Abelian variety isogeny classes: Statistics

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 2020, the data was complete for the following pairs $(g, q)$, where $q$ ranges over all primes and powers of a prime in the interval:

• for $g = 1$, data is available for all $2 \leq q \leq 499$ and $q \in \{512, 625, 729, 1024\}$,
• for $g = 2$, data is available for all $2 \leq q \leq 211$ and $q \in \{243, 256, 343, 512, 625, 729, 1024\}$,
• for $g = 3$, data is available for all $2 \leq q \leq 25$,
• for $g = 4$, data is available for $q = 2, 3, 4$ and $5$,
• for $g = 5$, data is available for $q = 2$ and $q=3$
• and for $g = 6$, only $q=2$ is available.

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%	8.68%	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
				50.17%		31.24%	97.95%	34.17%	27.79%	26.92%	98.65%	99.52%	99.66%	99.76%	99.40%
	yes	108	479	977	2611	5231		9629	21701	36374
		50.23%	70.75%	41.15%	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%