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Article Contents

# Certain sextics with many rational points

This research was partially supported by JSPS Grant-in-Aid for Young Scientists (B) 25800090.

• We construct a family of sextics from the Wiman and Edge sextics. We find a curve over $\mathbb{F}_{5^7}$ attaining the Serre bound, and update $9$ entries of genus $6$ in manYPoints.org by computer search on these sextics.

Mathematics Subject Classification: Primary: 11G20, 14G05; Secondary: 14G50.

 Citation:

• Table 1.  Maximal curves over $\mathbb{F}_{p^2}$ of genus $6$

 5 7 11 13 17 19 23 29 W W W W W 31 37 41 43 47 53 59 61 W S W S W S W 67 71 73 79 83 89 97 101 S W W W W S W 103 107 109 113 127 131 137 139 W W E S W W S W 149 151 157 163 167 173 179 181 E S S W S W S 191 193 197 199 W W S W

Table 2.  $S$ with many points over $\mathbb{F}_p$

 $\mathbb{F}_p$ $(a,b,c)$ $\#S(\mathbb{F}_p)$ old entry $19$ $(13,6,16)$ $56$ $[50 -68]$ $37$ $(29,28,14)$ $98$ $[86 -104]$ $43$ $(2,4,2)$ $104$ $[100-116]$ $53$ $(51,36,1)$ $132$ $[120-138 ]$ $61$ $(42,54,17)$ $140$ $[134-152]$ $67$ $(65,2,45)$ $152$ $[140-164]$ $71$ $(29,65,70)$ $156$ $[150-168]$

Table 3.  $S$ with many points over $\mathbb{F}_q$

 $\mathbb{F}_q$ $(a,b,c)$ $\#S(\mathbb{F}_q)$ old entry $5^3$ $(\beta^4,\beta^{56},\beta^{38})$ $240$ $[210-255]$ $u^3+3u+3=0$ $7^3$ $(\beta^{22},\beta^{94},\beta^{8})$ $542$ $[512-564]$ $u^3-u^2+4=0$
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