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The database currently contains 2,945,722 isogeny classes of abelian varieties of dimension up to 6 over finite fields. In addition to the statistics below, you can also create your own.

For each dimension $g$ and field $\mathbb{F}_q$ with a nonempty collection of isogeny classes in the database, the collection of isogeny classes is complete. As of October 2016, the data was complete for the following pairs $(g, q)$, where $q$ ranges over all primes and powers of a prime in the interval:

Here are some further comments on the extent of the data.

Distribution of dimension and base field

q
2 3 4 5-8 9-16 17-32 37-64 67-128 131-211 223-1024 Total
g
1 5 7 9 29 54 149 243 637 882 4169 6184
0.00% 0.00% 0.00% 0.00% 0.00% 0.01% 0.01% 0.02% 0.03% 0.14% 0.21%
2 35 63 91 503 1656 9898 32456 168187 404083 636925 1253897
0.00% 0.00% 0.00% 0.02% 0.06% 0.34% 1.10% 5.71% 13.72% 21.62% 42.57%
3 215 677 1397 18555 155908 878555 1055307
0.01% 0.02% 0.05% 0.63% 5.29% 29.82% 35.83%
4 1645 10963 38160 132839 183607
0.06% 0.37% 1.30% 4.51% 6.23%
5 14325 267465 281790
0.49% 9.08% 9.57%
6 164937 164937
5.60% 5.60%
Total 181162 279175 39657 151926 157618 888602 32699 168824 404965 641094 2945722
6.15% 9.48% 1.35% 5.16% 5.35% 30.17% 1.11% 5.73% 13.75% 21.76%

Distribution of dimension and $p$-rank

p rank
0 1 2 3 4 5 6 Total
g
1 184 6000 6184
2.98% 97.02% 0.21%
2 541 29248 1224108 1253897
0.04% 2.33% 97.62% 42.57%
3 779 5918 96784 951826 1055307
0.07% 0.56% 9.17% 90.19% 35.83%
4 1065 2260 9350 36254 134678 183607
0.58% 1.23% 5.09% 19.75% 73.35% 6.23%
5 1692 3000 8568 23636 67638 177256 281790
0.60% 1.06% 3.04% 8.39% 24.00% 62.90% 9.57%
6 2829 3108 6446 12178 24128 42126 74122 164937
1.72% 1.88% 3.91% 7.38% 14.63% 25.54% 44.94% 5.60%
Total 7090 49534 1345256 1023894 226444 219382 74122 2945722
0.24% 1.68% 45.67% 34.76% 7.69% 7.45% 2.52%

Distribution of dimension and number of twists

# twists
1 2 3-8 9-24 25-64 65-390 Total
g
1 95 5512 577 6184
1.54% 89.13% 9.33% 0.21%
2 1191286 54963 7648 1253897
95.01% 4.38% 0.61% 42.57%
3 963400 71385 16941 3155 426 1055307
91.29% 6.76% 1.61% 0.30% 0.04% 35.83%
4 140850 27971 9312 3738 1736 183607
76.71% 15.23% 5.07% 2.04% 0.95% 6.23%
5 196512 59600 17956 4506 3216 281790
69.74% 21.15% 6.37% 1.60% 1.14% 9.57%
6 88100 43896 17832 8370 6739 164937
53.41% 26.61% 10.81% 5.07% 4.09% 5.60%
Total 95 2585660 258392 69689 19769 12117 2945722
0.00% 87.78% 8.77% 2.37% 0.67% 0.41%

Distribution of dimension and maximum twist degree

max twist degree
1 2 3 4 5 6 8 10 12 14 15 16 18 20 22 24 28 30 36 40 42 44 48 56 60 66 70 72 80 84 88 110 120 126 132 140 168 180 210 240 252 280 336 360 420 504 840 Total
g
1 95 5512 2 202 331 15 27 6184
1.54% 89.13% 0.03% 3.27% 5.35% 0.24% 0.44% 0.21%
2 1206080 28148 9199 2935 123 6697 32 537 75 16 55 1253897
96.19% 2.24% 0.73% 0.23% 0.01% 0.53% 0.00% 0.04% 0.01% 0.00% 0.00% 42.57%
3 1012240 12531 17373 1935 130 8069 31 204 102 1851 13 245 72 45 65 1 303 5 23 39 2 4 12 12 1055307
95.92% 1.19% 1.65% 0.18% 0.01% 0.76% 0.00% 0.02% 0.01% 0.18% 0.00% 0.02% 0.01% 0.00% 0.01% 0.00% 0.03% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 35.83%
4 154321 11730 2 5273 721 38 6138 18 1 58 288 2498 12 450 24 190 42 9 20 1066 4 20 11 84 481 24 39 33 12 183607
84.05% 6.39% 0.00% 2.87% 0.39% 0.02% 3.34% 0.01% 0.00% 0.03% 0.16% 1.36% 0.01% 0.25% 0.01% 0.10% 0.02% 0.00% 0.01% 0.58% 0.00% 0.01% 0.01% 0.05% 0.26% 0.01% 0.02% 0.02% 0.01% 6.23%
5 230742 1782 26974 5199 114 4657 32 5 106 76 6 9546 16 194 260 182 74 6 116 30 275 14 452 13 90 2 4 490 10 4 192 44 12 27 30 14 281790
81.88% 0.63% 9.57% 1.84% 0.04% 1.65% 0.01% 0.00% 0.04% 0.03% 0.00% 3.39% 0.01% 0.07% 0.09% 0.06% 0.03% 0.00% 0.04% 0.01% 0.10% 0.00% 0.16% 0.00% 0.03% 0.00% 0.00% 0.17% 0.00% 0.00% 0.07% 0.02% 0.00% 0.01% 0.01% 0.00% 9.57%
6 107234 1899 4166 24413 82 5056 333 140 24 49 15818 202 111 36 304 677 210 152 290 16 14 248 1796 1242 8 42 57 2 48 4 28 66 4 166 164937
65.02% 1.15% 2.53% 14.80% 0.05% 3.07% 0.20% 0.08% 0.01% 0.03% 9.59% 0.12% 0.07% 0.02% 0.18% 0.41% 0.13% 0.09% 0.18% 0.01% 0.01% 0.15% 1.09% 0.75% 0.00% 0.03% 0.03% 0.00% 0.03% 0.00% 0.02% 0.04% 0.00% 0.10% 5.60%
Total 95 2716129 2 56292 2 63316 35218 487 30644 414 6 304 228 547 6 30250 243 1075 392 737 858 6 335 203 1989 14 20 477 38 445 2 4 2806 2 10 4 1462 103 66 117 2 78 4 54 66 4 166 2945722
0.00% 92.21% 0.00% 1.91% 0.00% 2.15% 1.20% 0.02% 1.04% 0.01% 0.00% 0.01% 0.01% 0.02% 0.00% 1.03% 0.01% 0.04% 0.01% 0.03% 0.03% 0.00% 0.01% 0.01% 0.07% 0.00% 0.00% 0.02% 0.00% 0.02% 0.00% 0.00% 0.10% 0.00% 0.00% 0.00% 0.05% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.01%

Distribution of dimension and degree of Endomorphism field

End. degree
1 2 3-8 9-24 25-64 65-168 Total
g
1 6030 114 40 6184
97.51% 1.84% 0.65% 0.21%
2 1215744 31748 6373 32 1253897
96.96% 2.53% 0.51% 0.00% 42.57%
3 1031374 12107 11481 337 8 1055307
97.73% 1.15% 1.09% 0.03% 0.00% 35.83%
4 165632 8052 8683 1162 78 183607
90.21% 4.39% 4.73% 0.63% 0.04% 6.23%
5 234172 14884 30004 2550 170 10 281790
83.10% 5.28% 10.65% 0.90% 0.06% 0.00% 9.57%
6 110718 14870 31734 6927 580 108 164937
67.13% 9.02% 19.24% 4.20% 0.35% 0.07% 5.60%
Total 2763670 81775 88315 11008 836 118 2945722
93.82% 2.78% 3.00% 0.37% 0.03% 0.00%

Distribution of geometrically simple isogeny classes among those that are simple

geom. simple
no yes Total
g
1 6184 6184
100.00% 0.24%
2 33172 1165130 1198302
2.77% 97.23% 46.17%
3 1996 960820 962816
0.21% 99.79% 37.10%
4 1975 139255 141230
1.40% 98.60% 5.44%
5 52 197052 197104
0.03% 99.97% 7.59%
6 1016 88686 89702
1.13% 98.87% 3.46%
Total 38211 2557127 2595338
1.47% 98.53%

Distribution of principally polarizable abelian surfaces

q
2 3 4 5 7 8 9 11 13 16 17 19 23 25
princ. polarizable
no 2 2 1 2 2 1 2 4 1 4
5.71% 1.55% 0.48% 1.20% 0.50% 0.19% 0.44% 0.52% 0.11% 0.34%
yes 33 63 91 127 206 165 285 399 512 455 761 896 1189 1273
94.29% 100.00% 100.00% 98.45% 99.52% 98.80% 100.00% 99.50% 99.81% 99.56% 99.48% 99.89% 99.66% 100.00%

Distribution of Jacobians among isogeny classes of abelian surfaces

q
2 3 4 5 7 8 9 11 13 16 17 19 23 25
Jacobian
no 15 13 26 14 15 12 35 16 19 29 24 19 24 42
42.86% 20.63% 28.57% 10.85% 7.25% 7.19% 12.28% 3.99% 3.70% 6.35% 3.14% 2.12% 2.01% 3.30%
yes 20 50 65 115 192 155 250 385 494 428 741 878 1169 1231
57.14% 79.37% 71.43% 89.15% 92.75% 92.81% 87.72% 96.01% 96.30% 93.65% 96.86% 97.88% 97.99% 96.70%

Distribution of Jacobians among isogeny classes of abelian threefolds

q
2 3 4 5 7 8 9 11 13 16 17 19 23 25
Jacobian
no 107 198 206 342 255 156 547 355 438 803 538 540 666 1992
49.77% 29.25% 14.75% 11.58% 3.20% 2.05% 3.54% 1.16% 0.87% 1.35% 0.48% 0.34% 0.24% 0.60%
unknown 1191 2493 7467 5283 8487 13559 58732 111745 156049 276851 330174
85.25% 31.24% 97.95% 34.17% 27.79% 26.92% 98.65% 99.52% 99.66% 99.76% 99.40%
yes 108 479 2611 5231 9629 21701 36374
50.23% 70.75% 88.42% 65.56% 62.29% 71.05% 72.21%

Distribution of Jacobian counts among isogeny classes of abelian surfaces

q
2 3 4 5 7 8 9 11 13 16 17 19 23 25
# Jacobians
0 15 13 26 14 15 12 35 16 19 29 24 19 24 42
42.86% 20.63% 28.57% 10.85% 7.25% 7.19% 12.28% 3.99% 3.70% 6.35% 3.14% 2.12% 2.01% 3.30%
1 20 31 16 34 23 7 14 15 14 9 11 8 5 5
57.14% 49.21% 17.58% 26.36% 11.11% 4.19% 4.91% 3.74% 2.73% 1.97% 1.44% 0.89% 0.42% 0.39%
2 19 33 36 42 43 40 24 6 24 28 27 18
30.16% 36.26% 27.91% 20.29% 15.09% 9.98% 4.68% 1.31% 3.14% 3.12% 2.26% 1.41%
3-8 16 45 121 98 136 210 246 129 225 220 201 239
17.58% 34.88% 58.45% 58.68% 47.72% 52.37% 47.95% 28.23% 29.41% 24.53% 16.85% 18.77%
9-16 6 37 50 102 154 113 283 313 333 235
2.90% 22.16% 17.54% 25.44% 30.02% 24.73% 36.99% 34.89% 27.91% 18.46%
17-256 13 7 18 56 171 198 309 603 734
7.78% 2.46% 4.49% 10.92% 37.42% 25.88% 34.45% 50.54% 57.66%

Distribution of Jacobian counts among isogeny classes of abelian threefolds

q
2 3 5
# Jacobians
0 107 198 342
49.77% 29.25% 11.58%
1 75 159 218
34.88% 23.49% 7.38%
2 21 112 216
9.77% 16.54% 7.31%
3-8 12 198 1196
5.58% 29.25% 40.50%
9-16 10 638
1.48% 21.61%
17-256 343
11.62%
257-6375

Distribution of hyperelliptic Jacobian counts among isogeny classes of abelian threefolds

q
2 3 5 7 9 11 13
# Hyp. curves
0 156 380 1230 2748 5830 8842 13997
72.56% 56.13% 41.65% 34.44% 37.71% 28.95% 27.79%
1 44 166 438 705 56 1056 1204
20.47% 24.52% 14.83% 8.84% 0.36% 3.46% 2.39%
2 13 76 374 723 1662 1122 1324
6.05% 11.23% 12.67% 9.06% 10.75% 3.67% 2.63%
3-8 2 55 767 2531 3510 7281 8815
0.93% 8.12% 25.97% 31.72% 22.71% 23.84% 17.50%
9-16 119 921 2262 5860 9514
4.03% 11.54% 14.63% 19.19% 18.89%
17-256 25 351 2139 6380 15507
0.85% 4.40% 13.84% 20.89% 30.79%
257-6375 2 10
0.01% 0.02%

Distribution of primitive abelian surfaces

q
4 8 9 16 25 27 32 49 64 81 125 128 243 256 343 512 625 729 1024
primitive
no 19 27 32 15 65 45 35 103 14 26 115 35 63 11 161 27 50 25 19
20.88% 16.17% 11.23% 3.28% 5.11% 4.00% 3.18% 2.94% 0.45% 0.45% 0.93% 0.43% 0.23% 0.05% 0.27% 0.04% 0.04% 0.02% 0.01%
yes 72 140 253 442 1208 1080 1066 3399 3093 5751 12312 8190 27902 23238 59146 63718 137525 144840 180200
79.12% 83.83% 88.77% 96.72% 94.89% 96.00% 96.82% 97.06% 99.55% 99.55% 99.07% 99.57% 99.77% 99.95% 99.73% 99.96% 99.96% 99.98% 99.99%

Distribution of primitive abelian threefolds

q
4 8 9 16 25
primitive
no 89 175 282 76 1301
6.37% 2.30% 1.82% 0.13% 0.39%
yes 1308 7448 15177 59459 330865
93.63% 97.70% 98.18% 99.87% 99.61%