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Degree	Group	Orders	$[a,b,c]$ triple

Genus	Orbit size	Passport size	Base field

Geometry type	Primitive	Primitivization

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Label	Degree	Group	abc	Ramification type	Genus	Orbit Size	Base field	Triples	Primitivization
1T1-1_1_1-a	$1$	1T1	$[1, 1, 1]$	$[[1], [1], [1]]$	$0$	$1$	$\Q$	$[[[1], [1], [1]]]$	1T1-1_1_1-a
2T1-2_2_1.1-a	$2$	2T1	$[2, 2, 1]$	$[[2], [2], [1, 1]]$	$0$	$1$	$\Q$	$[[[2, 1], [2, 1], [1, 2]]]$	2T1-2_2_1.1-a
3T1-3_3_1.1.1-a	$3$	3T1	$[3, 3, 1]$	$[[3], [3], [1, 1, 1]]$	$0$	$1$	$\Q$	$[[[2, 3, 1], [3, 1, 2], [1, 2, 3]]]$	3T1-3_3_1.1.1-a
3T1-3_3_3-a	$3$	3T1	$[3, 3, 3]$	$[[3], [3], [3]]$	$1$	$1$	$\Q(\sqrt{-3}) $	$[[[2, 3, 1], [2, 3, 1], [2, 3, 1]]]$	3T1-3_3_3-a
3T2-3_2.1_2.1-a	$3$	3T2	$[3, 2, 2]$	$[[3], [2, 1], [2, 1]]$	$0$	$1$	$\Q$	$[[[2, 3, 1], [1, 3, 2], [2, 1, 3]]]$	3T2-3_2.1_2.1-a
4T1-4_4_1.1.1.1-a	$4$	4T1	$[4, 4, 1]$	$[[4], [4], [1, 1, 1, 1]]$	$0$	$1$	$\Q$	$[[[2, 3, 4, 1], [4, 1, 2, 3], [1, 2, 3, 4]]]$	2T1-2_2_1.1-a
4T1-4_4_2.2-a	$4$	4T1	$[4, 4, 2]$	$[[4], [4], [2, 2]]$	$1$	$1$	$\Q$	$[[[2, 3, 4, 1], [2, 3, 4, 1], [3, 4, 1, 2]]]$	2T1-2_2_1.1-a
4T2-2.2_2.2_2.2-a	$4$	4T2	$[2, 2, 2]$	$[[2, 2], [2, 2], [2, 2]]$	$0$	$1$	$\Q$	$[[[4, 3, 2, 1], [2, 1, 4, 3], [3, 4, 1, 2]]]$	2T1-2_2_1.1-a
4T3-4_2.2_2.1.1-a	$4$	4T3	$[4, 2, 2]$	$[[4], [2, 2], [2, 1, 1]]$	$0$	$1$	$\Q$	$[[[2, 3, 4, 1], [2, 1, 4, 3], [3, 2, 1, 4]]]$	2T1-2_2_1.1-a
4T4-3.1_3.1_2.2-a	$4$	4T4	$[3, 3, 2]$	$[[3, 1], [3, 1], [2, 2]]$	$0$	$1$	$\Q$	$[[[2, 3, 1, 4], [2, 4, 3, 1], [3, 4, 1, 2]]]$	4T4-3.1_3.1_2.2-a
4T4-3.1_3.1_3.1-a	$4$	4T4	$[3, 3, 3]$	$[[3, 1], [3, 1], [3, 1]]$	$0$	$1$	$\Q$	$[[[2, 3, 1, 4], [3, 2, 4, 1], [1, 4, 2, 3]]]$	4T4-3.1_3.1_3.1-a
4T5-4_3.1_2.1.1-a	$4$	4T5	$[4, 3, 2]$	$[[4], [3, 1], [2, 1, 1]]$	$0$	$1$	$\Q$	$[[[2, 3, 4, 1], [1, 4, 2, 3], [2, 1, 3, 4]]]$	4T5-4_3.1_2.1.1-a
4T5-4_4_3.1-a	$4$	4T5	$[4, 4, 3]$	$[[4], [4], [3, 1]]$	$1$	$1$	$\Q$	$[[[2, 3, 4, 1], [3, 1, 4, 2], [3, 2, 4, 1]]]$	4T5-4_4_3.1-a
5T1-5_5_1.1.1.1.1-a	$5$	5T1	$[5, 5, 1]$	$[[5], [5], [1, 1, 1, 1, 1]]$	$0$	$1$	$\Q$	$[[[2, 3, 4, 5, 1], [5, 1, 2, 3, 4], [1, 2, 3, 4, 5]]]$	5T1-5_5_1.1.1.1.1-a
5T2-5_2.2.1_2.2.1-a	$5$	5T2	$[5, 2, 2]$	$[[5], [2, 2, 1], [2, 2, 1]]$	$0$	$1$	$\Q$	$[[[2, 3, 4, 5, 1], [3, 2, 1, 5, 4], [4, 3, 2, 1, 5]]]$	5T2-5_2.2.1_2.2.1-a
5T4-5_2.2.1_3.1.1-a	$5$	5T4	$[5, 2, 3]$	$[[5], [2, 2, 1], [3, 1, 1]]$	$0$	$1$	$\Q$	$[[[3, 5, 4, 2, 1], [1, 5, 4, 3, 2], [2, 3, 1, 4, 5]]]$	5T4-5_2.2.1_3.1.1-a
5T4-5_3.1.1_3.1.1-a	$5$	5T4	$[5, 3, 3]$	$[[5], [3, 1, 1], [3, 1, 1]]$	$0$	$1$	$\Q$	$[[[3, 5, 2, 1, 4], [2, 3, 1, 4, 5], [4, 2, 3, 5, 1]]]$	5T4-5_3.1.1_3.1.1-a
5T4-5_5_2.2.1-a	$5$	5T4	$[5, 5, 2]$	$[[5], [5], [2, 2, 1]]$	$1$	$1$	$\Q$	$[[[2, 3, 4, 5, 1], [3, 1, 4, 5, 2], [4, 2, 5, 1, 3]]]$	5T4-5_5_2.2.1-a
5T4-5_5_5-a	$5$	5T4	$[5, 5, 5]$	$[[5], [5], [5]]$	$2$	$1$	$\Q$	$[[[2, 3, 4, 5, 1], [3, 5, 4, 2, 1], [2, 5, 4, 1, 3]]]$	5T4-5_5_5-a
5T5-3.2_3.2_2.2.1-a	$5$	5T5	$[6, 6, 2]$	$[[3, 2], [3, 2], [2, 2, 1]]$	$0$	$1$	$\Q$	$[[[2, 3, 1, 5, 4], [5, 1, 4, 3, 2], [4, 2, 5, 1, 3]]]$	5T5-3.2_3.2_2.2.1-a
5T5-3.2_3.2_3.1.1-a	$5$	5T5	$[6, 6, 3]$	$[[3, 2], [3, 2], [3, 1, 1]]$	$0$	$1$	$\Q$	$[[[2, 3, 1, 5, 4], [4, 3, 2, 5, 1], [2, 5, 3, 4, 1]]]$	5T5-3.2_3.2_3.1.1-a
5T5-4.1_2.2.1_3.2-a	$5$	5T5	$[4, 2, 6]$	$[[4, 1], [2, 2, 1], [3, 2]]$	$0$	$1$	$\Q$	$[[[2, 4, 1, 3, 5], [5, 3, 2, 4, 1], [2, 5, 4, 3, 1]]]$	5T5-4.1_2.2.1_3.2-a
5T5-4.1_4.1_3.1.1-a	$5$	5T5	$[4, 4, 3]$	$[[4, 1], [4, 1], [3, 1, 1]]$	$0$	$1$	$\Q$	$[[[3, 2, 4, 5, 1], [1, 5, 2, 3, 4], [2, 3, 1, 4, 5]]]$	5T5-4.1_4.1_3.1.1-a
5T5-5_2.1.1.1_3.2-a	$5$	5T5	$[5, 2, 6]$	$[[5], [2, 1, 1, 1], [3, 2]]$	$0$	$1$	$\Q$	$[[[3, 5, 2, 1, 4], [2, 1, 3, 4, 5], [4, 3, 2, 5, 1]]]$	5T5-5_2.1.1.1_3.2-a
5T5-5_4.1_2.1.1.1-a	$5$	5T5	$[5, 4, 2]$	$[[5], [4, 1], [2, 1, 1, 1]]$	$0$	$1$	$\Q$	$[[[2, 3, 4, 5, 1], [1, 5, 2, 3, 4], [2, 1, 3, 4, 5]]]$	5T5-5_4.1_2.1.1.1-a
5T5-5_3.2_3.2-a	$5$	5T5	$[5, 6, 6]$	$[[5], [3, 2], [3, 2]]$	$1$	$1$	$\Q$	$[[[4, 5, 1, 2, 3], [2, 3, 1, 5, 4], [2, 5, 4, 3, 1]]]$	5T5-5_3.2_3.2-a
5T5-5_4.1_4.1-a	$5$	5T5	$[5, 4, 4]$	$[[5], [4, 1], [4, 1]]$	$1$	$1$	$\Q$	$[[[4, 5, 1, 2, 3], [2, 3, 4, 1, 5], [2, 3, 5, 4, 1]]]$	5T5-5_4.1_4.1-a
6T1-6_6_1.1.1.1.1.1-a	$6$	6T1	$[6, 6, 1]$	$[[6], [6], [1, 1, 1, 1, 1, 1]]$	$0$	$1$	$\Q$	$[[[2, 3, 4, 5, 6, 1], [6, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 6]]]$	2T1-2_2_1.1-a
6T1-6_2.2.2_3.3-a	$6$	6T1	$[6, 2, 3]$	$[[6], [2, 2, 2], [3, 3]]$	$1$	$1$	$\Q$	$[[[2, 3, 4, 5, 6, 1], [4, 5, 6, 1, 2, 3], [3, 4, 5, 6, 1, 2]]]$	2T1-2_2_1.1-a
6T1-6_6_3.3-a	$6$	6T1	$[6, 6, 3]$	$[[6], [6], [3, 3]]$	$2$	$1$	$\Q$	$[[[2, 3, 4, 5, 6, 1], [2, 3, 4, 5, 6, 1], [5, 6, 1, 2, 3, 4]]]$	2T1-2_2_1.1-a
6T2-3.3_2.2.2_2.2.2-a	$6$	6T2	$[3, 2, 2]$	$[[3, 3], [2, 2, 2], [2, 2, 2]]$	$0$	$1$	$\Q$	$[[[3, 4, 5, 6, 1, 2], [2, 1, 6, 5, 4, 3], [4, 3, 2, 1, 6, 5]]]$	3T2-3_2.1_2.1-a
6T3-6_2.2.2_2.2.1.1-a	$6$	6T3	$[6, 2, 2]$	$[[6], [2, 2, 2], [2, 2, 1, 1]]$	$0$	$1$	$\Q$	$[[[2, 3, 4, 5, 6, 1], [4, 3, 2, 1, 6, 5], [5, 4, 3, 2, 1, 6]]]$	3T2-3_2.1_2.1-a
6T4-3.3_3.3_2.2.1.1-a	$6$	6T4	$[3, 3, 2]$	$[[3, 3], [3, 3], [2, 2, 1, 1]]$	$0$	$1$	$\Q$	$[[[3, 4, 5, 6, 1, 2], [2, 6, 4, 5, 3, 1], [4, 2, 6, 1, 5, 3]]]$	3T1-3_3_1.1.1-a
6T4-3.3_3.3_3.3-a	$6$	6T4	$[3, 3, 3]$	$[[3, 3], [3, 3], [3, 3]]$	$1$	$1$	$\Q$	$[[[3, 4, 5, 6, 1, 2], [6, 4, 2, 3, 1, 5], [6, 1, 5, 3, 4, 2]]]$	3T1-3_3_3-a
6T5-6_2.2.2_3.1.1.1-a	$6$	6T5	$[6, 2, 3]$	$[[6], [2, 2, 2], [3, 1, 1, 1]]$	$0$	$1$	$\Q$	$[[[4, 3, 6, 5, 2, 1], [4, 5, 6, 1, 2, 3], [3, 2, 5, 4, 1, 6]]]$	2T1-2_2_1.1-a
6T5-6_6_3.1.1.1-a	$6$	6T5	$[6, 6, 3]$	$[[6], [6], [3, 1, 1, 1]]$	$1$	$1$	$\Q$	$[[[6, 5, 2, 1, 4, 3], [2, 3, 4, 5, 6, 1], [3, 2, 5, 4, 1, 6]]]$	2T1-2_2_1.1-a
6T5-6_6_3.3-a	$6$	6T5	$[6, 6, 3]$	$[[6], [6], [3, 3]]$	$2$	$1$	$\Q$	$[[[6, 5, 2, 1, 4, 3], [6, 1, 2, 3, 4, 5], [5, 4, 1, 6, 3, 2]]]$	2T1-2_2_1.1-a
6T6-6_3.3_2.1.1.1.1-a	$6$	6T6	$[6, 3, 2]$	$[[6], [3, 3], [2, 1, 1, 1, 1]]$	$0$	$1$	$\Q$	$[[[6, 4, 5, 3, 1, 2], [2, 6, 4, 5, 3, 1], [4, 2, 3, 1, 5, 6]]]$	3T1-3_3_1.1.1-a
6T6-6_6_2.2.1.1-a	$6$	6T6	$[6, 6, 2]$	$[[6], [6], [2, 2, 1, 1]]$	$1$	$1$	$\Q$	$[[[6, 4, 5, 3, 1, 2], [2, 3, 4, 5, 6, 1], [4, 5, 3, 1, 2, 6]]]$	3T1-3_3_1.1.1-a
6T6-6_6_3.3-a	$6$	6T6	$[6, 6, 3]$	$[[6], [6], [3, 3]]$	$2$	$1$	$\Q$	$[[[6, 4, 5, 3, 1, 2], [3, 1, 5, 6, 4, 2], [3, 4, 5, 6, 1, 2]]]$	3T1-3_3_3-a
6T7-4.2_3.3_2.2.1.1-a	$6$	6T7	$[4, 3, 2]$	$[[4, 2], [3, 3], [2, 2, 1, 1]]$	$0$	$1$	$\Q$	$[[[3, 5, 4, 6, 2, 1], [2, 3, 1, 5, 6, 4], [5, 4, 3, 2, 1, 6]]]$	3T2-3_2.1_2.1-a
6T7-4.2_4.2_3.3-a	$6$	6T7	$[4, 4, 3]$	$[[4, 2], [4, 2], [3, 3]]$	$1$	$1$	$\Q$	$[[[5, 1, 6, 2, 4, 3], [3, 5, 4, 6, 2, 1], [5, 3, 4, 2, 6, 1]]]$	3T2-3_2.1_2.1-a
6T8-4.1.1_3.3_2.2.2-a	$6$	6T8	$[4, 3, 2]$	$[[4, 1, 1], [3, 3], [2, 2, 2]]$	$0$	$1$	$\Q$	$[[[3, 2, 4, 6, 5, 1], [5, 3, 4, 2, 6, 1], [5, 4, 6, 2, 1, 3]]]$	3T2-3_2.1_2.1-a
6T8-4.1.1_4.1.1_3.3-a	$6$	6T8	$[4, 4, 3]$	$[[4, 1, 1], [4, 1, 1], [3, 3]]$	$0$	$1$	$\Q$	$[[[5, 1, 3, 2, 4, 6], [3, 2, 4, 6, 5, 1], [2, 3, 1, 5, 6, 4]]]$	3T2-3_2.1_2.1-a
6T9-6_6_2.2.1.1-a	$6$	6T9	$[6, 6, 2]$	$[[6], [6], [2, 2, 1, 1]]$	$1$	$1$	$\Q$	$[[[6, 5, 2, 1, 4, 3], [2, 5, 6, 3, 4, 1], [5, 4, 3, 2, 1, 6]]]$	2T1-2_2_1.1-a
6T10-4.2_4.2_3.1.1.1-a	$6$	6T10	$[4, 4, 3]$	$[[4, 2], [4, 2], [3, 1, 1, 1]]$	$0$	$1$	$\Q$	$[[[4, 1, 6, 5, 2, 3], [2, 3, 6, 5, 4, 1], [1, 4, 3, 6, 5, 2]]]$	2T1-2_2_1.1-a
6T10-4.2_4.2_3.3-a	$6$	6T10	$[4, 4, 3]$	$[[4, 2], [4, 2], [3, 3]]$	$1$	$1$	$\Q$	$[[[4, 1, 6, 5, 2, 3], [4, 1, 2, 3, 6, 5], [3, 6, 5, 2, 1, 4]]]$	2T1-2_2_1.1-a
6T11-6_2.2.1.1_4.1.1-a	$6$	6T11	$[6, 2, 4]$	$[[6], [2, 2, 1, 1], [4, 1, 1]]$	$0$	$1$	$\Q$	$[[[5, 6, 4, 2, 3, 1], [5, 4, 3, 2, 1, 6], [6, 2, 1, 3, 5, 4]]]$	3T2-3_2.1_2.1-a
6T11-6_4.1.1_4.2-a	$6$	6T11	$[6, 4, 4]$	$[[6], [4, 1, 1], [4, 2]]$	$1$	$1$	$\Q$	$[[[6, 1, 2, 3, 4, 5], [6, 2, 1, 3, 5, 4], [2, 4, 6, 5, 1, 3]]]$	3T2-3_2.1_2.1-a
6T11-6_4.2_2.2.2-a	$6$	6T11	$[6, 4, 2]$	$[[6], [4, 2], [2, 2, 2]]$	$1$	$1$	$\Q$	$[[[3, 4, 2, 6, 1, 5], [2, 4, 6, 5, 1, 3], [4, 6, 5, 1, 3, 2]]]$	3T2-3_2.1_2.1-a

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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
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This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.