Global number field 17.17.2200187128095499475530336818113367454680001.1

• Label: 17.17.2200187128...0001.1
• Degree: $17$
• Signature: $[17, 0]$
• Discriminant: $443^{16}$
• Root discriminant: $309.55$
• Ramified prime: $443$
• Class number: $1$ (GRH)
• Class group: Trivial (GRH)
• Galois group: $C_{17}$ (as 17T1)

Normalized defining polynomial

\( x^{17} - x^{16} - 208 x^{15} - 17 x^{14} + 15287 x^{13} + 13881 x^{12} - 487578 x^{11} - 703261 x^{10} + 6754359 x^{9} + 10540902 x^{8} - 41136753 x^{7} - 57683825 x^{6} + 92010954 x^{5} + 95287840 x^{4} - 17501435 x^{3} - 25026156 x^{2} - 563260 x + 1246103 \)

Invariants

Degree:		$17$
Signature:		$[17, 0]$
Discriminant:		\(2200187128095499475530336818113367454680001=443^{16}\)
Root discriminant:		$309.55$
Ramified primes:		$443$
This field is Galois and abelian over $\Q$.
Conductor:		\(443\)
Dirichlet character group:		$\lbrace$$\chi_{443}(1,·)$, $\chi_{443}(67,·)$, $\chi_{443}(324,·)$, $\chi_{443}(267,·)$, $\chi_{443}(13,·)$, $\chi_{443}(270,·)$, $\chi_{443}(209,·)$, $\chi_{443}(409,·)$, $\chi_{443}(225,·)$, $\chi_{443}(123,·)$, $\chi_{443}(425,·)$, $\chi_{443}(428,·)$, $\chi_{443}(370,·)$, $\chi_{443}(169,·)$, $\chi_{443}(248,·)$, $\chi_{443}(59,·)$, $\chi_{443}(380,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{4013480870186551123112427443978541538197334827965530746341757461} a^{16} + \frac{30664611433497702720987163865545312304482917972317611699004451}{4013480870186551123112427443978541538197334827965530746341757461} a^{15} - \frac{234236602191203470251879063625934586476098163260364116994871423}{4013480870186551123112427443978541538197334827965530746341757461} a^{14} - \frac{1538525439450223170396067485610034784683093539704487494351943862}{4013480870186551123112427443978541538197334827965530746341757461} a^{13} + \frac{55342538172956023481629299767885008096663009551709221443365116}{4013480870186551123112427443978541538197334827965530746341757461} a^{12} + \frac{753763549332543509880444461910703558803979931791746488591544835}{4013480870186551123112427443978541538197334827965530746341757461} a^{11} + \frac{449079695525428969055400104617954013904982282494392624345855044}{4013480870186551123112427443978541538197334827965530746341757461} a^{10} + \frac{1356331504173895330061633890226502143904723258261751560217500635}{4013480870186551123112427443978541538197334827965530746341757461} a^{9} - \frac{258115246454719936497475617167093245823083651779523923072083989}{4013480870186551123112427443978541538197334827965530746341757461} a^{8} - \frac{1439096506119506255900602412328539151888891788935030943061379179}{4013480870186551123112427443978541538197334827965530746341757461} a^{7} - \frac{264370601676178270686168837198796059220002990634243712724781297}{4013480870186551123112427443978541538197334827965530746341757461} a^{6} - \frac{180010373031027196207070542203811906918368022587662836705394813}{4013480870186551123112427443978541538197334827965530746341757461} a^{5} + \frac{782240170953201337633372345915642504015417842262032178961326220}{4013480870186551123112427443978541538197334827965530746341757461} a^{4} - \frac{425224591713287603242465941484685733198764997139724614627529825}{4013480870186551123112427443978541538197334827965530746341757461} a^{3} + \frac{1591411055048919701518357482737603500517050089451817421319409739}{4013480870186551123112427443978541538197334827965530746341757461} a^{2} + \frac{1107332544781032139606978167678448335717904050624033843189535906}{4013480870186551123112427443978541538197334827965530746341757461} a - \frac{1274193145411167372061357994658018891769389950257661519381602369}{4013480870186551123112427443978541538197334827965530746341757461}$

Class group and class number

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

Unit group

Rank:		$16$
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:		\( 2360552434044002.5 \) (assuming GRH)

Galois group

$C_{17}$ (as 17T1):

A cyclic group of order 17

The 17 conjugacy class representatives for $C_{17}$

Character table for $C_{17}$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Frobenius cycle types

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59
Cycle type	$17$	$17$	$17$	$17$	$17$	$17$	$17$	$17$	$17$	$17$	$17$	$17$	$17$	$17$	$17$	$17$	$17$

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
443	Data not computed