Global number field 36.0.2063964752380648518006363619171361060603216996551622656.1: \(\Q(\zeta_{76})\)

• Label: 36.0.20639647523...2656.1
• Degree: $36$
• Signature: $[0, 18]$
• Discriminant: $2^{36}\cdot 19^{34}$
• Root discriminant: $32.27$
• Ramified primes: $2, 19$
• Class number: $19$ (GRH)
• Class group: $[19]$ (GRH)
• Galois group: $C_2\times C_{18}$ (as 36T2)

Normalized defining polynomial

\( x^{36} - x^{34} + x^{32} - x^{30} + x^{28} - x^{26} + x^{24} - x^{22} + x^{20} - x^{18} + x^{16} - x^{14} + x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1 \)

Invariants

Degree:		$36$
Signature:		$[0, 18]$
Discriminant:		\(2063964752380648518006363619171361060603216996551622656=2^{36}\cdot 19^{34}\)
Root discriminant:		$32.27$
Ramified primes:		$2, 19$
This field is Galois and abelian over $\Q$.
Conductor:		\(76=2^{2}\cdot 19\)
Dirichlet character group:		$\lbrace$$\chi_{76}(1,·)$, $\chi_{76}(3,·)$, $\chi_{76}(5,·)$, $\chi_{76}(7,·)$, $\chi_{76}(9,·)$, $\chi_{76}(11,·)$, $\chi_{76}(13,·)$, $\chi_{76}(15,·)$, $\chi_{76}(17,·)$, $\chi_{76}(21,·)$, $\chi_{76}(23,·)$, $\chi_{76}(25,·)$, $\chi_{76}(27,·)$, $\chi_{76}(29,·)$, $\chi_{76}(31,·)$, $\chi_{76}(33,·)$, $\chi_{76}(35,·)$, $\chi_{76}(37,·)$, $\chi_{76}(39,·)$, $\chi_{76}(41,·)$, $\chi_{76}(43,·)$, $\chi_{76}(45,·)$, $\chi_{76}(47,·)$, $\chi_{76}(49,·)$, $\chi_{76}(51,·)$, $\chi_{76}(53,·)$, $\chi_{76}(55,·)$, $\chi_{76}(59,·)$, $\chi_{76}(61,·)$, $\chi_{76}(63,·)$, $\chi_{76}(65,·)$, $\chi_{76}(67,·)$, $\chi_{76}(69,·)$, $\chi_{76}(71,·)$, $\chi_{76}(73,·)$, $\chi_{76}(75,·)$$\rbrace$
This is a CM field.

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

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

Class group and class number

$C_{19}$, which has order $19$ (assuming GRH)

Unit group

Rank:		$17$
Torsion generator:		\( a \) (order $76$)
Fundamental units:		\( a^{4} + 1 \),  \( a^{8} + 1 \),  \( a^{28} - a^{14} + a^{4} \),  \( a^{8} + a^{4} + 1 \),  \( a^{22} + a^{6} \),  \( a^{28} - a^{18} + 1 \),  \( a^{22} - a^{20} + a^{2} - 1 \),  \( a^{18} - a^{16} \),  \( a - 1 \),  \( a^{3} - 1 \),  \( a^{9} - 1 \),  \( a^{11} - 1 \),  \( a^{5} - 1 \),  \( a^{31} + a^{16} \),  \( a^{7} - 1 \),  \( a^{13} - 1 \),  \( a^{17} - 1 \) (assuming GRH)
Regulator:		\( 2595333985839.583 \) (assuming GRH)

Galois group

$C_2\times C_{18}$ (as 36T2):

An abelian group of order 36

The 36 conjugacy class representatives for $C_2\times C_{18}$

Character table for $C_2\times C_{18}$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-19}) \), \(\Q(\sqrt{19}) \), 3.3.361.1, \(\Q(i, \sqrt{19})\), 6.0.8340544.1, 6.0.2476099.1, 6.6.158470336.1, \(\Q(\zeta_{19})^+\), 12.0.25112847391952896.1, 18.0.75613185918270483380568064.1, \(\Q(\zeta_{19})\), \(\Q(\zeta_{76})^+\)

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	R	$18^{2}$	${\href{/LocalNumberField/5.9.0.1}{9} }^{4}$	${\href{/LocalNumberField/7.6.0.1}{6} }^{6}$	${\href{/LocalNumberField/11.6.0.1}{6} }^{6}$	$18^{2}$	${\href{/LocalNumberField/17.9.0.1}{9} }^{4}$	R	$18^{2}$	$18^{2}$	${\href{/LocalNumberField/31.6.0.1}{6} }^{6}$	${\href{/LocalNumberField/37.2.0.1}{2} }^{18}$	$18^{2}$	$18^{2}$	$18^{2}$	$18^{2}$	$18^{2}$

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
2	Data not computed
19	Data not computed