{5} + \frac{\sqrt{- 26}}{130}, - \frac{5}{31} + \frac{\sqrt{- 26}}{806}$
27	1	none
28	2	W₄, W₂₈	1	2	None
		W₇	0	6	$\pm \frac{1}{4} + \frac{\sqrt{- 7}}{28}, \pm \frac{1}{8} + \frac{\sqrt{- 7}}{56}, \pm \frac{3}{8} + \frac{\sqrt{- 7}}{56}$
29	2	W₂₉	0	6	$\frac{\sqrt{- 29}}{29}, \frac{1}{2} + \frac{\sqrt{- 29}}{58}, \pm \frac{1}{3} + \frac{\sqrt{- 29}}{87}, \pm \frac{1}{5} + \frac{\sqrt{- 29}}{145}$
30	3	W₅, W₆, W₃₀	1	4	None
		W₂, W₃, W₁₀	2	0	None
		W₁₅	0	8	$- \frac{1}{2} + \frac{\sqrt{- 15}}{30}, \pm \frac{1}{4} + \frac{\sqrt{- 15}}{60}, \pm \frac{1}{6} + \frac{\sqrt{- 15}}{120}, - \frac{3}{8} + \frac{\sqrt{- 15}}{120}, - \frac{1}{16} + \frac{\sqrt{- 15}}{240}, - \frac{3}{34} + \frac{\sqrt{- 15}}{510}$
31	2	W₃₁	0	6	$\frac{\sqrt{- 31}}{31}, \frac{1}{2} + \frac{\sqrt{- 31}}{62}, \pm \frac{1}{4} + \frac{\sqrt{- 31}}{124}, \pm \frac{2}{5} + \frac{\sqrt{- 31}}{155}$
32	1	None
33	3	W₃	2	0	None
		W₁₁	0	8	$\pm \frac{1}{3} + \frac{\sqrt{- 11}}{33}, \pm \frac{1}{6} + \frac{\sqrt{- 11}}{66}, \pm \frac{2}{9} + \frac{\sqrt{- 11}}{99}, \pm \frac{1}{12} + \frac{\sqrt{- 11}}{132}$
		W₃₃	1	4	None
34	3	W₂, W₁₇	1	4	None
		W₃₄
35	3	W₅	1	4	None
		W₇	2	0	None
		W₃₅	0	8	$\frac{\sqrt{- 35}}{35}, \pm \frac{1}{4} + \frac{\sqrt{- 35}}{140}, - \frac{1}{6} + \frac{\sqrt{- 35}}{210}, \pm \frac{1}{12} + \frac{\sqrt{- 35}}{420}, - \frac{5}{12} + \frac{\sqrt{- 35}}{420}, - \frac{1}{18} + \frac{\sqrt{- 35}}{630}$
36	1	none
37	2	W₃₇	1	2	none
38	4	W₂	2	2	none
		W₁₉	1	6	$\pm \frac{1}{2} + \frac{\sqrt{- 76}}{38}, \pm \frac{1}{4} + \frac{\sqrt{- 76}}{152}, \pm \frac{1}{10} + \frac{\sqrt{- 76}}{380}$
		W₃₈	1	6	$\frac{\sqrt{- 38}}{38}, \frac{2}{39} + \frac{\sqrt{- 38}}{741}, \frac{2}{51} + \frac{\sqrt{- 38}}{1938}, \frac{20}{273} + \frac{\sqrt{- 38}}{546}, \frac{128}{2737} + \frac{\sqrt{- 38}}{2737}, \frac{88}{1297} + \frac{\sqrt{- 38}}{1297}$
39	3	W₃	1	4	none
		W₁₃	2	0	none
		W₃₉	0	8	$\frac{\sqrt{- 39}}{39}, \pm \frac{1}{4} + \frac{\sqrt{- 39}}{156}, \pm \frac{1}{8} + \frac{\sqrt{- 39}}{312}, - \frac{3}{8} + \frac{\sqrt{- 39}}{312}, - \frac{5}{16} + \frac{\sqrt{- 39}}{624}, - \frac{1}{20} + \frac{\sqrt{- 39}}{780}$
40	3	W₅, W₈	2	0	none
		W₄₀	1	4	none
		$(\begin{matrix} - 10 & 1 \\ - 120 & 10 \end{matrix})$	0	8	$\pm \frac{1}{4} + \frac{\sqrt{- 5}}{20}, \pm \frac{1}{12} + \frac{\sqrt{- 5}}{60}, \pm \frac{5}{12} + \frac{\sqrt{- 5}}{60}, \pm \frac{5}{36} + \frac{\sqrt{- 5}}{180}$
41	3	W₄₁	0	8	$\frac{\sqrt{- 41}}{41}, \frac{1}{2} + \frac{\sqrt{- 41}}{82}, \pm \frac{1}{3} + \frac{\sqrt{- 41}}{123}, \pm \frac{2}{5} + \frac{\sqrt{- 41}}{205}, \pm \frac{1}{6} + \frac{\sqrt{- 41}}{246}$
42	5	W₂, W₇	3	0	none
		W₃, W₆	2	4	none
		W₂₁, W₄₂
		W₁₄	1	8	$\pm \frac{1}{3} + \frac{\sqrt{- 14}}{42}, \pm \frac{1}{15} + \frac{\sqrt{- 14}}{210}, \frac{2}{57} + \frac{\sqrt{- 14}}{798}, - \frac{4}{75} + \frac{\sqrt{- 14}}{1050}, - \frac{4}{225} + \frac{\sqrt{- 14}}{3150}, - \frac{2}{3} + \frac{\sqrt{- 14}}{42}$
43	3	W₄₃	1	4	none
44	4	W₄	2	2	none
		W₁₁	1	6	$\pm \frac{1}{12} + \frac{\sqrt{- 11}}{132}, \pm \frac{1}{4} + \frac{\sqrt{- 11}}{44}, - \frac{3}{4} + \frac{\sqrt{- 11}}{44}, - \frac{3}{100} + \frac{\sqrt{- 11}}{1100}$
		W₄₄	1	6	$\frac{\sqrt{- 44}}{44}, \frac{52}{687} + \frac{\sqrt{- 44}}{687}, \frac{2}{59} + \frac{\sqrt{- 44}}{2596}, \frac{1}{15} + \frac{\sqrt{- 44}}{660}, \frac{4}{141} + \frac{\sqrt{- 44}}{6204}, \frac{1}{9} + \frac{\sqrt{- 44}}{396}$
45	3	W₅, W₉, W₄₅	1	4	none
46	5	W₂	3	0	none
		W₂₃	0	12	$- \frac{1}{2} + \frac{\sqrt{- 23}}{46}, \pm \frac{1}{4} + \frac{\sqrt{- 23}}{92}, \pm \frac{1}{6} + \frac{\sqrt{- 23}}{138}, \pm \frac{1}{8} + \frac{\sqrt{- 23}}{184}, \pm \frac{3}{8} + \frac{\sqrt{- 23}}{184}, \pm \frac{1}{12} + \frac{\sqrt{- 23}}{276}, - \frac{1}{24} + \frac{\sqrt{- 23}}{552}$
		W₄₆	2	4	none
47	4	W₄₇	0	10	$\frac{\sqrt{- 47}}{47}, \frac{1}{2} + \frac{\sqrt{- 47}}{94}, \pm \frac{1}{3} + \frac{\sqrt{- 47}}{141}, \pm \frac{1}{4} + \frac{\sqrt{- 47}}{188}, \pm \frac{1}{6} + \frac{\sqrt{- 47}}{282}, \pm \frac{2}{7} + \frac{\sqrt{- 47}}{329}$
48	3	W₃, W₁₆	2	0	none
		W₄₈	1	4	none
		$(\begin{matrix} - 6 & - 1 \\ - 48 & 6 \end{matrix})$	0	8	$\pm \frac{1}{8} + \frac{\sqrt{- 3}}{24}, \pm \frac{3}{8} + \frac{\sqrt{- 3}}{24}, \pm \frac{3}{56} + \frac{\sqrt{- 3}}{168}, \pm \frac{17}{56} + \frac{\sqrt{- 3}}{168}$
49	1	none
50	2	W₂, W₂₅	1	2	none
		W₅₀	0	6	$\frac{\sqrt{- 2}}{10}, \pm \frac{3}{11} + \frac{\sqrt{- 2}}{110}, \pm \frac{1}{17} + \frac{\sqrt{- 2}}{170}, - \frac{7}{43} + \frac{\sqrt{- 2}}{430}$

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