Global number field 18.0.75855608471185389708554302866739.2

• Label: 18.0.75855608471...6739.2
• Degree: $18$
• Signature: $[0, 9]$
• Discriminant: $-\,7^{12}\cdot 19^{17}$
• Root discriminant: $59.04$
• Ramified primes: $7, 19$
• Class number: $81$ (GRH)
• Class group: $[3, 3, 3, 3]$ (GRH)
• Galois group: $C_{18}$ (as 18T1)

Normalized defining polynomial

\( x^{18} - x^{17} + x^{16} - 39 x^{15} + 39 x^{14} - 39 x^{13} + 476 x^{12} - 818 x^{11} + 5739 x^{10} - 10185 x^{9} + 13605 x^{8} - 52973 x^{7} + 10679 x^{6} - 11572 x^{5} + 80200 x^{4} + 256689 x^{3} + 421421 x^{2} + 316197 x + 116851 \)

Invariants

Degree:		$18$
Signature:		$[0, 9]$
Discriminant:		\(-75855608471185389708554302866739=-\,7^{12}\cdot 19^{17}\)
Root discriminant:		$59.04$
Ramified primes:		$7, 19$
This field is Galois and abelian over $\Q$.
Conductor:		\(133=7\cdot 19\)
Dirichlet character group:		$\lbrace$$\chi_{133}(64,·)$, $\chi_{133}(1,·)$, $\chi_{133}(130,·)$, $\chi_{133}(8,·)$, $\chi_{133}(9,·)$, $\chi_{133}(74,·)$, $\chi_{133}(81,·)$, $\chi_{133}(86,·)$, $\chi_{133}(23,·)$, $\chi_{133}(106,·)$, $\chi_{133}(44,·)$, $\chi_{133}(109,·)$, $\chi_{133}(113,·)$, $\chi_{133}(50,·)$, $\chi_{133}(51,·)$, $\chi_{133}(116,·)$, $\chi_{133}(72,·)$, $\chi_{133}(60,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{7} a^{15} + \frac{3}{7} a^{14} + \frac{3}{7} a^{13} - \frac{2}{7} a^{12} - \frac{2}{7} a^{11} - \frac{2}{7} a^{10} + \frac{2}{7} a^{8} + \frac{2}{7} a^{7} + \frac{2}{7} a^{6} - \frac{3}{7} a^{5} - \frac{3}{7} a^{4} - \frac{1}{7} a^{3}$, $\frac{1}{7763} a^{16} + \frac{214}{7763} a^{15} - \frac{687}{7763} a^{14} - \frac{2638}{7763} a^{13} - \frac{3385}{7763} a^{12} + \frac{185}{7763} a^{11} - \frac{3803}{7763} a^{10} - \frac{2259}{7763} a^{9} - \frac{297}{7763} a^{8} + \frac{2965}{7763} a^{7} + \frac{3576}{7763} a^{6} + \frac{2969}{7763} a^{5} + \frac{2530}{7763} a^{4} - \frac{2451}{7763} a^{3} + \frac{378}{1109} a^{2} + \frac{261}{1109} a - \frac{18}{1109}$, $\frac{1}{1006471927455797471152731670797337761578552639} a^{17} + \frac{25147584665119662691274443508255158985110}{1006471927455797471152731670797337761578552639} a^{16} - \frac{47058711495313395615826652757440301359341831}{1006471927455797471152731670797337761578552639} a^{15} - \frac{304988773523617002539972625444490447152761576}{1006471927455797471152731670797337761578552639} a^{14} + \frac{165940421262753828794737398779234497286652753}{1006471927455797471152731670797337761578552639} a^{13} - \frac{161804524106540399209107742394334102513790703}{1006471927455797471152731670797337761578552639} a^{12} - \frac{50758637040752966505194359971931342079225807}{143781703922256781593247381542476823082650377} a^{11} - \frac{206581344077976027651371496353862788745067040}{1006471927455797471152731670797337761578552639} a^{10} - \frac{27108251771115371337368381406476486619900551}{1006471927455797471152731670797337761578552639} a^{9} + \frac{376244712728795691511873265398267476575776078}{1006471927455797471152731670797337761578552639} a^{8} - \frac{73566479163088264792774123257229822659350658}{1006471927455797471152731670797337761578552639} a^{7} - \frac{23654871270223996343881748193382698171506306}{1006471927455797471152731670797337761578552639} a^{6} - \frac{93773022326736400731107924015614477931541360}{1006471927455797471152731670797337761578552639} a^{5} - \frac{45568910712456393152301453514622610903365277}{143781703922256781593247381542476823082650377} a^{4} - \frac{49434874096517183223346613061828487474196356}{143781703922256781593247381542476823082650377} a^{3} - \frac{14647651863820053196395938724712422466004695}{143781703922256781593247381542476823082650377} a^{2} - \frac{60733681959399806165204200212885510985737178}{143781703922256781593247381542476823082650377} a + \frac{61365233334818979143861255956650773046612790}{143781703922256781593247381542476823082650377}$

Class group and class number

$C_{3}\times C_{3}\times C_{3}\times C_{3}$, which has order $81$ (assuming GRH)

Unit group

Rank:		$8$
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:		\( 4369063.34801 \) (assuming GRH)

Galois group

$C_{18}$ (as 18T1):

A cyclic group of order 18

The 18 conjugacy class representatives for $C_{18}$

Character table for $C_{18}$

Intermediate fields

\(\Q(\sqrt{-19}) \), 3.3.361.1, 6.0.2476099.1, 9.9.1998099208210609.2

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	$18$	$18$	${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$	R	${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$	$18$	${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$	R	${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$	$18$	${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$	${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$	$18$	${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$	${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$	$18$	$18$

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
$7$	7.3.2.1	$x^{3} + 14$	$3$	$1$	$2$	$C_3$	$[\ ]_{3}$
	7.3.2.1	$x^{3} + 14$	$3$	$1$	$2$	$C_3$	$[\ ]_{3}$
	7.3.2.1	$x^{3} + 14$	$3$	$1$	$2$	$C_3$	$[\ ]_{3}$
	7.3.2.1	$x^{3} + 14$	$3$	$1$	$2$	$C_3$	$[\ ]_{3}$
	7.3.2.1	$x^{3} + 14$	$3$	$1$	$2$	$C_3$	$[\ ]_{3}$
	7.3.2.1	$x^{3} + 14$	$3$	$1$	$2$	$C_3$	$[\ ]_{3}$
19	Data not computed