Number field 33.33.19453503693937104587930129781385805482525288081047228454446873116938739522375281.1

• Label: 33.33.194...281.1
• Degree: $33$
• Signature: $[33, 0]$
• Discriminant: $1.945\times 10^{79}$
• Root discriminant: \(252.75\)
• Ramified primes: $13,67$
• Class number: not computed
• Class group: not computed
• Galois group: $C_{33}$ (as 33T1)

Normalized defining polynomial

x^33 - 8*x^32 - 108*x^31 + 826*x^30 + 5517*x^29 - 37410*x^28 - 174850*x^27 + 974941*x^26 + 3753002*x^25 - 16170687*x^24 - 56209081*x^23 + 178946863*x^22 + 592568182*x^21 - 1351731560*x^20 - 4397140947*x^19 + 7038388451*x^18 + 22835380012*x^17 - 25350973354*x^16 - 82027847149*x^15 + 63440275941*x^14 + 199922687695*x^13 - 111827183164*x^12 - 320812557227*x^11 + 142179324189*x^10 + 323210228406*x^9 - 130110920902*x^8 - 188148415319*x^7 + 77322457297*x^6 + 52579210647*x^5 - 22741173483*x^4 - 3948105037*x^3 + 1212304894*x^2 + 227068790*x + 8672509

Invariants

Degree:		$33$
Signature:		$[33, 0]$
Discriminant:		\(19453503693937104587930129781385805482525288081047228454446873116938739522375281\) \(\medspace = 13^{22}\cdot 67^{30}\)
Root discriminant:		\(252.75\)
Galois root discriminant:		$13^{2/3}67^{10/11}\approx 252.75334042094067$
Ramified primes:		\(13\), \(67\)
Discriminant root field:		\(\Q\)
$\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$:		$C_{33}$
This field is Galois and abelian over $\Q$.
Conductor:		\(871=13\cdot 67\)
Dirichlet character group:		$\lbrace$$\chi_{871}(1,·)$, $\chi_{871}(131,·)$, $\chi_{871}(263,·)$, $\chi_{871}(9,·)$, $\chi_{871}(269,·)$, $\chi_{871}(14,·)$, $\chi_{871}(399,·)$, $\chi_{871}(22,·)$, $\chi_{871}(68,·)$, $\chi_{871}(282,·)$, $\chi_{871}(796,·)$, $\chi_{871}(159,·)$, $\chi_{871}(417,·)$, $\chi_{871}(679,·)$, $\chi_{871}(40,·)$, $\chi_{871}(685,·)$, $\chi_{871}(560,·)$, $\chi_{871}(308,·)$, $\chi_{871}(692,·)$, $\chi_{871}(828,·)$, $\chi_{871}(196,·)$, $\chi_{871}(198,·)$, $\chi_{871}(484,·)$, $\chi_{871}(464,·)$, $\chi_{871}(81,·)$, $\chi_{871}(729,·)$, $\chi_{871}(92,·)$, $\chi_{871}(612,·)$, $\chi_{871}(360,·)$, $\chi_{871}(107,·)$, $\chi_{871}(625,·)$, $\chi_{871}(627,·)$, $\chi_{871}(126,·)$$\rbrace$
This is not a CM field.
This field has no CM subfields.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $\frac{1}{239}a^{28}+\frac{52}{239}a^{27}-\frac{3}{239}a^{26}-\frac{75}{239}a^{25}-\frac{89}{239}a^{24}-\frac{19}{239}a^{23}+\frac{38}{239}a^{22}+\frac{111}{239}a^{21}-\frac{111}{239}a^{20}+\frac{29}{239}a^{19}+\frac{117}{239}a^{18}-\frac{51}{239}a^{17}-\frac{21}{239}a^{16}-\frac{56}{239}a^{15}+\frac{15}{239}a^{14}-\frac{65}{239}a^{13}+\frac{80}{239}a^{12}-\frac{2}{239}a^{11}+\frac{70}{239}a^{10}+\frac{88}{239}a^{9}+\frac{101}{239}a^{8}+\frac{87}{239}a^{7}+\frac{77}{239}a^{6}+\frac{84}{239}a^{5}-\frac{63}{239}a^{4}-\frac{27}{239}a^{3}-\frac{36}{239}a^{2}+\frac{17}{239}a+\frac{32}{239}$, $\frac{1}{239}a^{29}-\frac{78}{239}a^{27}+\frac{81}{239}a^{26}-\frac{13}{239}a^{25}+\frac{68}{239}a^{24}+\frac{70}{239}a^{23}+\frac{47}{239}a^{22}+\frac{92}{239}a^{21}+\frac{65}{239}a^{20}+\frac{43}{239}a^{19}+\frac{79}{239}a^{18}+\frac{2}{239}a^{17}+\frac{80}{239}a^{16}+\frac{59}{239}a^{15}+\frac{111}{239}a^{14}+\frac{114}{239}a^{13}-\frac{99}{239}a^{12}-\frac{65}{239}a^{11}+\frac{33}{239}a^{10}+\frac{66}{239}a^{9}+\frac{93}{239}a^{8}+\frac{94}{239}a^{7}-\frac{96}{239}a^{6}+\frac{110}{239}a^{5}-\frac{97}{239}a^{4}-\frac{66}{239}a^{3}-\frac{23}{239}a^{2}+\frac{104}{239}a+\frac{9}{239}$, $\frac{1}{256447}a^{30}+\frac{423}{256447}a^{29}+\frac{113}{256447}a^{28}+\frac{58757}{256447}a^{27}-\frac{4802}{256447}a^{26}+\frac{18962}{256447}a^{25}+\frac{36691}{256447}a^{24}-\frac{125498}{256447}a^{23}-\frac{92030}{256447}a^{22}+\frac{41301}{256447}a^{21}+\frac{91660}{256447}a^{20}-\frac{102624}{256447}a^{19}+\frac{16092}{256447}a^{18}+\frac{111880}{256447}a^{17}+\frac{94657}{256447}a^{16}-\frac{110386}{256447}a^{15}+\frac{2958}{8843}a^{14}-\frac{101000}{256447}a^{13}-\frac{102425}{256447}a^{12}-\frac{9441}{256447}a^{11}-\frac{113615}{256447}a^{10}-\frac{85675}{256447}a^{9}-\frac{114551}{256447}a^{8}-\frac{2277}{6931}a^{7}+\frac{4562}{256447}a^{6}-\frac{30255}{256447}a^{5}-\frac{1267}{256447}a^{4}+\frac{3947}{256447}a^{3}+\frac{69778}{256447}a^{2}+\frac{74255}{256447}a-\frac{35491}{256447}$, $\frac{1}{256447}a^{31}+\frac{375}{256447}a^{29}+\frac{228}{256447}a^{28}+\frac{98967}{256447}a^{27}-\frac{72186}{256447}a^{26}-\frac{20429}{256447}a^{25}+\frac{58637}{256447}a^{24}+\frac{90432}{256447}a^{23}+\frac{1471}{6931}a^{22}-\frac{32545}{256447}a^{21}+\frac{121235}{256447}a^{20}+\frac{43581}{256447}a^{19}+\frac{50915}{256447}a^{18}+\frac{91936}{256447}a^{17}+\frac{1897}{8843}a^{16}-\frac{4813}{256447}a^{15}+\frac{11520}{256447}a^{14}-\frac{3776}{8843}a^{13}+\frac{99113}{256447}a^{12}-\frac{52617}{256447}a^{11}+\frac{51144}{256447}a^{10}+\frac{78539}{256447}a^{9}+\frac{96554}{256447}a^{8}+\frac{6486}{256447}a^{7}+\frac{14339}{256447}a^{6}+\frac{59}{239}a^{5}-\frac{9488}{256447}a^{4}-\frac{57902}{256447}a^{3}-\frac{95289}{256447}a^{2}+\frac{86895}{256447}a+\frac{125891}{256447}$, $\frac{1}{16\cdots 59}a^{32}-\frac{11\cdots 59}{16\cdots 59}a^{31}-\frac{64\cdots 62}{16\cdots 59}a^{30}-\frac{30\cdots 41}{16\cdots 59}a^{29}+\frac{16\cdots 17}{16\cdots 59}a^{28}+\frac{35\cdots 80}{16\cdots 59}a^{27}-\frac{70\cdots 92}{16\cdots 59}a^{26}+\frac{41\cdots 67}{16\cdots 59}a^{25}+\frac{74\cdots 87}{16\cdots 59}a^{24}+\frac{60\cdots 69}{45\cdots 07}a^{23}+\frac{54\cdots 57}{16\cdots 59}a^{22}+\frac{12\cdots 16}{16\cdots 59}a^{21}-\frac{32\cdots 31}{16\cdots 59}a^{20}-\frac{22\cdots 35}{58\cdots 71}a^{19}-\frac{50\cdots 00}{16\cdots 59}a^{18}-\frac{18\cdots 72}{16\cdots 59}a^{17}-\frac{72\cdots 46}{16\cdots 59}a^{16}-\frac{39\cdots 91}{16\cdots 59}a^{15}-\frac{79\cdots 80}{16\cdots 59}a^{14}+\frac{78\cdots 61}{16\cdots 59}a^{13}-\frac{49\cdots 31}{16\cdots 59}a^{12}-\frac{82\cdots 47}{16\cdots 59}a^{11}-\frac{23\cdots 85}{16\cdots 59}a^{10}-\frac{15\cdots 62}{16\cdots 59}a^{9}-\frac{47\cdots 53}{16\cdots 59}a^{8}+\frac{14\cdots 33}{16\cdots 59}a^{7}-\frac{75\cdots 52}{16\cdots 59}a^{6}+\frac{74\cdots 69}{16\cdots 59}a^{5}+\frac{54\cdots 33}{16\cdots 59}a^{4}+\frac{15\cdots 10}{16\cdots 59}a^{3}-\frac{59\cdots 34}{16\cdots 59}a^{2}+\frac{14\cdots 50}{16\cdots 59}a+\frac{13\cdots 12}{16\cdots 59}$

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

Class group and class number

Ideal class group:		not computed
Narrow class group:		not computed

Unit group

Rank:		$32$
Torsion generator:		\( -1 \)  (order $2$)
Fundamental units:		not computed
Regulator:		not computed

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr = \mathstrut &\frac{2^{33}\cdot(2\pi)^{0}\cdot R \cdot h}{2\cdot\sqrt{19453503693937104587930129781385805482525288081047228454446873116938739522375281}}\cr\mathstrut & \text{ some values not computed } \end{aligned}\]

Galois group

$C_{33}$ (as 33T1):

A cyclic group of order 33

The 33 conjugacy class representatives for $C_{33}$

Character table for $C_{33}$

Intermediate fields

3.3.169.1, 11.11.1822837804551761449.1

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	$33$	$33$	${\href{/padicField/5.11.0.1}{11} }^{3}$	$33$	$33$	R	$33$	$33$	$33$	${\href{/padicField/29.3.0.1}{3} }^{11}$	${\href{/padicField/31.11.0.1}{11} }^{3}$	${\href{/padicField/37.3.0.1}{3} }^{11}$	$33$	$33$	${\href{/padicField/47.11.0.1}{11} }^{3}$	${\href{/padicField/53.11.0.1}{11} }^{3}$	$33$

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
\(13\)		Deg $33$	$3$	$11$	$22$
\(67\)		Deg $33$	$11$	$3$	$30$

Spectrum of ring of integers