Number field search results

Degree	Signature	Class number	Class group structure	CM field
Ramified prime count	Ramified	Unramified primes	Galois group	Is Galois
Discriminant	Root discriminant	Regulator	Intermediate field	$p$-adic completions
Monogenic	Index	Ines. \(p\)

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Results (1-50 of 1326465 matches)

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Label	Polynomial	Degree	Signature	Discriminant	Ram. prime count	Root discriminant	CM field	Galois	Monogenic	Galois group	Class group	Unit group torsion	Unit group rank	Regulator
5.1.1609.1	x^5 - x^3 - x^2 + x + 1	$5$	[1,2]	$1609$	$$	$4.37835739203$			✓	$S_5$ (as 5T5)	trivial	$2$	$2$	$0.268355550838$
5.1.1649.1	x^5 - x^4 + x^2 - x + 1	$5$	[1,2]	$17\cdot 97$	$$	$4.39991342464$			✓	$S_5$ (as 5T5)	trivial	$2$	$2$	$0.273599709102$
5.1.1777.1	x^5 - x^4 + x^3 - 2x^2 + x - 1	$5$	[1,2]	$1777$	$$	$4.46619298991$			✓	$S_5$ (as 5T5)	trivial	$2$	$2$	$0.290415099662$
5.1.2209.1	x^5 - 2x^4 + 2x^3 - x^2 + 1	$5$	[1,2]	$47^{2}$	$$	$4.66486567972$			✓	$D_{5}$ (as 5T2)	trivial	$2$	$2$	$0.346947612066$
5.1.2297.1	x^5 - x^4 + x^3 - x^2 - 1	$5$	[1,2]	$2297$	$$	$4.70145408445$			✓	$S_5$ (as 5T5)	trivial	$2$	$2$	$0.357352114929$
5.1.2617.1	x^5 + x^3 - 2x^2 - 1	$5$	[1,2]	$2617$	$$	$4.82570472203$			✓	$S_5$ (as 5T5)	trivial	$2$	$2$	$0.394462762943$
5.1.2665.1	x^5 - x^4 + x^3 - x^2 + 2x - 1	$5$	[1,2]	$5\cdot 13\cdot 41$	$$	$4.84327849939$			✓	$S_5$ (as 5T5)	trivial	$2$	$2$	$0.399660818464$
5.1.2869.1	x^5 - x - 1	$5$	[1,2]	$19\cdot 151$	$$	$4.91525563382$			✓	$S_5$ (as 5T5)	trivial	$2$	$2$	$0.432343878825$
5.1.3017.1	x^5 - x^2 + 1	$5$	[1,2]	$7\cdot 431$	$$	$4.96495208964$			✓	$S_5$ (as 5T5)	trivial	$2$	$2$	$0.437122673287$
5.1.3089.1	x^5 - x^3 + 2x - 1	$5$	[1,2]	$3089$	$$	$4.98842654594$			✓	$S_5$ (as 5T5)	trivial	$2$	$2$	$0.486630116907$
5.1.3233.1	x^5 - x^2 - 1	$5$	[1,2]	$53\cdot 61$	$$	$5.0340919158$			✓	$S_5$ (as 5T5)	trivial	$2$	$2$	$0.503137602609$
5.1.3369.1	x^5 + x^3 - x^2 - x - 1	$5$	[1,2]	$3\cdot 1123$	$$	$5.07574975343$			✓	$S_5$ (as 5T5)	trivial	$2$	$2$	$0.518409869312$
5.1.3857.1	x^5 - x^4 + x^3 - x^2 + 1	$5$	[1,2]	$7\cdot 19\cdot 29$	$$	$5.2149473454$			✓	$S_5$ (as 5T5)	trivial	$2$	$2$	$0.574676824693$
5.1.3889.1	x^5 - x^4 + x^3 + x - 1	$5$	[1,2]	$3889$	$$	$5.22357204058$			✓	$S_5$ (as 5T5)	trivial	$2$	$2$	$0.532749828479$
5.1.4169.1	x^5 + 2x^3 - x^2 + 2x - 1	$5$	[1,2]	$11\cdot 379$	$$	$5.29671223777$			✓	$S_5$ (as 5T5)	trivial	$2$	$2$	$0.607298353079$
5.1.4261.1	x^5 - 2x^2 - 2x - 1	$5$	[1,2]	$4261$	$$	$5.31988576671$			✓	$S_5$ (as 5T5)	trivial	$2$	$2$	$0.758429941403$
5.1.4409.1	x^5 - x^4 + x^2 + x - 1	$5$	[1,2]	$4409$	$$	$5.3563385594$			✓	$S_5$ (as 5T5)	trivial	$2$	$2$	$0.631438453903$
5.1.4417.1	x^5 - 2x^4 + 2x^3 - x^2 - 1	$5$	[1,2]	$7\cdot 631$	$$	$5.35828093372$			✓	$S_5$ (as 5T5)	trivial	$2$	$2$	$0.571446122147$
5.1.4429.1	x^5 - x^4 + 2x^3 - x^2 + 1	$5$	[1,2]	$43\cdot 103$	$$	$5.36118922487$			✓	$S_5$ (as 5T5)	trivial	$2$	$2$	$0.780070782411$
5.1.4432.1	x^5 - x^4 + 2x^2 - x + 1	$5$	[1,2]	$2^{4}\cdot 277$	$$	$5.36191531255$			✓	$S_5$ (as 5T5)	trivial	$2$	$2$	$0.780720256768$
5.1.4477.1	x^5 + x^3 + x - 1	$5$	[1,2]	$11^{2}\cdot 37$	$$	$5.37275972428$			✓	$S_5$ (as 5T5)	trivial	$2$	$2$	$0.786371046393$
5.3.4511.1	x^5 - x^3 - 2x^2 + 1	$5$	[3,1]	$-\,13\cdot 347$	$$	$5.38089559333$			✓	$S_5$ (as 5T5)	trivial	$2$	$3$	$0.628579167485$
5.1.4549.1	x^5 - x^4 + 2x^3 - 2x^2 - 1	$5$	[1,2]	$4549$	$$	$5.38993077448$			✓	$S_5$ (as 5T5)	trivial	$2$	$2$	$0.601042087852$
5.1.4597.1	x^5 + x^3 - 2x^2 - x - 1	$5$	[1,2]	$4597$	$$	$5.40125772864$			✓	$S_5$ (as 5T5)	trivial	$2$	$2$	$0.801495712181$
5.1.4757.1	x^5 - x^4 + 2x^3 - x^2 + 2x - 1	$5$	[1,2]	$67\cdot 71$	$$	$5.43834343727$			✓	$S_5$ (as 5T5)	trivial	$2$	$2$	$0.822644320451$
5.1.4817.1	x^5 - 2x^4 + 2x^3 - x^2 + 2x - 1	$5$	[1,2]	$4817$	$$	$5.45199349845$			✓	$S_5$ (as 5T5)	trivial	$2$	$2$	$0.687711449121$
5.1.4897.1	x^5 - x^4 - x^3 + x^2 + 1	$5$	[1,2]	$59\cdot 83$	$$	$5.46998355669$			✓	$S_5$ (as 5T5)	trivial	$2$	$2$	$0.612023201458$
5.3.4903.1	x^5 - x^4 - x^3 + 2x^2 - x - 1	$5$	[3,1]	$-\,4903$	$$	$5.47132330872$			✓	$S_5$ (as 5T5)	trivial	$2$	$3$	$0.668925630977$
5.1.5025.1	x^5 - x^4 - x^2 + x + 1	$5$	[1,2]	$3\cdot 5^{2}\cdot 67$	$$	$5.49828456648$			✓	$S_5$ (as 5T5)	trivial	$2$	$2$	$0.708326486922$
5.1.5164.1	x^5 - x^4 - x^3 + 2x + 1	$5$	[1,2]	$2^{2}\cdot 1291$	$$	$5.5283718475$			✓	$S_5$ (as 5T5)	trivial	$2$	$2$	$0.873888115674$
5.1.5437.1	x^5 - 2x^4 + 2x^3 - 2x^2 - 1	$5$	[1,2]	$5437$	$$	$5.58562617452$			✓	$S_5$ (as 5T5)	trivial	$2$	$2$	$0.909810356182$
5.1.5501.1	x^5 - 2x^4 + 2x^2 - x + 1	$5$	[1,2]	$5501$	$$	$5.59871459382$			✓	$S_5$ (as 5T5)	trivial	$2$	$2$	$0.68402903196$
5.3.5519.1	x^5 - x^4 - x^3 + 3x^2 - 1	$5$	[3,1]	$-\,5519$	$$	$5.60237375466$			✓	$S_5$ (as 5T5)	trivial	$2$	$3$	$0.732128382974$
5.1.5584.1	x^5 - x^4 + x + 1	$5$	[1,2]	$2^{4}\cdot 349$	$$	$5.61550840903$			✓	$S_5$ (as 5T5)	trivial	$2$	$2$	$0.927152539126$
5.1.5653.1	x^5 - x^4 + 2x - 1	$5$	[1,2]	$5653$	$$	$5.62931818739$			✓	$S_5$ (as 5T5)	trivial	$2$	$2$	$0.714858266315$
5.1.5753.1	x^5 - x^4 - x^3 + 2x^2 - x + 1	$5$	[1,2]	$11\cdot 523$	$$	$5.64909495501$			✓	$S_5$ (as 5T5)	trivial	$2$	$2$	$0.777786743614$
5.3.5783.1	x^5 - 2x^4 + x^3 + 2x^2 - 2x - 1	$5$	[3,1]	$-\,5783$	$$	$5.65497433803$			✓	$S_5$ (as 5T5)	trivial	$2$	$3$	$0.757141624325$
5.1.5864.1	x^5 - x^4 + x^3 + 2x^2 - 2x + 1	$5$	[1,2]	$2^{3}\cdot 733$	$$	$5.67072768331$			✓	$S_5$ (as 5T5)	trivial	$2$	$2$	$0.957732052495$
5.1.5913.1	x^5 - x^4 + 2x^3 - x^2 + x + 1	$5$	[1,2]	$3^{4}\cdot 73$	$$	$5.68017316557$			✓	$S_5$ (as 5T5)	trivial	$2$	$2$	$0.794503605626$
5.1.6241.1	x^5 - x^4 + x^3 - 2x^2 + 3x - 1	$5$	[1,2]	$79^{2}$	$$	$5.74183669575$			✓	$D_{5}$ (as 5T2)	trivial	$2$	$2$	$0.803647538428$
5.1.6449.1	x^5 - x^4 + x^3 - 3x^2 - 1	$5$	[1,2]	$6449$	$$	$5.77960922231$			✓	$S_5$ (as 5T5)	trivial	$2$	$2$	$0.826566357784$
5.1.6581.1	x^5 - 2x^2 - 1	$5$	[1,2]	$6581$	$$	$5.80307760104$			✓	$S_5$ (as 5T5)	trivial	$2$	$2$	$1.04188879242$
5.1.6757.1	x^5 - x^4 + 3x^2 - 2x + 1	$5$	[1,2]	$29\cdot 233$	$$	$5.83378989382$			✓	$S_5$ (as 5T5)	trivial	$2$	$2$	$1.08518392663$
5.1.6793.1	x^5 - x^4 + x^3 + x + 1	$5$	[1,2]	$6793$	$$	$5.83999295111$			✓	$S_5$ (as 5T5)	trivial	$2$	$2$	$0.97237060045$
5.3.7031.1	x^5 - x^3 - x^2 - x + 1	$5$	[3,1]	$-\,79\cdot 89$	$$	$5.88035329589$			✓	$S_5$ (as 5T5)	trivial	$2$	$3$	$0.888380795899$
5.1.7096.1	x^5 - x^2 + x + 1	$5$	[1,2]	$2^{3}\cdot 887$	$$	$5.89118581837$			✓	$S_5$ (as 5T5)	trivial	$2$	$2$	$1.10002131304$
5.1.7177.1	x^5 - x^4 + 3x^3 - x^2 + 2x - 1	$5$	[1,2]	$7177$	$$	$5.90457426388$			✓	$S_5$ (as 5T5)	trivial	$2$	$2$	$1.01016967835$
5.1.7265.1	x^5 - x^4 + x^3 + 2x^2 - x + 1	$5$	[1,2]	$5\cdot 1453$	$$	$5.91898342382$			✓	$S_5$ (as 5T5)	trivial	$2$	$2$	$0.920865082729$
5.1.7333.1	x^5 - 2x^4 + 2x^3 - 2x^2 + 3x - 1	$5$	[1,2]	$7333$	$$	$5.93002244277$			✓	$S_5$ (as 5T5)	trivial	$2$	$2$	$0.84949301581$
5.3.7367.1	x^5 - 2x^4 + 3x^2 - 2x - 1	$5$	[3,1]	$-\,53\cdot 139$	$$	$5.93551127049$			✓	$S_5$ (as 5T5)	trivial	$2$	$3$	$0.904106825927$

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