LMFDB - Number field search results

Note: Search results may be incomplete. Given $p$-adic completions contain an unramified field and completions are only searched for ramified primes.

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Degree	Signature	Class number	Class group structure	CM field

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$p$-adic completions	Minimal sibling	Monogenic	Index	Ines. $p$

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Results (1-50 of 83429 matches)

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Label	Polynomial	Degree	Signature	Discriminant	Ram. prime count	Root discriminant	Galois root discriminant	Monogenic	Galois group	Class group	Unit group torsion	Unit group rank	Regulator
4.4.1957.1	$x^{4} - 4 x^{2} - x + 1$	$4$	[4,0]	$19\cdot 103$	$2$	$6.65116476425$	$44.23799272118933$	✓	$S_4$ (as 4T5)	trivial	$2$	$3$	$1.91836302339$
4.0.2781.1	$x^{4} - x^{3} - 6 x^{2} + 8 x + 19$	$4$	[0,2]	$3^{3}\cdot 103$	$2$	$7.26189971659$	$23.134469942896533$		$D_{4}$ (as 4T3)	trivial	$6$	$1$	$4.80415373901$
4.0.6489.2	$x^{4} - x^{3} - 11 x^{2} + 48$	$4$	[0,2]	$3^{2}\cdot 7\cdot 103$	$3$	$8.97520637604$	$46.50806381693394$		$D_{4}$ (as 4T3)	trivial	$6$	$1$	$11.971170067$
4.2.6592.1	$x^{4} - 2 x^{3} - x^{2} + 2 x - 7$	$4$	[2,1]	$-\,2^{6}\cdot 103$	$2$	$9.01061221676$	$28.705400188814647$		$D_{4}$ (as 4T3)	trivial	$2$	$2$	$3.19105476131$
4.0.6901.1	$x^{4} - x^{3} - x^{2} - x + 7$	$4$	[0,2]	$67\cdot 103$	$2$	$9.11439837311$	$83.07225770376053$		$S_4$ (as 4T5)	trivial	$2$	$1$	$2.10527656751$
4.2.9888.1	$x^{4} - x^{2} - 4 x + 2$	$4$	[2,1]	$-\,2^{5}\cdot 3\cdot 103$	$3$	$9.97188162571$	$70.31358332498779$	✓	$S_4$ (as 4T5)	trivial	$2$	$2$	$10.349393288$
4.0.10712.1	$x^{4} - x^{3} - 2 x^{2} + x + 11$	$4$	[0,2]	$2^{3}\cdot 13\cdot 103$	$3$	$10.1734356175$	$103.49879226348489$		$S_4$ (as 4T5)	trivial	$2$	$1$	$5.65602335603$
4.0.11124.1	$x^{4} - 2 x^{3} + 6 x^{2} + 3$	$4$	[0,2]	$2^{2}\cdot 3^{3}\cdot 103$	$3$	$10.2698770678$	$58.042584187050004$		$S_4$ (as 4T5)	trivial	$2$	$1$	$11.09873647$
4.2.11948.1	$x^{4} - x^{3} + 8 x - 16$	$4$	[2,1]	$-\,2^{2}\cdot 29\cdot 103$	$3$	$10.4549943747$	$86.75694987764365$	✓	$S_4$ (as 4T5)	trivial	$2$	$2$	$9.50322896125$
4.2.12875.1	$x^{4} - x^{3} + 6 x^{2} - 6 x + 1$	$4$	[2,1]	$-\,5^{3}\cdot 103$	$2$	$10.65213862$	$33.93486420206204$	✓	$S_4$ (as 4T5)	$[2]$	$2$	$2$	$2.60214327115$
4.0.14832.1	$x^{4} - 20 x^{2} + 103$	$4$	[0,2]	$2^{4}\cdot 3^{2}\cdot 103$	$3$	$11.0357011006$	$49.71921157862421$		$D_{4}$ (as 4T3)	$[3]$	$6$	$1$	$5.89901286945$
4.2.15759.1	$x^{4} - x^{3} - x^{2} + 4 x - 5$	$4$	[2,1]	$-\,3^{2}\cdot 17\cdot 103$	$3$	$11.20423383$	$72.47758274114831$	✓	$S_4$ (as 4T5)	trivial	$2$	$2$	$8.31126515547$
4.0.17613.1	$x^{4} - 2 x^{3} - 20 x^{2} + 21 x + 117$	$4$	[0,2]	$3^{2}\cdot 19\cdot 103$	$3$	$11.5201553012$	$76.62245101796209$		$D_{4}$ (as 4T3)	trivial	$6$	$1$	$11.9639935561$
4.0.21424.1	$x^{4} - 2 x^{3} + 7 x^{2} - 4 x + 3$	$4$	[0,2]	$2^{4}\cdot 13\cdot 103$	$3$	$12.0983220204$	$103.49879226348489$	✓	$S_4$ (as 4T5)	trivial	$2$	$1$	$4.5879455764$
4.2.25132.1	$x^{4} - x^{3} - 3 x^{2} + 5 x - 18$	$4$	[2,1]	$-\,2^{2}\cdot 61\cdot 103$	$3$	$12.5908996547$	$125.82594292505412$	✓	$S_4$ (as 4T5)	trivial	$2$	$2$	$22.7073097251$
4.2.26471.1	$x^{4} - x^{3} - 4 x^{2} - 3 x + 11$	$4$	[2,1]	$-\,103\cdot 257$	$2$	$12.7553560823$	$162.69910878674168$		$S_4$ (as 4T5)	trivial	$2$	$2$	$9.97381304902$
4.0.26677.1	$x^{4} - 2 x^{2} - 5 x + 8$	$4$	[0,2]	$7\cdot 37\cdot 103$	$3$	$12.7800998573$	$163.33095236359824$	✓	$S_4$ (as 4T5)	trivial	$2$	$1$	$16.5446396303$
4.0.27192.1	$x^{4} - x^{3} - x^{2} + 6 x + 6$	$4$	[0,2]	$2^{3}\cdot 3\cdot 11\cdot 103$	$4$	$12.8413383134$	$164.8999696785903$	✓	$S_4$ (as 4T5)	$[5]$	$2$	$1$	$2.17690615083$
4.2.28531.1	$x^{4} - x^{3} + 5 x^{2} - 14 x + 8$	$4$	[2,1]	$-\,103\cdot 277$	$2$	$12.9965849085$	$168.9112192839777$		$S_4$ (as 4T5)	$[3]$	$2$	$2$	$6.56953264707$
4.2.28943.1	$x^{4} - x^{3} + 2 x^{2} + 3 x - 4$	$4$	[2,1]	$-\,103\cdot 281$	$2$	$13.0432520324$	$170.12642357964268$	✓	$S_4$ (as 4T5)	trivial	$2$	$2$	$13.1424315808$
4.0.31312.2	$x^{4} - 2 x^{3} + 7 x^{2} - 2 x + 2$	$4$	[0,2]	$2^{4}\cdot 19\cdot 103$	$3$	$13.3023295285$	$88.47598544237866$	✓	$S_4$ (as 4T5)	$[2]$	$2$	$1$	$8.00766400539$
4.4.34196.1	$x^{4} - 8 x^{2} - 2 x + 4$	$4$	[4,0]	$2^{2}\cdot 83\cdot 103$	$3$	$13.5985883463$	$146.77237516338877$	✓	$S_4$ (as 4T5)	trivial	$2$	$3$	$24.7275247018$
4.0.36153.1	$x^{4} - 2 x^{3} + 33 x^{2} - 32 x + 259$	$4$	[0,2]	$3^{3}\cdot 13\cdot 103$	$3$	$13.7891052114$	$83.41251760979391$		$D_{4}$ (as 4T3)	$[2]$	$6$	$1$	$12.4104211409$
4.0.36153.2	$x^{4} - 2 x^{3} - 27 x^{2} + 28 x + 244$	$4$	[0,2]	$3^{3}\cdot 13\cdot 103$	$3$	$13.7891052114$	$83.41251760979391$		$D_{4}$ (as 4T3)	$[6]$	$6$	$1$	$4.17706921404$
4.0.37389.1	$x^{4} - x^{3} - 6 x^{2} + 10 x + 23$	$4$	[0,2]	$3\cdot 11^{2}\cdot 103$	$3$	$13.9054792849$	$58.30094338859364$		$D_{4}$ (as 4T3)	trivial	$2$	$1$	$7.77277010477$
4.0.37389.2	$x^{4} - x^{3} - 7 x^{2} + 5 x + 25$	$4$	[0,2]	$3\cdot 11^{2}\cdot 103$	$3$	$13.9054792849$	$58.30094338859364$		$D_{4}$ (as 4T3)	trivial	$2$	$1$	$9.07144063593$
4.2.38831.1	$x^{4} - x^{3} + 2 x^{2} - 3 x - 4$	$4$	[2,1]	$-\,13\cdot 29\cdot 103$	$3$	$14.0376575557$	$197.05582965241095$	✓	$S_4$ (as 4T5)	trivial	$2$	$2$	$14.9858596768$
4.0.39861.1	$x^{4} - x^{3} + 34 x^{2} - 18 x + 273$	$4$	[0,2]	$3^{2}\cdot 43\cdot 103$	$3$	$14.1298336007$	$115.26925001924842$		$D_{4}$ (as 4T3)	trivial	$6$	$1$	$25.3771407042$
4.0.39861.2	$x^{4} - x^{3} - 31 x^{2} + 7 x + 283$	$4$	[0,2]	$3^{2}\cdot 43\cdot 103$	$3$	$14.1298336007$	$115.26925001924842$		$D_{4}$ (as 4T3)	trivial	$6$	$1$	$14.1586341476$
4.2.41200.1	$x^{4} + x^{2} - 4 x - 8$	$4$	[2,1]	$-\,2^{4}\cdot 5^{2}\cdot 103$	$3$	$14.2470288587$	$59.351077919057175$		$S_4$ (as 4T5)	trivial	$2$	$2$	$23.294439494$
4.2.41612.1	$x^{4} - x^{3} - x^{2} + 9 x + 4$	$4$	[2,1]	$-\,2^{2}\cdot 101\cdot 103$	$3$	$14.2825136388$	$161.9071257360612$		$S_4$ (as 4T5)	trivial	$2$	$2$	$23.2027277225$
4.2.42127.1	$x^{4} - x^{3} - 5 x^{2} + 16 x + 16$	$4$	[2,1]	$-\,103\cdot 409$	$2$	$14.3265009585$	$205.24862971527972$		$S_4$ (as 4T5)	trivial	$2$	$2$	$15.0749780511$
4.4.42848.1	$x^{4} - 9 x^{2} - 2 x + 2$	$4$	[4,0]	$2^{5}\cdot 13\cdot 103$	$3$	$14.3874106262$	$146.3693957082559$		$S_4$ (as 4T5)	trivial	$2$	$3$	$26.2950347425$
4.2.44187.1	$x^{4} - x^{3} - x^{2} - x - 11$	$4$	[2,1]	$-\,3\cdot 11\cdot 13\cdot 103$	$4$	$14.4985185724$	$210.20704079549762$		$S_4$ (as 4T5)	trivial	$2$	$2$	$10.1494388317$
4.0.44496.1	$x^{4} - 2 x^{3} + 12 x^{2} - 8 x + 31$	$4$	[0,2]	$2^{4}\cdot 3^{3}\cdot 103$	$3$	$14.5237994332$	$82.08460975250826$		$S_4$ (as 4T5)	$[4]$	$2$	$1$	$3.43667674077$
4.2.45423.1	$x^{4} - x^{3} - 19 x + 10$	$4$	[2,1]	$-\,3^{2}\cdot 7^{2}\cdot 103$	$3$	$14.5988603277$	$64.32472423832594$		$S_4$ (as 4T5)	trivial	$2$	$2$	$16.311360958$
4.4.46144.1	$x^{4} - 2 x^{3} - 13 x^{2} + 12 x + 36$	$4$	[4,0]	$2^{6}\cdot 7\cdot 103$	$3$	$14.6564506383$	$75.9473501841901$		$D_{4}$ (as 4T3)	trivial	$2$	$3$	$28.3394413069$
4.0.46453.1	$x^{4} - x^{3} + 5 x^{2} + 14 x + 8$	$4$	[0,2]	$11\cdot 41\cdot 103$	$3$	$14.6809257316$	$215.5295803364355$		$S_4$ (as 4T5)	$[9]$	$2$	$1$	$2.56099053955$
4.0.47277.1	$x^{4} - 5 x + 12$	$4$	[0,2]	$3^{3}\cdot 17\cdot 103$	$3$	$14.7456009787$	$150.75944739376848$		$S_4$ (as 4T5)	trivial	$2$	$1$	$28.486930628$
4.2.48204.1	$x^{4} - x^{3} + x^{2} - 3 x - 24$	$4$	[2,1]	$-\,2^{2}\cdot 3^{2}\cdot 13\cdot 103$	$4$	$14.8173578469$	$100.60917311485773$		$S_4$ (as 4T5)	trivial	$2$	$2$	$36.0282841705$
4.0.48513.1	$x^{4} - 2 x^{3} - 5 x^{2} + 3 x + 17$	$4$	[0,2]	$3\cdot 103\cdot 157$	$3$	$14.841046744$	$220.25666845750663$	✓	$S_4$ (as 4T5)	trivial	$2$	$1$	$13.5673505095$
4.2.54075.1	$x^{4} - 4 x^{2} - 9 x - 3$	$4$	[2,1]	$-\,3\cdot 5^{2}\cdot 7\cdot 103$	$4$	$15.2492727395$	$135.9904035706508$	✓	$S_4$ (as 4T5)	trivial	$2$	$2$	$20.8315898833$
4.2.59328.1	$x^{4} - 2 x^{3} + 9 x^{2} - 8 x - 56$	$4$	[2,1]	$-\,2^{6}\cdot 3^{2}\cdot 103$	$3$	$15.6068381667$	$49.71921157862421$		$D_{4}$ (as 4T3)	trivial	$2$	$2$	$28.0566350547$
4.0.59328.1	$x^{4} - 40 x^{2} + 412$	$4$	[0,2]	$2^{6}\cdot 3^{2}\cdot 103$	$3$	$15.6068381667$	$70.31358332498779$		$D_{4}$ (as 4T3)	trivial	$6$	$1$	$14.2733331006$
4.2.60976.1	$x^{4} - 2 x^{3} - 8 x^{2} + 2 x + 10$	$4$	[2,1]	$-\,2^{4}\cdot 37\cdot 103$	$3$	$15.7141079717$	$138.58656675377108$	✓	$S_4$ (as 4T5)	trivial	$2$	$2$	$25.5321590654$
4.0.62109.1	$x^{4} - 2 x^{3} + 2 x^{2} + 5 x + 4$	$4$	[0,2]	$3^{2}\cdot 67\cdot 103$	$3$	$15.7866010627$	$143.88537104236832$	✓	$S_4$ (as 4T5)	$[2, 2]$	$2$	$1$	$5.34018997746$
4.2.62315.1	$x^{4} - 10 x^{2} - 11 x + 3$	$4$	[2,1]	$-\,5\cdot 11^{2}\cdot 103$	$3$	$15.7996748636$	$75.26619427073486$	✓	$S_4$ (as 4T5)	trivial	$2$	$2$	$11.234354705$
4.0.63860.1	$x^{4} - x^{3} + 7 x^{2} + x + 2$	$4$	[0,2]	$2^{2}\cdot 5\cdot 31\cdot 103$	$4$	$15.8967091573$	$200.5723787642339$	✓	$S_4$ (as 4T5)	trivial	$2$	$1$	$22.9144247452$
4.2.63963.1	$x^{4} - 2 x^{3} - 6 x^{2} - 5 x + 1$	$4$	[2,1]	$-\,3^{3}\cdot 23\cdot 103$	$3$	$15.9031152473$	$110.9490202272738$	✓	$S_4$ (as 4T5)	trivial	$2$	$2$	$8.75469281824$
4.0.66744.1	$x^{4} + 9 x^{2} - 12 x + 18$	$4$	[0,2]	$2^{3}\cdot 3^{4}\cdot 103$	$3$	$16.073226238$	$124.20105326346038$		$S_4$ (as 4T5)	trivial	$2$	$1$	$54.2839692893$

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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.