Global number field 36.36.164550840223975716663655069866834081172656515609690871995791535257.1: \(\Q(\zeta_{73})^+\)

• Label: 36.36.1645508402...5257.1
• Degree: $36$
• Signature: $[36, 0]$
• Discriminant: $73^{35}$
• Root discriminant: $64.80$
• Ramified prime: $73$
• Class number: $1$ (GRH)
• Class group: Trivial (GRH)
• Galois group: $C_{36}$ (as 36T1)

Normalized defining polynomial

\( x^{36} - x^{35} - 35 x^{34} + 34 x^{33} + 561 x^{32} - 528 x^{31} - 5456 x^{30} + 4960 x^{29} + 35960 x^{28} - 31465 x^{27} - 169911 x^{26} + 142506 x^{25} + 593775 x^{24} - 475020 x^{23} - 1560780 x^{22} + 1184040 x^{21} + 3108105 x^{20} - 2220075 x^{19} - 4686825 x^{18} + 3124550 x^{17} + 5311735 x^{16} - 3268760 x^{15} - 4457400 x^{14} + 2496144 x^{13} + 2704156 x^{12} - 1352078 x^{11} - 1144066 x^{10} + 497420 x^{9} + 319770 x^{8} - 116280 x^{7} - 54264 x^{6} + 15504 x^{5} + 4845 x^{4} - 969 x^{3} - 171 x^{2} + 18 x + 1 \)

Invariants

Degree:		$36$
Signature:		$[36, 0]$
Discriminant:		\(164550840223975716663655069866834081172656515609690871995791535257=73^{35}\)
Root discriminant:		$64.80$
Ramified primes:		$73$
This field is Galois and abelian over $\Q$.
Conductor:		\(73\)
Dirichlet character group:		$\lbrace$$\chi_{73}(1,·)$, $\chi_{73}(2,·)$, $\chi_{73}(3,·)$, $\chi_{73}(4,·)$, $\chi_{73}(6,·)$, $\chi_{73}(8,·)$, $\chi_{73}(9,·)$, $\chi_{73}(12,·)$, $\chi_{73}(16,·)$, $\chi_{73}(18,·)$, $\chi_{73}(19,·)$, $\chi_{73}(23,·)$, $\chi_{73}(24,·)$, $\chi_{73}(25,·)$, $\chi_{73}(27,·)$, $\chi_{73}(32,·)$, $\chi_{73}(35,·)$, $\chi_{73}(36,·)$, $\chi_{73}(37,·)$, $\chi_{73}(38,·)$, $\chi_{73}(41,·)$, $\chi_{73}(46,·)$, $\chi_{73}(48,·)$, $\chi_{73}(49,·)$, $\chi_{73}(50,·)$, $\chi_{73}(54,·)$, $\chi_{73}(55,·)$, $\chi_{73}(57,·)$, $\chi_{73}(61,·)$, $\chi_{73}(64,·)$, $\chi_{73}(65,·)$, $\chi_{73}(67,·)$, $\chi_{73}(69,·)$, $\chi_{73}(70,·)$, $\chi_{73}(71,·)$, $\chi_{73}(72,·)$$\rbrace$
This is not a CM field.

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

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

Class group and class number

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

Unit group

Rank:		$35$
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:		\( 1217682649213958000000 \) (assuming GRH)

Galois group

$C_{36}$ (as 36T1):

A cyclic group of order 36

The 36 conjugacy class representatives for $C_{36}$

Character table for $C_{36}$ is not computed

Intermediate fields

\(\Q(\sqrt{73}) \), 3.3.5329.1, 4.4.389017.1, 6.6.2073071593.1, 9.9.806460091894081.1, 12.12.313726685568359708377.1, 18.18.47477585226700098686074966922953.1

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	${\href{/LocalNumberField/2.9.0.1}{9} }^{4}$	${\href{/LocalNumberField/3.6.0.1}{6} }^{6}$	$36$	${\href{/LocalNumberField/7.12.0.1}{12} }^{3}$	$36$	$36$	${\href{/LocalNumberField/17.12.0.1}{12} }^{3}$	$18^{2}$	$18^{2}$	$36$	$36$	${\href{/LocalNumberField/37.9.0.1}{9} }^{4}$	${\href{/LocalNumberField/41.9.0.1}{9} }^{4}$	${\href{/LocalNumberField/43.12.0.1}{12} }^{3}$	$36$	$36$	$36$

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
73	Data not computed