Global number field 27.27.93991579198394673195940551085616659931617170424829641.1

• Label: 27.27.9399157919...9641.1
• Degree: $27$
• Signature: $[27, 0]$
• Discriminant: $109^{26}$
• Root discriminant: $91.61$
• Ramified prime: $109$
• Class number: $1$ (GRH)
• Class group: Trivial (GRH)
• Galois group: $C_{27}$ (as 27T1)

Normalized defining polynomial

\( x^{27} - x^{26} - 52 x^{25} + 47 x^{24} + 1128 x^{23} - 914 x^{22} - 13369 x^{21} + 9612 x^{20} + 95357 x^{19} - 60102 x^{18} - 425693 x^{17} + 231576 x^{16} + 1201391 x^{15} - 553157 x^{14} - 2121177 x^{13} + 810403 x^{12} + 2271851 x^{11} - 706862 x^{10} - 1399735 x^{9} + 342875 x^{8} + 461618 x^{7} - 78149 x^{6} - 74294 x^{5} + 4948 x^{4} + 4861 x^{3} + 271 x^{2} - 34 x - 1 \)

Invariants

Degree:		$27$
Signature:		$[27, 0]$
Discriminant:		\(93991579198394673195940551085616659931617170424829641=109^{26}\)
Root discriminant:		$91.61$
Ramified primes:		$109$
This field is Galois and abelian over $\Q$.
Conductor:		\(109\)
Dirichlet character group:		$\lbrace$$\chi_{109}(1,·)$, $\chi_{109}(66,·)$, $\chi_{109}(3,·)$, $\chi_{109}(5,·)$, $\chi_{109}(7,·)$, $\chi_{109}(9,·)$, $\chi_{109}(75,·)$, $\chi_{109}(78,·)$, $\chi_{109}(15,·)$, $\chi_{109}(16,·)$, $\chi_{109}(81,·)$, $\chi_{109}(21,·)$, $\chi_{109}(22,·)$, $\chi_{109}(89,·)$, $\chi_{109}(25,·)$, $\chi_{109}(26,·)$, $\chi_{109}(27,·)$, $\chi_{109}(80,·)$, $\chi_{109}(35,·)$, $\chi_{109}(38,·)$, $\chi_{109}(97,·)$, $\chi_{109}(105,·)$, $\chi_{109}(45,·)$, $\chi_{109}(48,·)$, $\chi_{109}(49,·)$, $\chi_{109}(73,·)$, $\chi_{109}(63,·)$$\rbrace$
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $\frac{1}{1123} a^{25} - \frac{34}{1123} a^{24} - \frac{442}{1123} a^{23} - \frac{216}{1123} a^{22} + \frac{514}{1123} a^{21} - \frac{109}{1123} a^{20} + \frac{283}{1123} a^{19} - \frac{131}{1123} a^{17} + \frac{371}{1123} a^{16} + \frac{458}{1123} a^{15} + \frac{271}{1123} a^{14} + \frac{217}{1123} a^{13} + \frac{202}{1123} a^{12} + \frac{54}{1123} a^{11} + \frac{93}{1123} a^{10} - \frac{470}{1123} a^{9} + \frac{175}{1123} a^{8} + \frac{267}{1123} a^{7} - \frac{162}{1123} a^{6} + \frac{372}{1123} a^{5} - \frac{455}{1123} a^{4} + \frac{399}{1123} a^{3} + \frac{326}{1123} a^{2} - \frac{519}{1123} a - \frac{485}{1123}$, $\frac{1}{38652025412215529314156951896530659073754985185677} a^{26} + \frac{7991820473358747625315273651387437262759229759}{38652025412215529314156951896530659073754985185677} a^{25} - \frac{5708654903781952966464696042478353027023479401775}{38652025412215529314156951896530659073754985185677} a^{24} - \frac{6186162281376923819715251090418470911910670210614}{38652025412215529314156951896530659073754985185677} a^{23} - \frac{12331912655244367618773764117871076690368317768979}{38652025412215529314156951896530659073754985185677} a^{22} - \frac{8936341473516665454352196413952403830083091750364}{38652025412215529314156951896530659073754985185677} a^{21} + \frac{15909583813273576205872374925441305547140334566048}{38652025412215529314156951896530659073754985185677} a^{20} - \frac{17124005522461010163166944449743236478851565131480}{38652025412215529314156951896530659073754985185677} a^{19} + \frac{1131243861880206622487224895748177119175431808507}{38652025412215529314156951896530659073754985185677} a^{18} + \frac{10131547728768713481593382213399429739843929575823}{38652025412215529314156951896530659073754985185677} a^{17} + \frac{15879872474710352589162011574401397002815852814420}{38652025412215529314156951896530659073754985185677} a^{16} - \frac{3419397289927642984606887966506756305904157241746}{38652025412215529314156951896530659073754985185677} a^{15} + \frac{7037614405128318262367167130071473911098952123333}{38652025412215529314156951896530659073754985185677} a^{14} + \frac{10813767970620400974639557399893721703234218600009}{38652025412215529314156951896530659073754985185677} a^{13} + \frac{18953064522385387700662742818921448718978909545762}{38652025412215529314156951896530659073754985185677} a^{12} - \frac{11601011163587359431900189699253017882181987437844}{38652025412215529314156951896530659073754985185677} a^{11} - \frac{4710553884336296256326384521486051964869216439896}{38652025412215529314156951896530659073754985185677} a^{10} - \frac{19165859196367498761124773927239874127297181311085}{38652025412215529314156951896530659073754985185677} a^{9} - \frac{16202373375016903866451445930276334458761116000900}{38652025412215529314156951896530659073754985185677} a^{8} - \frac{5865401031874208170021050225532510091646552586161}{38652025412215529314156951896530659073754985185677} a^{7} + \frac{15079756736058852639143887769929590418394852626071}{38652025412215529314156951896530659073754985185677} a^{6} + \frac{17096352634286255689097295666631034101339614755825}{38652025412215529314156951896530659073754985185677} a^{5} - \frac{19174887351527986774219457277635871403212823929290}{38652025412215529314156951896530659073754985185677} a^{4} - \frac{13604717269736156948755389097026605481882589706489}{38652025412215529314156951896530659073754985185677} a^{3} - \frac{16002487590744990129746618463002626536040244750707}{38652025412215529314156951896530659073754985185677} a^{2} + \frac{15156754393213303651162277391201253649836080967595}{38652025412215529314156951896530659073754985185677} a - \frac{892345826767498889908404178579862884658507885090}{38652025412215529314156951896530659073754985185677}$

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

Unit group

Rank:		$26$
Torsion generator:		\( -1 \) (order $2$)
Fundamental units:		Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
Regulator:		\( 457712909681794240 \) (assuming GRH)

Galois group

$C_{27}$ (as 27T1):

A cyclic group of order 27

The 27 conjugacy class representatives for $C_{27}$

Character table for $C_{27}$ is not computed

Intermediate fields

3.3.11881.1, 9.9.19925626416901921.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59
Cycle type	${\href{/LocalNumberField/2.9.0.1}{9} }^{3}$	$27$	$27$	$27$	$27$	$27$	${\href{/LocalNumberField/17.9.0.1}{9} }^{3}$	${\href{/LocalNumberField/19.9.0.1}{9} }^{3}$	${\href{/LocalNumberField/23.9.0.1}{9} }^{3}$	$27$	$27$	$27$	${\href{/LocalNumberField/41.3.0.1}{3} }^{9}$	${\href{/LocalNumberField/43.9.0.1}{9} }^{3}$	$27$	$27$	$27$

Cycle lengths which are repeated in a cycle type are indicated by exponents.

Local algebras for ramified primes

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
109	Data not computed