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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
number of curves 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
number of curves 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 \(\mathrm{avg}\ 4.25\)
number of curves 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 \(\mathrm{avg}\ 0.66\)
number of curves 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 \(D_6\) \(C_2\) \(C_4\) \(V_4\) \(C_6\) \(D_4\)
number of curves 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 \(C_{10}\) \(D_6\) \(C_2\) \(2D_6\) \(V_4\) \(\tilde{S}_4\) \(D_4\)
number of curves 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 \(\mathrm{avg}\ 1.21\)
number of curves 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 \(\mathrm{avg}\ 1.62\)
number of curves 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 Sha False True
number of curves 2284 63874
proportion 3.45% 96.55%

Distribution of local solvability

locally solvable False True
number of curves 3371 62787
proportion 5.10% 94.90%

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

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

Distribution of Sato-Tate group identity components

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

Distribution of torsion subgroup orders

torsion order 1 2 3 4 5 6 7 8 9 10
number of curves 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
number of curves 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 \(\mathrm{avg}\ 1.63\)
number of curves 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%