Global number field 42.42.98118980687896783910098639727548084722605054105289047332129555342401833439729.1: \(\Q(\zeta_{129})^+\)

• Label: 42.42.9811898068...9729.1
• Degree: $42$
• Signature: $[42, 0]$
• Discriminant: $3^{21}\cdot 43^{41}$
• Root discriminant: $68.10$
• Ramified primes: $3, 43$
• Class number: $1$ (GRH)
• Class group: Trivial (GRH)
• Galois group: $C_{42}$ (as 42T1)

Normalized defining polynomial

\( x^{42} - x^{41} - 42 x^{40} + 42 x^{39} + 818 x^{38} - 818 x^{37} - 9803 x^{36} + 9803 x^{35} + 80884 x^{34} - 80884 x^{33} - 487103 x^{32} + 487103 x^{31} + 2214673 x^{30} - 2214673 x^{29} - 7756167 x^{28} + 7756167 x^{27} + 21159269 x^{26} - 21159269 x^{25} - 45176143 x^{24} + 45176143 x^{23} + 75433697 x^{22} - 75433697 x^{21} - 97942948 x^{20} + 97942948 x^{19} + 97804877 x^{18} - 97804877 x^{17} - 73850908 x^{16} + 73850908 x^{15} + 41150012 x^{14} - 41150012 x^{13} - 16350448 x^{12} + 16350448 x^{11} + 4413607 x^{10} - 4413607 x^{9} - 753918 x^{8} + 753918 x^{7} + 72886 x^{6} - 72886 x^{5} - 3267 x^{4} + 3267 x^{3} + 44 x^{2} - 44 x + 1 \)

Invariants

Degree:		$42$
Signature:		$[42, 0]$
Discriminant:		\(98118980687896783910098639727548084722605054105289047332129555342401833439729=3^{21}\cdot 43^{41}\)
Root discriminant:		$68.10$
Ramified primes:		$3, 43$
This field is Galois and abelian over $\Q$.
Conductor:		\(129=3\cdot 43\)
Dirichlet character group:		$\lbrace$$\chi_{129}(128,·)$, $\chi_{129}(1,·)$, $\chi_{129}(2,·)$, $\chi_{129}(4,·)$, $\chi_{129}(5,·)$, $\chi_{129}(8,·)$, $\chi_{129}(10,·)$, $\chi_{129}(13,·)$, $\chi_{129}(16,·)$, $\chi_{129}(20,·)$, $\chi_{129}(25,·)$, $\chi_{129}(26,·)$, $\chi_{129}(29,·)$, $\chi_{129}(31,·)$, $\chi_{129}(32,·)$, $\chi_{129}(40,·)$, $\chi_{129}(49,·)$, $\chi_{129}(50,·)$, $\chi_{129}(52,·)$, $\chi_{129}(58,·)$, $\chi_{129}(62,·)$, $\chi_{129}(64,·)$, $\chi_{129}(65,·)$, $\chi_{129}(67,·)$, $\chi_{129}(71,·)$, $\chi_{129}(77,·)$, $\chi_{129}(79,·)$, $\chi_{129}(80,·)$, $\chi_{129}(89,·)$, $\chi_{129}(97,·)$, $\chi_{129}(98,·)$, $\chi_{129}(100,·)$, $\chi_{129}(103,·)$, $\chi_{129}(104,·)$, $\chi_{129}(109,·)$, $\chi_{129}(113,·)$, $\chi_{129}(116,·)$, $\chi_{129}(119,·)$, $\chi_{129}(121,·)$, $\chi_{129}(124,·)$, $\chi_{129}(125,·)$, $\chi_{129}(127,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $a^{40}$, $a^{41}$

Class group and class number

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

Unit group

Rank:		$41$
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:		\( 16535537450237775000000000 \) (assuming GRH)

Galois group

$C_{42}$ (as 42T1):

A cyclic group of order 42

The 42 conjugacy class representatives for $C_{42}$

Character table for $C_{42}$ is not computed

Intermediate fields

\(\Q(\sqrt{129}) \), 3.3.1849.1, 6.6.3969227961.1, 7.7.6321363049.1, 14.14.3757843639805369947326441.1, \(\Q(\zeta_{43})^+\)

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	${\href{/LocalNumberField/2.7.0.1}{7} }^{6}$	R	$21^{2}$	${\href{/LocalNumberField/7.6.0.1}{6} }^{7}$	${\href{/LocalNumberField/11.14.0.1}{14} }^{3}$	$21^{2}$	$42$	$42$	$42$	$21^{2}$	$21^{2}$	${\href{/LocalNumberField/37.6.0.1}{6} }^{7}$	${\href{/LocalNumberField/41.14.0.1}{14} }^{3}$	R	${\href{/LocalNumberField/47.14.0.1}{14} }^{3}$	$42$	${\href{/LocalNumberField/59.14.0.1}{14} }^{3}$

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
3	Data not computed
43	Data not computed