accessibility
Sato-Tate distributions in genus 2 (52 total, 9 maximal, 6 minimal)
See Sato-Tate distributions and Galois endomorphism modules in genus 2, F. Fité, K.S. Kedlaya, V. Rotger, and A.V. Sutherland, Compositio Mathematica 148 (2012), 1390-1442.
1.4.A.1.1a USp(4)
y2 = x5 - x + 1
a1: 
a2: 
s2: 
s3: 
s4: 
s5: 
M[a1] = (1. 0. 1. 0. 3. 0. 14. 0. 84. 0. 594, ...)
M[a2] = (1. 1. 2. 4. 10. 27. 82. 268. 940. 3476. 13448, ...)
1.4.B.1.1a G3,3
y2 = x6 + x2 + 1
a1: 
a2: 
s2: 
s3: 
s4: 
s5: 
M[a1] = (1. 0. 2. 0. 10. 0. 70. 0. 588. 0. 5544, ...)
M[a2] = (1. 2. 5. 14. 44. 152. 569. 2270. 9524. 41576. 187348, ...)
1.4.B.2.1a N(G3,3)
y2 = x6 + x5 + x - 1
a1: 
a2: 
s2: 
s3: 
s4: 
s5: 
M[a1] = (1. 0. 1. 0. 5. 0. 35. 0. 294. 0. 2772, ...)
M[a2] = (1. 1. 3. 7. 23. 76. 287. 1135. 4769. 20788. 93695, ...)
1.4.C.1.1a G1,3
y2 = x6 + 3x4 - 2 over Q(√-1)
a1: 
a2: 
s2: 
s3: 
s4: 
s5: 
M[a1] = (1. 0. 3. 0. 20. 0. 175. 0. 1764. 0. 19404, ...)
M[a2] = (1. 2. 6. 20. 76. 312. 1364. 6232. 29460. 142952. 708328, ...)
1.4.C.2.1a N(G1,3)
y2 = x6 + 3x4 - 2
a1: 
a2: 
s2: 
s3: 
s4: 
s5: 
M[a1] = (1. 0. 2. 0. 11. 0. 90. 0. 889. 0. 9723, ...)
M[a2] = (1. 2. 5. 14. 46. 172. 714. 3180. 14858. 71732. 354676, ...)
1.4.D.1.1a F
y2 = x6 + 3x4 + x2 - 1 over Q(√-1,√2)
a1: 
a2: 
s2: 
s3: 
s4: 
s5: 
M[a1] = (1. 0. 4. 0. 36. 0. 400. 0. 4900. 0. 63504, ...)
M[a2] = (1. 2. 8. 32. 148. 712. 3584. 18496. 97444. 521096. 2820448, ...)
1.4.D.2.1a Fab
y2 = x6 + 3x4 + x2 - 1 over Q(√2)
a1: 
a2: 
s2: 
s3: 
s4: 
s5: 
M[a1] = (1. 0. 2. 0. 18. 0. 200. 0. 2450. 0. 31752, ...)
M[a2] = (1. 2. 6. 20. 82. 372. 1824. 9312. 48850. 260804. 1410736, ...)
1.4.D.2.1b F_a
y2 = x6 + 3x4 + x2 - 1 over Q(√-1)
a1: 
a2: 
s2: 
s3: 
s4: 
s5: 
M[a1] = (1. 0. 3. 0. 21. 0. 210. 0. 2485. 0. 31878, ...)
M[a2] = (1. 2. 6. 20. 82. 372. 1824. 9312. 48850. 260804. 1410736, ...)
1.4.D.4.1a Fac
y2 = x5 + 1
a1: 
a2: 
s2: 
s3: 
s4: 
s5: 
M[a1] = (1. 0. 1. 0. 9. 0. 100. 0. 1225. 0. 15876, ...)
M[a2] = (1. 1. 3. 10. 41. 186. 912. 4656. 24425. 130402. 705368, ...)
1.4.D.4.2a Fa,b
y2 = x6 + 3x4 + x2 - 1
a1: 
a2: 
s2: 
s3: 
s4: 
s5: 
M[a1] = (1. 0. 2. 0. 12. 0. 110. 0. 1260. 0. 16002, ...)
M[a2] = (1. 2. 5. 14. 49. 202. 944. 4720. 24553. 130658. 705880, ...)
1.4.E.1.1a E1
y2 = x6 + x4 + x2 + 1
a1: 
a2: 
s2: 
s3: 
s4: 
s5: 
M[a1] = (1. 0. 4. 0. 32. 0. 320. 0. 3584. 0. 43008, ...)
M[a2] = (1. 3. 10. 37. 150. 654. 3012. 14445. 71398. 361114. 1859628, ...)
1.4.E.2.1a E2
y2 = x6 + x5 + 3x4 + 3x2 - x + 1
a1: 
a2: 
s2: 
s3: 
s4: 
s5: 
M[a1] = (1. 0. 2. 0. 16. 0. 160. 0. 1792. 0. 21504, ...)
M[a2] = (1. 1. 6. 17. 78. 322. 1516. 7205. 35734. 180494. 929940, ...)
1.4.E.2.1b J(E1)
y2 = x5 + x3 + x
a1: 
a2: 
s2: 
s3: 
s4: 
s5: 
M[a1] = (1. 0. 2. 0. 16. 0. 160. 0. 1792. 0. 21504, ...)
M[a2] = (1. 2. 6. 20. 78. 332. 1516. 7240. 35734. 180620. 929940, ...)
1.4.E.3.1a E3
y2 = x5 + x4 - 3x3 - 4x2 - x
a1: 
a2: 
s2: 
s3: 
s4: 
s5: 
M[a1] = (1. 0. 2. 0. 12. 0. 110. 0. 1204. 0. 14364, ...)
M[a2] = (1. 1. 4. 13. 52. 222. 1014. 4839. 23860. 120526. 620278, ...)
1.4.E.4.1a E4
y2 = x5 + x4 + x2 - x
a1: 
a2: 
s2: 
s3: 
s4: 
s5: 
M[a1] = (1. 0. 2. 0. 12. 0. 100. 0. 1008. 0. 11424, ...)
M[a2] = (1. 1. 4. 11. 46. 182. 824. 3817. 18582. 92678. 473368, ...)
1.4.E.4.2a J(E2)
y2 = x5 + x3 - x
a1: 
a2: 
s2: 
s3: 
s4: 
s5: 
M[a1] = (1. 0. 1. 0. 8. 0. 80. 0. 896. 0. 10752, ...)
M[a2] = (1. 1. 4. 10. 42. 166. 768. 3620. 17902. 90310. 465096, ...)
1.4.E.6.1a J(E3)
y2 = x6 + x3 + 4
a1: 
a2: 
s2: 
s3: 
s4: 
s5: 
M[a1] = (1. 0. 1. 0. 6. 0. 55. 0. 602. 0. 7182, ...)
M[a2] = (1. 1. 3. 8. 29. 116. 517. 2437. 11965. 60326. 310265, ...)
1.4.E.6.2a E6
y2 = x5 + 2x4 - x3 - 3x2 - x
a1: 
a2: 
s2: 
s3: 
s4: 
s5: 
M[a1] = (1. 0. 2. 0. 12. 0. 100. 0. 980. 0. 10584, ...)
M[a2] = (1. 1. 4. 11. 44. 172. 754. 3397. 16020. 77516. 384578, ...)
1.4.E.8.3a J(E4)
y2 = x5 + x3 + 2x
a1: 
a2: 
s2: 
s3: 
s4: 
s5: 
M[a1] = (1. 0. 1. 0. 6. 0. 50. 0. 504. 0. 5712, ...)
M[a2] = (1. 1. 3. 7. 26. 96. 422. 1926. 9326. 46402. 236810, ...)
1.4.E.12.4a J(E6)
y2 = x6 + x3 - 2
a1: 
a2: 
s2: 
s3: 
s4: 
s5: 
M[a1] = (1. 0. 1. 0. 6. 0. 50. 0. 490. 0. 5292, ...)
M[a2] = (1. 1. 3. 7. 25. 91. 387. 1716. 8045. 38821. 192415, ...)
1.4.F.1.1a C1
y2 = x6 + 1 over Q(√-3)
a1: 
a2: 
s2: 
s3: 
s4: 
s5: 
M[a1] = (1. 0. 8. 0. 96. 0. 1280. 0. 17920. 0. 258048, ...)
M[a2] = (1. 4. 18. 88. 454. 2424. 13236. 73392. 411462. 2325976. 13233628, ...)
1.4.F.2.1a J(C1)
y2 = x5 - x over Q(√-1)
a1: 
a2: 
s2: 
s3: 
s4: 
s5: 
M[a1] = (1. 0. 4. 0. 48. 0. 640. 0. 8960. 0. 129024, ...)
M[a2] = (1. 1. 11. 40. 235. 1196. 6650. 36632. 205859. 1162732. 6617326, ...)
1.4.F.2.1b C2
y2 = x5 - x over Q(√-2)
a1: 
a2: 
s2: 
s3: 
s4: 
s5: 
M[a1] = (1. 0. 4. 0. 48. 0. 640. 0. 8960. 0. 129024, ...)
M[a2] = (1. 2. 10. 44. 230. 1212. 6628. 36696. 205766. 1162988. 6616940, ...)
1.4.F.2.1c C2,1
y2 = x6 + 1
a1: 
a2: 
s2: 
s3: 
s4: 
s5: 
M[a1] = (1. 0. 4. 0. 48. 0. 640. 0. 8960. 0. 129024, ...)
M[a2] = (1. 3. 11. 48. 235. 1228. 6650. 36760. 205859. 1163244. 6617326, ...)
1.4.F.3.1a C3
y2 = x6 + 4 over Q(√-3)
a1: 
a2: 
s2: 
s3: 
s4: 
s5: 
M[a1] = (1. 0. 4. 0. 36. 0. 440. 0. 6020. 0. 86184, ...)
M[a2] = (1. 2. 8. 34. 164. 842. 4506. 24726. 137892. 777418. 4417178, ...)
1.4.F.4.1a C4,1
y2 = x5 + 2x over Q(√-1)
a1: 
a2: 
s2: 
s3: 
s4: 
s5: 
M[a1] = (1. 0. 2. 0. 24. 0. 320. 0. 4480. 0. 64512, ...)
M[a2] = (1. 1. 5. 22. 115. 606. 3314. 18348. 102883. 581494. 3308470, ...)
1.4.F.4.1b C4
y2 = x6 + x5 - 5x4 - 5x2 - x + 1 over Q(√-2)
a1: 
a2: 
s2: 
s3: 
s4: 
s5: 
M[a1] = (1. 0. 4. 0. 36. 0. 400. 0. 5040. 0. 68544, ...)
M[a2] = (1. 2. 8. 32. 150. 732. 3776. 20064. 109318. 605804. 3400848, ...)
1.4.F.4.2a D2
y2 = x5 + 9x over Q(√-2)
a1: 
a2: 
s2: 
s3: 
s4: 
s5: 
M[a1] = (1. 0. 2. 0. 24. 0. 320. 0. 4480. 0. 64512, ...)
M[a2] = (1. 1. 6. 22. 118. 606. 3324. 18348. 102918. 581494. 3308596, ...)
1.4.F.4.2b J(C2)
y2 = x5 - x
a1: 
a2: 
s2: 
s3: 
s4: 
s5: 
M[a1] = (1. 0. 2. 0. 24. 0. 320. 0. 4480. 0. 64512, ...)
M[a2] = (1. 1. 7. 22. 123. 606. 3346. 18348. 103011. 581494. 3308982, ...)
1.4.F.4.2c D2,1
y2 = x5 + x
a1: 
a2: 
s2: 
s3: 
s4: 
s5: 
M[a1] = (1. 0. 2. 0. 24. 0. 320. 0. 4480. 0. 64512, ...)
M[a2] = (1. 2. 7. 26. 123. 622. 3346. 18412. 103011. 581750. 3308982, ...)
1.4.F.6.1a D3
y2 = x6 + 10x3 - 2 over Q(√-2)
a1: 
a2: 
s2: 
s3: 
s4: 
s5: 
M[a1] = (1. 0. 2. 0. 18. 0. 220. 0. 3010. 0. 43092, ...)
M[a2] = (1. 1. 5. 17. 85. 421. 2263. 12363. 68981. 388709. 2208715, ...)
1.4.F.6.1b D3,2
y2 = x6 + 4
a1: 
a2: 
s2: 
s3: 
s4: 
s5: 
M[a1] = (1. 0. 2. 0. 18. 0. 220. 0. 3010. 0. 43092, ...)
M[a2] = (1. 2. 6. 21. 90. 437. 2285. 12427. 69074. 388965. 2209101, ...)
1.4.F.6.2a J(C3)
y2 = x6 + 10x3 - 2 over Q(√-3)
a1: 
a2: 
s2: 
s3: 
s4: 
s5: 
M[a1] = (1. 0. 2. 0. 18. 0. 220. 0. 3010. 0. 43092, ...)
M[a2] = (1. 1. 5. 16. 85. 416. 2264. 12342. 68989. 388624. 2208760, ...)
1.4.F.6.2b C6,1
y2 = x6 + 6x5 - 30x4 + 20x3 + 15x2 - 12x + 1
a1: 
a2: 
s2: 
s3: 
s4: 
s5: 
M[a1] = (1. 0. 2. 0. 18. 0. 220. 0. 3010. 0. 43092, ...)
M[a2] = (1. 1. 5. 18. 85. 426. 2264. 12384. 68989. 388794. 2208760, ...)
1.4.F.6.2c C6
y2 = x6 + 2 over Q(√-3)
a1: 
a2: 
s2: 
s3: 
s4: 
s5: 
M[a1] = (1. 0. 4. 0. 36. 0. 400. 0. 4900. 0. 63504, ...)
M[a2] = (1. 2. 8. 32. 148. 712. 3586. 18524. 97796. 524744. 2854258, ...)
1.4.F.8.2a J(C4)
y2 = x6 + x5 - 5x4 - 5x2 - x + 1
a1: 
a2: 
s2: 
s3: 
s4: 
s5: 
M[a1] = (1. 0. 2. 0. 18. 0. 200. 0. 2520. 0. 34272, ...)
M[a2] = (1. 1. 5. 16. 79. 366. 1904. 10032. 54723. 302902. 1700680, ...)
1.4.F.8.3a D4,1
y2 = x5 + 2x
a1: 
a2: 
s2: 
s3: 
s4: 
s5: 
M[a1] = (1. 0. 1. 0. 12. 0. 160. 0. 2240. 0. 32256, ...)
M[a2] = (1. 1. 4. 13. 63. 311. 1678. 9206. 51523. 290875. 1654554, ...)
1.4.F.8.3b D4
y2 = x5 + 3x over Q(√-2)
a1: 
a2: 
s2: 
s3: 
s4: 
s5: 
M[a1] = (1. 0. 2. 0. 18. 0. 200. 0. 2520. 0. 34272, ...)
M[a2] = (1. 1. 5. 16. 78. 366. 1898. 10032. 54694. 302902. 1700550, ...)
1.4.F.8.3c D4,2
y2 = x6 + x5 + 10x3 + 5x2 + x - 2
a1: 
a2: 
s2: 
s3: 
s4: 
s5: 
M[a1] = (1. 0. 2. 0. 18. 0. 200. 0. 2520. 0. 34272, ...)
M[a2] = (1. 2. 6. 20. 83. 382. 1920. 10096. 54787. 303158. 1700936, ...)
1.4.F.8.5a J(D2)
y2 = x5 + 9x;
a1: 
a2: 
s2: 
s3: 
s4: 
s5: 
M[a1] = (1. 0. 1. 0. 12. 0. 160. 0. 2240. 0. 32256, ...)
M[a2] = (1. 1. 5. 13. 67. 311. 1694. 9206. 51587. 290875. 1654810, ...)
1.4.F.12.3a T
y2 = x6 + 6x5 - 20x4 + 20x3 - 20x2 - 8x + 8 over Q(√-2)
a1: 
a2: 
s2: 
s3: 
s4: 
s5: 
M[a1] = (1. 0. 2. 0. 12. 0. 120. 0. 1540. 0. 21672, ...)
M[a2] = (1. 1. 4. 12. 52. 236. 1202. 6378. 35044. 195924. 1108834, ...)
1.4.F.12.4a J(D3)
y2 = x6 + 10x3 - 2
a1: 
a2: 
s2: 
s3: 
s4: 
s5: 
M[a1] = (1. 0. 1. 0. 9. 0. 110. 0. 1505. 0. 21546, ...)
M[a2] = (1. 1. 4. 10. 48. 216. 1153. 6203. 34576. 194440. 1104699, ...)
1.4.F.12.4b D6,1
y2 = x6 + 6x5 - 30x4 - 40x3 + 60x2 +24x - 8
a1: 
a2: 
s2: 
s3: 
s4: 
s5: 
M[a1] = (1. 0. 1. 0. 9. 0. 110. 0. 1505. 0. 21546, ...)
M[a2] = (1. 1. 4. 11. 48. 221. 1153. 6224. 34576. 194525. 1104699, ...)
1.4.F.12.4c D6
y2 = x6 + 3x5 + 10x3 - 15x2 + 15x - 6 over Q(√-3)
a1: 
a2: 
s2: 
s3: 
s4: 
s5: 
M[a1] = (1. 0. 2. 0. 18. 0. 200. 0. 2450. 0. 31752, ...)
M[a2] = (1. 1. 5. 16. 77. 356. 1803. 9262. 48933. 262372. 1427255, ...)
1.4.F.12.4d D6,2
y2 = x6 + 2
a1: 
a2: 
s2: 
s3: 
s4: 
s5: 
M[a1] = (1. 0. 2. 0. 18. 0. 200. 0. 2450. 0. 31752, ...)
M[a2] = (1. 2. 6. 20. 82. 372. 1825. 9326. 49026. 262628. 1427641, ...)
1.4.F.12.5a J(C6)
y2 = x6 - 15x4 - 20x3 + 6x + 1
a1: 
a2: 
s2: 
s3: 
s4: 
s5: 
M[a1] = (1. 0. 2. 0. 18. 0. 200. 0. 2450. 0. 31752, ...)
M[a2] = (1. 1. 5. 16. 77. 356. 1804. 9262. 48941. 262372. 1427300, ...)
1.4.F.16.11a J(D4)
y2 = x5 + 3x
a1: 
a2: 
s2: 
s3: 
s4: 
s5: 
M[a1] = (1. 0. 1. 0. 9. 0. 100. 0. 1260. 0. 17136, ...)
M[a2] = (1. 1. 4. 10. 45. 191. 973. 5048. 27443. 151579. 850659, ...)
1.4.F.24.12a O1
y2 = x6 + 7x5 + 10x4 + 10x3 + 15x2 + 17x + 4
a1: 
a2: 
s2: 
s3: 
s4: 
s5: 
M[a1] = (1. 0. 1. 0. 6. 0. 60. 0. 770. 0. 10836, ...)
M[a2] = (1. 1. 3. 8. 30. 126. 617. 3221. 17586. 98090. 554673, ...)
1.4.F.24.12b O
y2 = x6 - 5x4 + 10x3 - 5x2 + 2x - 1 over Q(√-2)
a1: 
a2: 
s2: 
s3: 
s4: 
s5: 
M[a1] = (1. 0. 2. 0. 12. 0. 100. 0. 1050. 0. 12852, ...)
M[a2] = (1. 1. 4. 11. 45. 181. 837. 4047. 20757. 110117. 600669, ...)
1.4.F.24.13a J(T)
y2 = x6 + 6x5 - 20x4 + 20x3 - 20x2 - 8x + 8
a1: 
a2: 
s2: 
s3: 
s4: 
s5: 
M[a1] = (1. 0. 1. 0. 6. 0. 60. 0. 770. 0. 10836, ...)
M[a2] = (1. 1. 3. 7. 29. 121. 612. 3200. 17565. 98005. 554588, ...)
1.4.F.24.14a J(D6)
y2 = x6 + 3x5 + 10x3 - 15x2 + 15x - 6
a1: 
a2: 
s2: 
s3: 
s4: 
s5: 
M[a1] = (1. 0. 1. 0. 9. 0. 100. 0. 1225. 0. 15876, ...)
M[a2] = (1. 1. 4. 10. 44. 186. 923. 4663. 24552. 131314. 713969, ...)
1.4.F.48.48a J(O)
y2 = x6 - 5x4 + 10x3 - 5x2 + 2x - 1
a1: 
a2: 
s2: 
s3: 
s4: 
s5: 
M[a1] = (1. 0. 1. 0. 6. 0. 50. 0. 525. 0. 6426, ...)
M[a2] = (1. 1. 3. 7. 26. 96. 432. 2045. 10432. 55144. 300548, ...)
Acknowledgement: This work was partially supported by NSF grant DSM-1115455.
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