Global number field 42.42.1236219045653317330764639083558080790009887216550178193020957667462329030021.1: \(\Q(\zeta_{147})^+\)

• Label: 42.42.1236219045...0021.1
• Degree: $42$
• Signature: $[42, 0]$
• Discriminant: $3^{21}\cdot 7^{77}$
• Root discriminant: $61.36$
• Ramified primes: $3, 7$
• Class number: $1$ (GRH)
• Class group: Trivial (GRH)
• Galois group: $C_{42}$ (as 42T1)

Normalized defining polynomial

\( x^{42} - 42 x^{40} + 819 x^{38} - 9842 x^{36} - x^{35} + 81585 x^{34} + 35 x^{33} - 494802 x^{32} - 560 x^{31} + 2272424 x^{30} + 5425 x^{29} - 8069424 x^{28} - 35525 x^{27} + 22428252 x^{26} + 166257 x^{25} - 49085400 x^{24} - 573300 x^{23} + 84672315 x^{22} + 1480051 x^{21} - 114717330 x^{20} - 2877896 x^{19} + 121090515 x^{18} + 4206314 x^{17} - 98285670 x^{16} - 4577216 x^{15} + 60174899 x^{14} + 3643150 x^{13} - 27041546 x^{12} - 2063243 x^{11} + 8580418 x^{10} + 798357 x^{9} - 1816836 x^{8} - 198948 x^{7} + 235249 x^{6} + 29211 x^{5} - 15974 x^{4} - 2170 x^{3} + 392 x^{2} + 56 x + 1 \)

Invariants

Degree:		$42$
Signature:		$[42, 0]$
Discriminant:		\(1236219045653317330764639083558080790009887216550178193020957667462329030021=3^{21}\cdot 7^{77}\)
Root discriminant:		$61.36$
Ramified primes:		$3, 7$
This field is Galois and abelian over $\Q$.
Conductor:		\(147=3\cdot 7^{2}\)
Dirichlet character group:		$\lbrace$$\chi_{147}(1,·)$, $\chi_{147}(130,·)$, $\chi_{147}(131,·)$, $\chi_{147}(4,·)$, $\chi_{147}(5,·)$, $\chi_{147}(142,·)$, $\chi_{147}(143,·)$, $\chi_{147}(16,·)$, $\chi_{147}(17,·)$, $\chi_{147}(146,·)$, $\chi_{147}(20,·)$, $\chi_{147}(22,·)$, $\chi_{147}(25,·)$, $\chi_{147}(26,·)$, $\chi_{147}(37,·)$, $\chi_{147}(38,·)$, $\chi_{147}(41,·)$, $\chi_{147}(43,·)$, $\chi_{147}(46,·)$, $\chi_{147}(47,·)$, $\chi_{147}(58,·)$, $\chi_{147}(59,·)$, $\chi_{147}(62,·)$, $\chi_{147}(64,·)$, $\chi_{147}(67,·)$, $\chi_{147}(68,·)$, $\chi_{147}(79,·)$, $\chi_{147}(80,·)$, $\chi_{147}(83,·)$, $\chi_{147}(85,·)$, $\chi_{147}(88,·)$, $\chi_{147}(89,·)$, $\chi_{147}(100,·)$, $\chi_{147}(101,·)$, $\chi_{147}(104,·)$, $\chi_{147}(106,·)$, $\chi_{147}(109,·)$, $\chi_{147}(110,·)$, $\chi_{147}(121,·)$, $\chi_{147}(122,·)$, $\chi_{147}(125,·)$, $\chi_{147}(127,·)$$\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:		\( 1966641158581684500000000 \) (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

\(\Q(\sqrt{21}) \), \(\Q(\zeta_{7})^+\), \(\Q(\zeta_{21})^+\), 7.7.13841287201.1, 14.14.2932917071205091238064909.1, \(\Q(\zeta_{49})^+\)

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	$42$	R	$21^{2}$	R	$42$	${\href{/LocalNumberField/13.14.0.1}{14} }^{3}$	$21^{2}$	${\href{/LocalNumberField/19.6.0.1}{6} }^{7}$	$42$	${\href{/LocalNumberField/29.14.0.1}{14} }^{3}$	${\href{/LocalNumberField/31.6.0.1}{6} }^{7}$	$21^{2}$	${\href{/LocalNumberField/41.7.0.1}{7} }^{6}$	${\href{/LocalNumberField/43.7.0.1}{7} }^{6}$	$21^{2}$	$42$	$21^{2}$

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
7	Data not computed