Number field 17.17.397527879693854754943959770157821268619247041.1

• Label: 17.17.397...041.1
• Degree: $17$
• Signature: $[17, 0]$
• Discriminant: $3.975\times 10^{44}$
• Root discriminant: $420.23$
• Ramified prime: $613$
• Class number: $1$ (GRH)
• Class group: trivial (GRH)
• Galois group: $C_{17}$ (as 17T1)

Normalized defining polynomial

\(x^{17} - x^{16} - 288 x^{15} + 265 x^{14} + 26034 x^{13} - 40228 x^{12} - 875968 x^{11} + 2022008 x^{10} + 8464009 x^{9} - 27681440 x^{8} - 8855367 x^{7} + 101412811 x^{6} - 87313302 x^{5} - 38624139 x^{4} + 67164168 x^{3} - 7149746 x^{2} - 7878215 x - 664471\)  

Invariants

Degree:		$17$
Signature:		$[17, 0]$
Discriminant:		\(397527879693854754943959770157821268619247041\)\(\medspace = 613^{16}\)
Root discriminant:		$420.23$
Ramified primes:		$613$
$|\Gal(K/\Q)|$:		$17$
This field is Galois and abelian over $\Q$.
Conductor:		\(613\)
Dirichlet character group:		$\lbrace$$\chi_{613}(1,·)$, $\chi_{613}(387,·)$, $\chi_{613}(197,·)$, $\chi_{613}(198,·)$, $\chi_{613}(583,·)$, $\chi_{613}(585,·)$, $\chi_{613}(586,·)$, $\chi_{613}(143,·)$, $\chi_{613}(220,·)$, $\chi_{613}(287,·)$, $\chi_{613}(546,·)$, $\chi_{613}(227,·)$, $\chi_{613}(37,·)$, $\chi_{613}(171,·)$, $\chi_{613}(430,·)$, $\chi_{613}(116,·)$, $\chi_{613}(190,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{83} a^{15} - \frac{17}{83} a^{14} + \frac{3}{83} a^{13} - \frac{23}{83} a^{12} - \frac{18}{83} a^{11} - \frac{39}{83} a^{10} - \frac{36}{83} a^{9} - \frac{37}{83} a^{8} - \frac{8}{83} a^{7} - \frac{21}{83} a^{6} + \frac{4}{83} a^{5} - \frac{40}{83} a^{4} + \frac{11}{83} a^{3} - \frac{29}{83} a^{2} - \frac{4}{83} a - \frac{32}{83}$, $\frac{1}{8005016753506549989469084281029751031104417898106301188902501} a^{16} - \frac{21742052392920060370221755237911823333762807269570076890675}{8005016753506549989469084281029751031104417898106301188902501} a^{15} + \frac{1318262744276119314600470957188333247003568665005465830337918}{8005016753506549989469084281029751031104417898106301188902501} a^{14} + \frac{978985556839464408917368141669338015554258870105656961439672}{8005016753506549989469084281029751031104417898106301188902501} a^{13} + \frac{2769652849413101156955094782253666473242821723068975047662913}{8005016753506549989469084281029751031104417898106301188902501} a^{12} - \frac{3865963828740415503858859292080994092134756470880241715944650}{8005016753506549989469084281029751031104417898106301188902501} a^{11} + \frac{1566778693849231577261720198965935208866780018514555776469291}{8005016753506549989469084281029751031104417898106301188902501} a^{10} + \frac{3292053559812149492078265514315409205047286548695243019801671}{8005016753506549989469084281029751031104417898106301188902501} a^{9} + \frac{2015958968976941668569215230872253848562337373638239291297888}{8005016753506549989469084281029751031104417898106301188902501} a^{8} - \frac{3768311406065772447653212035143009470927710873927962180210625}{8005016753506549989469084281029751031104417898106301188902501} a^{7} - \frac{651578051308172587753476565966120883630751279431155212135290}{8005016753506549989469084281029751031104417898106301188902501} a^{6} + \frac{1211160314524927167956762805442153429806021925493763606465554}{8005016753506549989469084281029751031104417898106301188902501} a^{5} + \frac{1576750091352007627088216157619973684725771671970502479230746}{8005016753506549989469084281029751031104417898106301188902501} a^{4} - \frac{916811404478504719979165509046006355992462069581910473626098}{8005016753506549989469084281029751031104417898106301188902501} a^{3} - \frac{3795236126167884475140566458620786336071637918943472071349866}{8005016753506549989469084281029751031104417898106301188902501} a^{2} - \frac{1000132303391176795188573093161245333680963265630325848143544}{8005016753506549989469084281029751031104417898106301188902501} a + \frac{2324389487913912762466131974077911977648826063869399720738164}{8005016753506549989469084281029751031104417898106301188902501}$  

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:		\( 40860825172254010 \) (assuming GRH)

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{17}\cdot(2\pi)^{0}\cdot 40860825172254010 \cdot 1}{2\sqrt{397527879693854754943959770157821268619247041}}\approx 0.134308428412734$ (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
613	Data not computed