Genus 2 curves over $\Q$: Statistics

The database currently contains 66,158 genus 2 curves in 65,534 isogeny classes, with absolute discriminant at most 1,000,000.

Distribution of rational points

rational points	0	1	2	3	4	5	6	7	8	9
count	5140	6454	11575	8719	9434	5089	6561	2967	3911	1328
proportion	7.77%	9.76%	17.50%	13.18%	14.26%	7.69%	9.92%	4.48%	5.91%	2.01%
rational points	10	11	12	13	14	15	16	17	18	19
count	1926	578	1130	234	574	97	276	28	103	2
proportion	2.91%	0.87%	1.71%	0.35%	0.87%	0.15%	0.42%	0.04%	0.16%	0.00%
rational points	20	21	22	24	26	\(\mathrm{avg}\ 4.25\)
count	20	1	3	6	2	66158
proportion	0.03%	0.00%	0.00%	0.01%	0.00%	100.00%

Distribution of rational Weierstrass points

Weierstrass points	0	1	2	3	4	\(\mathrm{avg}\ 0.66\)
count	32616	24611	8005	886	40	66158
proportion	49.30%	37.20%	12.10%	1.34%	0.06%	100.00%

Distribution of $\mathrm{Aut}(X)$

automorphism group	$D_6$	$C_2$	$C_4$	$C_2^2$	$C_6$	$D_4$
count	4	63480	15	2597	58	4
proportion	0.01%	95.95%	0.02%	3.93%	0.09%	0.01%

Distribution of $\mathrm{Aut}(X_{\overline{\mathbb{Q}}})$

automorphism group	$C_{10}$	$D_6$	$C_2$	$C_3:D_4$	$C_2^2$	$\GL(2,3)$	$D_4$
count	5	88	63310	4	2698	3	50
proportion	0.01%	0.13%	95.70%	0.01%	4.08%	0.00%	0.08%

Distribution of analytic ranks

analytic rank	0	1	2	3	4	\(\mathrm{avg}\ 1.21\)
count	12131	30579	20561	2877	10	66158
proportion	18.34%	46.22%	31.08%	4.35%	0.02%	100.00%

Distribution of 2-Selmer ranks

2-Selmer rank	0	1	2	3	4	5	6	\(\mathrm{avg}\ 1.62\)
count	3878	25496	29403	6879	448	49	5	66158
proportion	5.86%	38.54%	44.44%	10.40%	0.68%	0.07%	0.01%	100.00%

Distribution of squareness of Ш

has square Ш	False	True
count	2284	63874
proportion	3.45%	96.55%

Distribution of analytic order of Ш

analytic order of Ш	1	2	4	8	9	16	18	32	36	49
count	61950	2233	1790	45	46	78	2	3	6	1
proportion	93.64%	3.38%	2.71%	0.07%	0.07%	0.12%	0.00%	0.00%	0.01%	0.00%
analytic order of Ш	50	64	256	\(\mathrm{avg}\ 1.16\)
count	1	2	1	66158
proportion	0.00%	0.00%	0.00%	100.00%

Distribution of local solvability

locally solvable	False	True
count	3371	62787
proportion	5.10%	94.90%

Distribution of $\mathrm{GL}_2$-type

is of GL2-type	False	True
count	63312	2846
proportion	95.70%	4.30%

Distribution of Sato-Tate group identity components

identity component	\(\mathrm{U}(1)\times\mathrm{U}(1)\)	\(\mathrm{U}(1)\times\mathrm{SU}(2)\)	\(\mathrm{U}(1)\)	\(\mathrm{SU}(2)\)	\(\mathrm{USp}(4)\)	\(\mathrm{SU}(2)\times\mathrm{SU}(2)\)
count	6	303	8	150	63107	2584
proportion	0.01%	0.46%	0.01%	0.23%	95.39%	3.91%

Distribution of Sato-Tate groups

Sato-Tate group	$D_{2,1}$	$D_{3,2}$	$D_{6,2}$	$E_1$	$E_2$	$E_3$	$E_4$	$E_6$	$F_{ac}$	$J(C_2)$
count	3	1	1	8	3	7	10	51	6	2
proportion	0.00%	0.00%	0.00%	0.01%	0.00%	0.01%	0.02%	0.08%	0.01%	0.00%
Sato-Tate group	$J(C_4)$	$J(E_1)$	$J(E_2)$	$J(E_3)$	$J(E_4)$	$J(E_6)$	$N(\mathrm{SU}(2)\times\mathrm{SU}(2))$	$N(\mathrm{U}(1)\times\mathrm{SU}(2))$	$\mathrm{SU}(2)\times\mathrm{SU}(2)$	$\mathrm{USp}(4)$
count	1	24	9	4	17	17	144	303	2440	63107
proportion	0.00%	0.04%	0.01%	0.01%	0.03%	0.03%	0.22%	0.46%	3.69%	95.39%

Distribution of torsion subgroup orders

torsion order	1	2	3	4	5	6	7	8	9	10
count	44190	14681	2295	2754	725	595	97	393	38	131
proportion	66.79%	22.19%	3.47%	4.16%	1.10%	0.90%	0.15%	0.59%	0.06%	0.20%
torsion order	11	12	13	14	15	16	17	18	19	20
count	8	125	7	12	17	40	1	9	1	13
proportion	0.01%	0.19%	0.01%	0.02%	0.03%	0.06%	0.00%	0.01%	0.00%	0.02%
torsion order	21	22	24	27	28	29	32	36	39	\(\mathrm{avg}\ 1.63\)
count	5	2	10	2	2	1	2	1	1	66158
proportion	0.01%	0.00%	0.02%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	100.00%