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Degree	Signature	Discriminant	Root discriminant	Galois root discriminant

Ramified prime count	Ramified	Unramified primes	Maximum ramified prime	Galois group

Class group structure	Class number	Regulator	Intermediate field	Field is

Narrow class group	Narrow class number	Relative class number	CM field	$p$-adic completions

Monogenic	Index	Ines. $p$	Minimal sibling	Unit signature rank

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Results (1-50 of at least 1000)

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Label	Polynomial	Degree	Signature	Discriminant	Ram. prime count	Root discriminant	Galois root discriminant	CM field	Monogenic	Galois group	Class group	Narrow class group	Unit group torsion	Unit group rank	Regulator	Unit signature rank	Max $p$
4.2.73167.1	$x^{4} - x^{3} + 4 x^{2} + 9 x - 6$	$4$	(2, 1)	$-\,3\cdot 29^{3}$	$2$	$16.4467015702$	$21.645076535339435$			$S_4$ (as 4T5)	$[2]$	$[4]$	$2$	$2$	$10.8936606631$	$1$	$29$
4.0.1756008.2	$x^{4} - x^{3} + 4 x^{2} + 9 x + 81$	$4$	(0, 2)	$2^{3}\cdot 3^{2}\cdot 29^{3}$	$3$	$36.4025345329$	$61.221521589761345$			$D_{4}$ (as 4T3)	$[12]$	$[12]$	$2$	$1$	$10.4181912513$	$0$	$29$
4.0.2634012.2	$x^{4} - x^{3} + 33 x^{2} + 125 x + 226$	$4$	(0, 2)	$2^{2}\cdot 3^{3}\cdot 29^{3}$	$3$	$40.2860267988$	$56.973045473056004$			$D_{4}$ (as 4T3)	$[12]$	$[12]$	$2$	$1$	$27.7226515789$	$0$	$29$
4.0.2634012.3	$x^{4} - x^{3} + 33 x^{2} + 9 x + 342$	$4$	(0, 2)	$2^{2}\cdot 3^{3}\cdot 29^{3}$	$3$	$40.2860267988$	$90.04714709975984$			$S_4$ (as 4T5)	$[4]$	$[4]$	$2$	$1$	$122.42543552$	$0$	$29$
4.2.2902291.1	$x^{4} - x^{3} + 4 x^{2} + 96 x - 296$	$4$	(2, 1)	$-\,7\cdot 17\cdot 29^{3}$	$3$	$41.2748248277$	$136.3238928046774$			$S_4$ (as 4T5)	$[2]$	$[4]$	$2$	$2$	$150.827612811$	$1$	$29$
4.0.3024236.1	$x^{4} - x^{3} + 33 x^{2} - 20 x + 52$	$4$	(0, 2)	$2^{2}\cdot 29^{3}\cdot 31$	$3$	$41.7017153684$	$139.15837253451795$			$S_4$ (as 4T5)	$[4]$	$[4]$	$2$	$1$	$28.8738517312$	$0$	$31$
4.2.3292515.1	$x^{4} - x^{3} + 33 x^{2} - 107 x - 151$	$4$	(2, 1)	$-\,3^{3}\cdot 5\cdot 29^{3}$	$3$	$42.5972870553$	$100.67577104749301$			$S_4$ (as 4T5)	$[2]$	$[4]$	$2$	$2$	$46.7861108129$	$1$	$29$
4.2.3609572.1	$x^{4} - x^{3} - 25 x^{2} + 96 x - 64$	$4$	(2, 1)	$-\,2^{2}\cdot 29^{3}\cdot 37$	$3$	$43.5876974051$	$152.03002122022247$			$S_4$ (as 4T5)	$[2]$	$[4]$	$2$	$2$	$83.9543385632$	$1$	$37$
4.0.3731517.1	$x^{4} - x^{3} + 4 x^{2} + 96 x + 168$	$4$	(0, 2)	$3^{2}\cdot 17\cdot 29^{3}$	$3$	$43.9512626448$	$89.24493682978284$			$D_{4}$ (as 4T3)	$[12]$	$[12]$	$2$	$1$	$31.1885685973$	$0$	$29$
4.0.4609521.1	$x^{4} - x^{3} + 33 x^{2} + 38 x + 52$	$4$	(0, 2)	$3^{3}\cdot 7\cdot 29^{3}$	$3$	$46.3355102075$	$75.36825487784023$			$D_{4}$ (as 4T3)	$[12]$	$[12]$	$2$	$1$	$12.0270524997$	$0$	$29$
4.2.4682688.1	$x^{4} - 2 x^{3} - 42 x^{2} + 14 x - 241$	$4$	(2, 1)	$-\,2^{6}\cdot 3\cdot 29^{3}$	$3$	$46.5182968338$	$61.221521589761345$			$S_4$ (as 4T5)	$[4]$	$[2, 4]$	$2$	$2$	$111.690600581$	$1$	$29$
4.2.4682688.3	$x^{4} - 2 x^{3} - 13 x^{2} + 14 x - 9$	$4$	(2, 1)	$-\,2^{6}\cdot 3\cdot 29^{3}$	$3$	$46.5182968338$	$61.221521589761345$		✓	$D_{4}$ (as 4T3)	$[2]$	$[4]$	$2$	$2$	$110.188886094$	$1$	$29$
4.0.7024032.1	$x^{4} + 29 x^{2} + 232$	$4$	(0, 2)	$2^{5}\cdot 3^{2}\cdot 29^{3}$	$3$	$51.4809580412$	$86.58030614135774$			$D_{4}$ (as 4T3)	$[12]$	$[12]$	$2$	$1$	$19.361959782$	$0$	$29$
4.2.8682484.1	$x^{4} - x^{3} + 62 x^{2} - 165 x + 81$	$4$	(2, 1)	$-\,2^{2}\cdot 29^{3}\cdot 89$	$3$	$54.2826445988$	$235.78897656348698$			$S_4$ (as 4T5)	$[2]$	$[4]$	$2$	$2$	$281.5094198253745$	$1$	$89$
4.0.8999541.1	$x^{4} - 2 x^{3} + 16 x^{2} - 15 x + 78$	$4$	(0, 2)	$3^{2}\cdot 29^{3}\cdot 41$	$3$	$54.7715573909$	$138.59611418452099$		✓	$D_{4}$ (as 4T3)	$[12]$	$[12]$	$2$	$1$	$49.2316187495$	$0$	$41$
4.0.9658044.1	$x^{4} - x^{3} + 4 x^{2} - 78 x + 168$	$4$	(0, 2)	$2^{2}\cdot 3^{2}\cdot 11\cdot 29^{3}$	$4$	$55.7471019095$	$143.57719485249348$			$D_{4}$ (as 4T3)	$[12]$	$[12]$	$2$	$1$	$90.0198723233$	$0$	$29$
4.0.9658044.4	$x^{4} + 29 x^{2} + 1276$	$4$	(0, 2)	$2^{2}\cdot 3^{2}\cdot 11\cdot 29^{3}$	$4$	$55.7471019095$	$143.57719485249348$			$D_{4}$ (as 4T3)	$[12]$	$[12]$	$2$	$1$	$29.9110656408$	$0$	$29$
4.4.10365325.1	$x^{4} - x^{3} - 25 x^{2} + 9 x + 139$	$4$	(4, 0)	$5^{2}\cdot 17\cdot 29^{3}$	$3$	$56.7408360894$	$150.66173416832322$		✓	$S_4$ (as 4T5)	$[2]$	$[2, 4]$	$2$	$3$	$123.56924665571732$	$2$	$29$
4.0.10365325.1	$x^{4} + 29 x^{2} - 58 x + 232$	$4$	(0, 2)	$5^{2}\cdot 17\cdot 29^{3}$	$3$	$56.7408360894$	$150.66173416832322$			$S_4$ (as 4T5)	$[2, 4]$	$[2, 4]$	$2$	$1$	$66.70767745402108$	$0$	$29$
4.2.10536048.2	$x^{4} - 2 x^{3} - 42 x^{2} - 218 x + 397$	$4$	(2, 1)	$-\,2^{4}\cdot 3^{3}\cdot 29^{3}$	$3$	$56.9730454731$	$56.973045473056004$			$D_{4}$ (as 4T3)	$[4]$	$[2, 4]$	$2$	$2$	$111.584578042$	$1$	$29$
4.4.10536048.1	$x^{4} - 2 x^{3} - 42 x^{2} + 14 x + 281$	$4$	(4, 0)	$2^{4}\cdot 3^{3}\cdot 29^{3}$	$3$	$56.9730454731$	$127.34589668148547$			$S_4$ (as 4T5)	$[2]$	$[2, 4]$	$2$	$3$	$186.482980575$	$2$	$29$
4.2.11121384.1	$x^{4} - x^{3} + 4 x^{2} - 78 x - 180$	$4$	(2, 1)	$-\,2^{3}\cdot 3\cdot 19\cdot 29^{3}$	$4$	$57.7483671688$	$266.8584257795633$			$S_4$ (as 4T5)	$[2]$	$[4]$	$2$	$2$	$898.1347621674136$	$1$	$29$
4.2.12096944.1	$x^{4} + 58 x^{2} - 58 x - 58$	$4$	(2, 1)	$-\,2^{4}\cdot 29^{3}\cdot 31$	$3$	$58.9751314482$	$156.1999918744938$			$S_4$ (as 4T5)	$[2]$	$[4]$	$2$	$2$	$125.59848740374241$	$1$	$31$
4.0.12292056.3	$x^{4} - x^{3} + 91 x^{2} + 9 x + 1212$	$4$	(0, 2)	$2^{3}\cdot 3^{2}\cdot 7\cdot 29^{3}$	$4$	$59.2115094576$	$161.97692101148021$			$D_{4}$ (as 4T3)	$[3, 12]$	$[3, 12]$	$2$	$1$	$22.6036861782$	$0$	$29$
4.0.12292056.4	$x^{4} - 58 x^{2} + 1624$	$4$	(0, 2)	$2^{3}\cdot 3^{2}\cdot 7\cdot 29^{3}$	$4$	$59.2115094576$	$161.97692101148021$			$D_{4}$ (as 4T3)	$[12]$	$[12]$	$2$	$1$	$64.6469759719$	$0$	$29$
4.2.13170060.1	$x^{4} - 58 x - 87$	$4$	(2, 1)	$-\,2^{2}\cdot 3^{3}\cdot 5\cdot 29^{3}$	$4$	$60.2416610739$	$159.8128248685$			$S_4$ (as 4T5)	$[2]$	$[4]$	$2$	$2$	$488.91900179407094$	$1$	$29$
4.2.16462575.1	$x^{4} - x^{3} - 54 x^{2} + 299 x + 226$	$4$	(2, 1)	$-\,3^{3}\cdot 5^{2}\cdot 29^{3}$	$3$	$63.6978012815$	$63.69780128146995$		✓	$S_4$ (as 4T5)	$[2, 4]$	$[2, 2, 4]$	$2$	$2$	$53.5043682476$	$1$	$29$
4.0.16901577.2	$x^{4} - x^{3} - 25 x^{2} + 154 x + 748$	$4$	(0, 2)	$3^{2}\cdot 7\cdot 11\cdot 29^{3}$	$4$	$64.1182716396$	$189.93477575998043$			$D_{4}$ (as 4T3)	$[12]$	$[12]$	$2$	$1$	$11.0589858902$	$0$	$29$
4.0.16901577.3	$x^{4} - 2 x^{3} + 45 x^{2} - 44 x + 1267$	$4$	(0, 2)	$3^{2}\cdot 7\cdot 11\cdot 29^{3}$	$4$	$64.1182716396$	$189.93477575998043$			$D_{4}$ (as 4T3)	$[12]$	$[12]$	$2$	$1$	$19.8302435847$	$0$	$29$
4.2.17169856.1	$x^{4} - 2 x^{3} + 45 x^{2} - 44 x - 966$	$4$	(2, 1)	$-\,2^{6}\cdot 11\cdot 29^{3}$	$3$	$64.371208588$	$117.23028869625483$			$D_{4}$ (as 4T3)	$[2]$	$[4]$	$2$	$2$	$556.827357882$	$1$	$29$
4.2.17340579.2	$x^{4} - x^{3} - 54 x^{2} + 125 x + 1009$	$4$	(2, 1)	$-\,3^{2}\cdot 29^{3}\cdot 79$	$3$	$64.530629041$	$192.38564842377264$			$S_4$ (as 4T5)	$[4]$	$[2, 4]$	$2$	$2$	$331.2303071906512$	$1$	$79$
4.2.18730752.1	$x^{4} + 58 x^{2} - 87$	$4$	(2, 1)	$-\,2^{8}\cdot 3\cdot 29^{3}$	$3$	$65.7868062808$	$86.58030614135774$			$D_{4}$ (as 4T3)	$[4]$	$[2, 4]$	$2$	$2$	$102.391707423$	$1$	$29$
4.0.19535589.1	$x^{4} - x^{3} + 62 x^{2} - 20 x + 400$	$4$	(0, 2)	$3^{2}\cdot 29^{3}\cdot 89$	$3$	$66.482390578$	$204.19924363631335$			$D_{4}$ (as 4T3)	$[12]$	$[12]$	$2$	$1$	$42.19444559271323$	$0$	$89$
4.2.21145263.2	$x^{4} - 2 x^{3} - 13 x^{2} + 14 x - 444$	$4$	(2, 1)	$-\,3\cdot 17^{2}\cdot 29^{3}$	$3$	$67.811487767$	$89.24493682978284$			$D_{4}$ (as 4T3)	$[2]$	$[4]$	$2$	$2$	$293.772073541$	$1$	$29$
4.2.22706159.1	$x^{4} - x^{3} + 4 x^{2} - 107 x - 93$	$4$	(2, 1)	$-\,7^{2}\cdot 19\cdot 29^{3}$	$3$	$69.0296893031$	$199.330608364449$			$S_4$ (as 4T5)	$[2]$	$[4]$	$2$	$2$	$197.56588375833292$	$1$	$29$
4.0.22828104.1	$x^{4} - x^{3} + 62 x^{2} + 67 x + 487$	$4$	(0, 2)	$2^{3}\cdot 3^{2}\cdot 13\cdot 29^{3}$	$4$	$69.1221854094$	$220.7373352538102$			$D_{4}$ (as 4T3)	$[2, 12]$	$[2, 12]$	$2$	$1$	$8.732929161100865$	$0$	$29$
4.0.22828104.2	$x^{4} + 29 x^{2} + 754$	$4$	(0, 2)	$2^{3}\cdot 3^{2}\cdot 13\cdot 29^{3}$	$4$	$69.1221854094$	$220.7373352538102$			$D_{4}$ (as 4T3)	$[4, 48]$	$[4, 48]$	$2$	$1$	$4.708710987277358$	$0$	$29$
4.0.23608552.1	$x^{4} - x^{3} + 62 x^{2} - 310 x + 400$	$4$	(0, 2)	$2^{3}\cdot 11^{2}\cdot 29^{3}$	$3$	$69.7055467057$	$117.23028869625483$			$D_{4}$ (as 4T3)	$[20]$	$[20]$	$2$	$1$	$52.6471780781$	$0$	$29$
4.4.24486556.1	$x^{4} - 58 x^{2} - 116 x + 232$	$4$	(4, 0)	$2^{2}\cdot 29^{3}\cdot 251$	$3$	$70.3447882424252$	$395.9728002332082$		?	$S_4$ (as 4T5)	$[2]$	$[4]$	$2$	$3$	$309.600536407284$	$3$	$251$
4.0.27218124.1	$x^{4} - x^{3} + 62 x^{2} - 20 x + 1096$	$4$	(0, 2)	$2^{2}\cdot 3^{2}\cdot 29^{3}\cdot 31$	$4$	$72.22948978081322$	$241.02937152838246$			$S_4$ (as 4T5)	$[2, 4]$	$[2, 4]$	$2$	$1$	$98.69615076383778$	$0$	$31$
4.0.28486352.1	$x^{4} - 2 x^{3} + 16 x^{2} + 72 x + 136$	$4$	(0, 2)	$2^{4}\cdot 29^{3}\cdot 73$	$3$	$73.0565583684$	$239.69644342417837$			$S_4$ (as 4T5)	$[4, 4]$	$[4, 4]$	$2$	$1$	$8.220906651227686$	$0$	$73$
4.0.28486352.2	$x^{4} - 58 x + 435$	$4$	(0, 2)	$2^{4}\cdot 29^{3}\cdot 73$	$3$	$73.0565583684$	$239.69644342417837$			$S_4$ (as 4T5)	$[2, 4]$	$[2, 4]$	$2$	$1$	$91.96244711832092$	$0$	$73$
4.2.28657075.1	$x^{4} - x^{3} + 4 x^{2} - 20 x - 35$	$4$	(2, 1)	$-\,5^{2}\cdot 29^{3}\cdot 47$	$3$	$73.1657729623$	$191.57217480788185$		✓	$S_4$ (as 4T5)	$[2, 2]$	$[2, 4]$	$2$	$2$	$127.55219198565682$	$1$	$47$
4.2.29632635.1	$x^{4} - x^{3} + 33 x^{2} + 38 x - 296$	$4$	(2, 1)	$-\,3^{5}\cdot 5\cdot 29^{3}$	$3$	$73.7806654444$	$209.41404272916748$			$S_4$ (as 4T5)	$[2]$	$[4]$	$2$	$2$	$862.5432101646876$	$1$	$29$
4.2.29657024.1	$x^{4} - 580 x - 609$	$4$	(2, 1)	$-\,2^{6}\cdot 19\cdot 29^{3}$	$3$	$73.7958419679$	$154.07078395935062$			$D_{4}$ (as 4T3)	$[2]$	$[4]$	$2$	$2$	$577.566043953$	$1$	$29$
4.2.30120415.1	$x^{4} - x^{3} + 33 x^{2} + 328 x + 139$	$4$	(2, 1)	$-\,5\cdot 13\cdot 19\cdot 29^{3}$	$4$	$74.0824331544$	$439.1693039905251$			$S_4$ (as 4T5)	$[2]$	$[4]$	$2$	$2$	$353.10340936164494$	$1$	$29$
4.2.30827696.1	$x^{4} - 2 x^{3} + 74 x^{2} + 130 x + 49$	$4$	(2, 1)	$-\,2^{4}\cdot 29^{3}\cdot 79$	$3$	$74.5135520957$	$222.14781181137164$			$S_4$ (as 4T5)	$[4]$	$[2, 4]$	$2$	$2$	$88.79174291571346$	$1$	$79$
4.2.32266647.2	$x^{4} - 2 x^{3} + 45 x^{2} - 44 x - 125$	$4$	(2, 1)	$-\,3^{3}\cdot 7^{2}\cdot 29^{3}$	$3$	$75.3682548778$	$75.36825487784023$			$D_{4}$ (as 4T3)	$[4]$	$[2, 4]$	$2$	$2$	$168.466280135$	$1$	$29$
4.4.32778816.1	$x^{4} - 2 x^{3} - 71 x^{2} - 102 x - 9$	$4$	(4, 0)	$2^{6}\cdot 3\cdot 7\cdot 29^{3}$	$4$	$75.6655713199$	$161.97692101148021$			$D_{4}$ (as 4T3)	$[2]$	$[2, 4]$	$2$	$3$	$219.383211847$	$2$	$29$
4.0.32778816.2	$x^{4} - 116 x + 783$	$4$	(0, 2)	$2^{6}\cdot 3\cdot 7\cdot 29^{3}$	$4$	$75.6655713199$	$161.97692101148021$	✓		$D_{4}$ (as 4T3)	$[2, 12]$	$[2, 12]$	$2$	$1$	$10.5765850746$	$0$	$29$

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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.