Number field 34.34.12625853536261659455801827449124759267995559092128354245353083706292044373141887109743780036608.1

• Label: 34.34.126...608.1
• Degree: $34$
• Signature: $[34, 0]$
• Discriminant: $1.263\times 10^{94}$
• Root discriminant: \(585.71\)
• Ramified primes: $2,17$
• Class number: not computed
• Class group: not computed
• Galois group: $C_{34}$ (as 34T1)

Normalized defining polynomial

x^34 - 306*x^32 - 170*x^31 + 38964*x^30 + 39950*x^29 - 2736575*x^28 - 3896570*x^27 + 117983060*x^26 + 208049400*x^25 - 3285052273*x^24 - 6754932490*x^23 + 60547426236*x^22 + 140226759674*x^21 - 745857271805*x^20 - 1907878745466*x^19 + 6138846980317*x^18 + 17205240198006*x^17 - 33506139159644*x^16 - 103250052122532*x^15 + 119320834330186*x^14 + 411737087894594*x^13 - 267177797973454*x^12 - 1079717606164642*x^11 + 342032267449207*x^10 + 1815135781705466*x^9 - 170798444831476*x^8 - 1859368458613010*x^7 - 110445188516547*x^6 + 1048522170841052*x^5 + 176656014973095*x^4 - 256256238773310*x^3 - 62202168423886*x^2 + 8768628925908*x + 2196254131873

Invariants

Degree:		$34$
Signature:		$[34, 0]$
Discriminant:		12625853536261659455801827449124759267995559092128354245353083706292044373141887109743780036608
Root discriminant:		\(585.71\)
Galois root discriminant:		$2^{3/2}17^{32/17}\approx 585.7120239053163$
Ramified primes:		\(2\), \(17\)
Discriminant root field:		\(\Q(\sqrt{2}) \)
$\card{ \Gal(K/\Q) }$:		$34$
This field is Galois and abelian over $\Q$.
Conductor:		\(2312=2^{3}\cdot 17^{2}\)
Dirichlet character group:		$\lbrace$$\chi_{2312}(1,·)$, $\chi_{2312}(1157,·)$, $\chi_{2312}(2177,·)$, $\chi_{2312}(137,·)$, $\chi_{2312}(1293,·)$, $\chi_{2312}(273,·)$, $\chi_{2312}(1429,·)$, $\chi_{2312}(409,·)$, $\chi_{2312}(1565,·)$, $\chi_{2312}(69,·)$, $\chi_{2312}(545,·)$, $\chi_{2312}(1701,·)$, $\chi_{2312}(681,·)$, $\chi_{2312}(1837,·)$, $\chi_{2312}(817,·)$, $\chi_{2312}(1973,·)$, $\chi_{2312}(953,·)$, $\chi_{2312}(2109,·)$, $\chi_{2312}(1089,·)$, $\chi_{2312}(2245,·)$, $\chi_{2312}(1225,·)$, $\chi_{2312}(205,·)$, $\chi_{2312}(1361,·)$, $\chi_{2312}(341,·)$, $\chi_{2312}(1497,·)$, $\chi_{2312}(477,·)$, $\chi_{2312}(1633,·)$, $\chi_{2312}(613,·)$, $\chi_{2312}(1769,·)$, $\chi_{2312}(749,·)$, $\chi_{2312}(1905,·)$, $\chi_{2312}(885,·)$, $\chi_{2312}(2041,·)$, $\chi_{2312}(1021,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $\frac{1}{131}a^{30}-\frac{36}{131}a^{29}-\frac{17}{131}a^{28}+\frac{34}{131}a^{27}-\frac{33}{131}a^{26}+\frac{5}{131}a^{25}-\frac{58}{131}a^{24}-\frac{33}{131}a^{23}+\frac{14}{131}a^{22}-\frac{40}{131}a^{21}-\frac{3}{131}a^{20}-\frac{43}{131}a^{19}-\frac{26}{131}a^{18}+\frac{33}{131}a^{17}+\frac{61}{131}a^{16}+\frac{25}{131}a^{15}-\frac{50}{131}a^{14}-\frac{45}{131}a^{13}-\frac{48}{131}a^{12}+\frac{11}{131}a^{11}-\frac{49}{131}a^{10}-\frac{62}{131}a^{9}-\frac{28}{131}a^{8}+\frac{28}{131}a^{7}-\frac{25}{131}a^{6}-\frac{62}{131}a^{5}-\frac{58}{131}a^{4}+\frac{61}{131}a^{3}-\frac{34}{131}a^{2}-\frac{22}{131}a+\frac{22}{131}$, $\frac{1}{131}a^{31}-\frac{3}{131}a^{29}-\frac{54}{131}a^{28}+\frac{12}{131}a^{27}-\frac{4}{131}a^{26}-\frac{9}{131}a^{25}-\frac{25}{131}a^{24}+\frac{5}{131}a^{23}-\frac{60}{131}a^{22}-\frac{2}{131}a^{21}-\frac{20}{131}a^{20}-\frac{2}{131}a^{19}+\frac{14}{131}a^{18}-\frac{61}{131}a^{17}-\frac{6}{131}a^{16}+\frac{64}{131}a^{15}-\frac{11}{131}a^{14}+\frac{35}{131}a^{13}-\frac{14}{131}a^{12}-\frac{46}{131}a^{11}+\frac{8}{131}a^{10}-\frac{33}{131}a^{9}-\frac{63}{131}a^{8}-\frac{65}{131}a^{7}-\frac{45}{131}a^{6}-\frac{63}{131}a^{5}-\frac{62}{131}a^{4}-\frac{65}{131}a^{3}+\frac{64}{131}a^{2}+\frac{16}{131}a+\frac{6}{131}$, $\frac{1}{39\!\cdots\!23}a^{32}-\frac{12\!\cdots\!72}{39\!\cdots\!23}a^{31}-\frac{16\!\cdots\!90}{39\!\cdots\!23}a^{30}-\frac{41\!\cdots\!65}{39\!\cdots\!23}a^{29}+\frac{17\!\cdots\!54}{39\!\cdots\!23}a^{28}-\frac{80\!\cdots\!28}{39\!\cdots\!23}a^{27}+\frac{69\!\cdots\!25}{39\!\cdots\!23}a^{26}+\frac{41\!\cdots\!00}{39\!\cdots\!23}a^{25}-\frac{12\!\cdots\!75}{39\!\cdots\!23}a^{24}+\frac{84\!\cdots\!31}{39\!\cdots\!23}a^{23}-\frac{14\!\cdots\!79}{39\!\cdots\!23}a^{22}+\frac{17\!\cdots\!01}{39\!\cdots\!23}a^{21}+\frac{98\!\cdots\!90}{39\!\cdots\!23}a^{20}-\frac{79\!\cdots\!94}{39\!\cdots\!23}a^{19}-\frac{11\!\cdots\!65}{39\!\cdots\!23}a^{18}-\frac{26\!\cdots\!06}{39\!\cdots\!23}a^{17}-\frac{16\!\cdots\!49}{39\!\cdots\!23}a^{16}-\frac{16\!\cdots\!03}{39\!\cdots\!23}a^{15}+\frac{56\!\cdots\!29}{39\!\cdots\!23}a^{14}+\frac{18\!\cdots\!43}{39\!\cdots\!23}a^{13}-\frac{18\!\cdots\!29}{39\!\cdots\!23}a^{12}+\frac{51\!\cdots\!85}{39\!\cdots\!23}a^{11}+\frac{17\!\cdots\!87}{39\!\cdots\!23}a^{10}+\frac{22\!\cdots\!27}{39\!\cdots\!23}a^{9}-\frac{29\!\cdots\!59}{39\!\cdots\!23}a^{8}-\frac{18\!\cdots\!39}{39\!\cdots\!23}a^{7}-\frac{11\!\cdots\!28}{39\!\cdots\!23}a^{6}-\frac{84\!\cdots\!67}{39\!\cdots\!23}a^{5}-\frac{12\!\cdots\!40}{39\!\cdots\!23}a^{4}-\frac{11\!\cdots\!84}{39\!\cdots\!23}a^{3}+\frac{13\!\cdots\!72}{39\!\cdots\!23}a^{2}-\frac{14\!\cdots\!87}{39\!\cdots\!23}a-\frac{37\!\cdots\!60}{39\!\cdots\!23}$, $\frac{1}{13\!\cdots\!03}a^{33}+\frac{16\!\cdots\!05}{13\!\cdots\!03}a^{32}-\frac{11\!\cdots\!74}{13\!\cdots\!03}a^{31}-\frac{14\!\cdots\!49}{13\!\cdots\!03}a^{30}+\frac{11\!\cdots\!26}{13\!\cdots\!03}a^{29}-\frac{20\!\cdots\!20}{13\!\cdots\!03}a^{28}-\frac{17\!\cdots\!35}{13\!\cdots\!03}a^{27}+\frac{66\!\cdots\!27}{13\!\cdots\!03}a^{26}-\frac{32\!\cdots\!56}{13\!\cdots\!03}a^{25}-\frac{62\!\cdots\!09}{13\!\cdots\!03}a^{24}+\frac{34\!\cdots\!51}{13\!\cdots\!03}a^{23}+\frac{65\!\cdots\!85}{13\!\cdots\!03}a^{22}+\frac{42\!\cdots\!53}{13\!\cdots\!03}a^{21}+\frac{42\!\cdots\!97}{13\!\cdots\!03}a^{20}+\frac{47\!\cdots\!86}{13\!\cdots\!03}a^{19}+\frac{63\!\cdots\!22}{13\!\cdots\!03}a^{18}+\frac{37\!\cdots\!02}{13\!\cdots\!03}a^{17}+\frac{55\!\cdots\!12}{13\!\cdots\!03}a^{16}-\frac{19\!\cdots\!30}{13\!\cdots\!03}a^{15}-\frac{22\!\cdots\!88}{13\!\cdots\!03}a^{14}+\frac{32\!\cdots\!78}{13\!\cdots\!03}a^{13}+\frac{84\!\cdots\!09}{13\!\cdots\!03}a^{12}+\frac{42\!\cdots\!84}{13\!\cdots\!03}a^{11}+\frac{56\!\cdots\!63}{13\!\cdots\!03}a^{10}+\frac{37\!\cdots\!31}{13\!\cdots\!03}a^{9}-\frac{43\!\cdots\!87}{13\!\cdots\!03}a^{8}-\frac{58\!\cdots\!85}{13\!\cdots\!03}a^{7}+\frac{55\!\cdots\!66}{13\!\cdots\!03}a^{6}+\frac{45\!\cdots\!64}{13\!\cdots\!03}a^{5}+\frac{59\!\cdots\!30}{13\!\cdots\!03}a^{4}-\frac{26\!\cdots\!00}{13\!\cdots\!03}a^{3}-\frac{65\!\cdots\!08}{13\!\cdots\!03}a^{2}-\frac{21\!\cdots\!79}{13\!\cdots\!03}a-\frac{31\!\cdots\!72}{13\!\cdots\!03}$

Monogenic:		Not computed
Index:		$1$
Inessential primes:		None

Class group and class number

not computed

Unit group

Rank:		$33$
Torsion generator:		\( -1 \)  (order $2$)
Fundamental units:		not computed
Regulator:		not computed

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr $ not computed \end{aligned}\]

Galois group

$C_{34}$ (as 34T1):

A cyclic group of order 34

The 34 conjugacy class representatives for $C_{34}$

Character table for $C_{34}$

Intermediate fields

\(\Q(\sqrt{2}) \), 17.17.2367911594760467245844106297320951247361.1

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	R	$34$	$34$	$17^{2}$	$34$	$34$	R	$34$	$17^{2}$	$34$	$17^{2}$	$34$	$17^{2}$	$34$	$17^{2}$	$34$	$34$

In the table, R denotes a ramified prime. 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
\(2\)		Deg $34$	$2$	$17$	$51$
\(17\)	17.17.32.1	$x^{17} + 272 x^{16} + 17$	$17$	$1$	$32$	$C_{17}$	$[2]$
	17.17.32.1	$x^{17} + 272 x^{16} + 17$	$17$	$1$	$32$	$C_{17}$	$[2]$