Global number field 33.33.27189028279553414235049966267283185807800188603627566700161.1: \(\Q(\zeta_{67})^+\)

• Label: 33.33.2718902827...0161.1
• Degree: $33$
• Signature: $[33, 0]$
• Discriminant: $67^{32}$
• Root discriminant: $58.98$
• Ramified prime: $67$
• Class number: $1$ (GRH)
• Class group: Trivial (GRH)
• Galois group: $C_{33}$ (as 33T1)

Normalized defining polynomial

\( x^{33} - x^{32} - 32 x^{31} + 31 x^{30} + 465 x^{29} - 435 x^{28} - 4060 x^{27} + 3654 x^{26} + 23751 x^{25} - 20475 x^{24} - 98280 x^{23} + 80730 x^{22} + 296010 x^{21} - 230230 x^{20} - 657800 x^{19} + 480700 x^{18} + 1081575 x^{17} - 735471 x^{16} - 1307504 x^{15} + 817190 x^{14} + 1144066 x^{13} - 646646 x^{12} - 705432 x^{11} + 352716 x^{10} + 293930 x^{9} - 125970 x^{8} - 77520 x^{7} + 27132 x^{6} + 11628 x^{5} - 3060 x^{4} - 816 x^{3} + 136 x^{2} + 17 x - 1 \)

Invariants

Degree:		$33$
Signature:		$[33, 0]$
Discriminant:		\(27189028279553414235049966267283185807800188603627566700161=67^{32}\)
Root discriminant:		$58.98$
Ramified primes:		$67$
This field is Galois and abelian over $\Q$.
Conductor:		\(67\)
Dirichlet character group:		$\lbrace$$\chi_{67}(1,·)$, $\chi_{67}(4,·)$, $\chi_{67}(6,·)$, $\chi_{67}(9,·)$, $\chi_{67}(10,·)$, $\chi_{67}(14,·)$, $\chi_{67}(15,·)$, $\chi_{67}(16,·)$, $\chi_{67}(17,·)$, $\chi_{67}(19,·)$, $\chi_{67}(21,·)$, $\chi_{67}(22,·)$, $\chi_{67}(23,·)$, $\chi_{67}(24,·)$, $\chi_{67}(25,·)$, $\chi_{67}(26,·)$, $\chi_{67}(29,·)$, $\chi_{67}(33,·)$, $\chi_{67}(35,·)$, $\chi_{67}(36,·)$, $\chi_{67}(37,·)$, $\chi_{67}(39,·)$, $\chi_{67}(40,·)$, $\chi_{67}(47,·)$, $\chi_{67}(49,·)$, $\chi_{67}(54,·)$, $\chi_{67}(55,·)$, $\chi_{67}(56,·)$, $\chi_{67}(59,·)$, $\chi_{67}(60,·)$, $\chi_{67}(62,·)$, $\chi_{67}(64,·)$, $\chi_{67}(65,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$

Class group and class number

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

Unit group

Rank:		$32$
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:		\( 3985748844865106400 \) (assuming GRH)

Galois group

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

A cyclic group of order 33

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

Character table for $C_{33}$ is not computed

Intermediate fields

3.3.4489.1, 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	$33$	${\href{/LocalNumberField/3.11.0.1}{11} }^{3}$	${\href{/LocalNumberField/5.11.0.1}{11} }^{3}$	$33$	$33$	$33$	$33$	$33$	$33$	${\href{/LocalNumberField/29.3.0.1}{3} }^{11}$	$33$	${\href{/LocalNumberField/37.3.0.1}{3} }^{11}$	$33$	${\href{/LocalNumberField/43.11.0.1}{11} }^{3}$	$33$	${\href{/LocalNumberField/53.11.0.1}{11} }^{3}$	${\href{/LocalNumberField/59.11.0.1}{11} }^{3}$

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
67	Data not computed