Global number field 42.0.9380082945933081406113456619151991432292083579779389915131296484043.1: \(\Q(\zeta_{43})\)

• Label: 42.0.93800829459...4043.1
• Degree: $42$
• Signature: $[0, 21]$
• Discriminant: $-\,43^{41}$
• Root discriminant: $39.32$
• Ramified prime: $43$
• Class number: $211$ (GRH)
• Class group: $[211]$ (GRH)
• Galois group: $C_{42}$ (as 42T1)

Normalized defining polynomial

\( x^{42} - x^{41} + x^{40} - x^{39} + x^{38} - x^{37} + x^{36} - x^{35} + x^{34} - x^{33} + x^{32} - x^{31} + x^{30} - x^{29} + x^{28} - x^{27} + x^{26} - x^{25} + x^{24} - x^{23} + x^{22} - x^{21} + x^{20} - x^{19} + x^{18} - x^{17} + x^{16} - x^{15} + x^{14} - x^{13} + x^{12} - x^{11} + x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \)

Invariants

Degree:		$42$
Signature:		$[0, 21]$
Discriminant:		\(-9380082945933081406113456619151991432292083579779389915131296484043=-\,43^{41}\)
Root discriminant:		$39.32$
Ramified primes:		$43$
This field is Galois and abelian over $\Q$.
Conductor:		\(43\)
Dirichlet character group:		$\lbrace$$\chi_{43}(1,·)$, $\chi_{43}(2,·)$, $\chi_{43}(3,·)$, $\chi_{43}(4,·)$, $\chi_{43}(5,·)$, $\chi_{43}(6,·)$, $\chi_{43}(7,·)$, $\chi_{43}(8,·)$, $\chi_{43}(9,·)$, $\chi_{43}(10,·)$, $\chi_{43}(11,·)$, $\chi_{43}(12,·)$, $\chi_{43}(13,·)$, $\chi_{43}(14,·)$, $\chi_{43}(15,·)$, $\chi_{43}(16,·)$, $\chi_{43}(17,·)$, $\chi_{43}(18,·)$, $\chi_{43}(19,·)$, $\chi_{43}(20,·)$, $\chi_{43}(21,·)$, $\chi_{43}(22,·)$, $\chi_{43}(23,·)$, $\chi_{43}(24,·)$, $\chi_{43}(25,·)$, $\chi_{43}(26,·)$, $\chi_{43}(27,·)$, $\chi_{43}(28,·)$, $\chi_{43}(29,·)$, $\chi_{43}(30,·)$, $\chi_{43}(31,·)$, $\chi_{43}(32,·)$, $\chi_{43}(33,·)$, $\chi_{43}(34,·)$, $\chi_{43}(35,·)$, $\chi_{43}(36,·)$, $\chi_{43}(37,·)$, $\chi_{43}(38,·)$, $\chi_{43}(39,·)$, $\chi_{43}(40,·)$, $\chi_{43}(41,·)$, $\chi_{43}(42,·)$$\rbrace$
This is 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

$C_{211}$, which has order $211$ (assuming GRH)

Unit group

Rank:		$20$
Torsion generator:		\( a \) (order $86$)
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:		\( 2748021948787771.5 \) (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

\(\Q(\sqrt{-43}) \), 3.3.1849.1, 6.0.147008443.1, 7.7.6321363049.1, 14.0.1718264124282290785243.1, \(\Q(\zeta_{43})^+\)

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	${\href{/LocalNumberField/2.14.0.1}{14} }^{3}$	$42$	$42$	${\href{/LocalNumberField/7.6.0.1}{6} }^{7}$	${\href{/LocalNumberField/11.7.0.1}{7} }^{6}$	$21^{2}$	$21^{2}$	$42$	$21^{2}$	$42$	$21^{2}$	${\href{/LocalNumberField/37.6.0.1}{6} }^{7}$	${\href{/LocalNumberField/41.7.0.1}{7} }^{6}$	R	${\href{/LocalNumberField/47.7.0.1}{7} }^{6}$	$21^{2}$	${\href{/LocalNumberField/59.7.0.1}{7} }^{6}$

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
43	Data not computed