Global number field 18.0.144968523435705866934593522684771083.1

• Label: 18.0.14496852343...1083.1
• Degree: $18$
• Signature: $[0, 9]$
• Discriminant: $-\,19^{16}\cdot 43^{9}$
• Root discriminant: $89.83$
• Ramified primes: $19, 43$
• Class number: $228253$ (GRH)
• Class group: $[228253]$ (GRH)
• Galois group: $C_{18}$ (as 18T1)

Normalized defining polynomial

\( x^{18} - 7 x^{17} + 103 x^{16} - 546 x^{15} + 4715 x^{14} - 20109 x^{13} + 128861 x^{12} - 451187 x^{11} + 2334770 x^{10} - 6708098 x^{9} + 29183672 x^{8} - 67479711 x^{7} + 252196324 x^{6} - 448042532 x^{5} + 1455886589 x^{4} - 1795886216 x^{3} + 5108474617 x^{2} - 3332816606 x + 8337560089 \)

Invariants

Degree:		$18$
Signature:		$[0, 9]$
Discriminant:		\(-144968523435705866934593522684771083=-\,19^{16}\cdot 43^{9}\)
Root discriminant:		$89.83$
Ramified primes:		$19, 43$
This field is Galois and abelian over $\Q$.
Conductor:		\(817=19\cdot 43\)
Dirichlet character group:		$\lbrace$$\chi_{817}(1,·)$, $\chi_{817}(130,·)$, $\chi_{817}(644,·)$, $\chi_{817}(517,·)$, $\chi_{817}(343,·)$, $\chi_{817}(386,·)$, $\chi_{817}(216,·)$, $\chi_{817}(558,·)$, $\chi_{817}(214,·)$, $\chi_{817}(87,·)$, $\chi_{817}(472,·)$, $\chi_{817}(42,·)$, $\chi_{817}(44,·)$, $\chi_{817}(429,·)$, $\chi_{817}(302,·)$, $\chi_{817}(560,·)$, $\chi_{817}(689,·)$, $\chi_{817}(85,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{1618253869516156476561948521682864225382107503223044525494007} a^{17} - \frac{64689984925412840844633111252082028338558407513713608650818}{1618253869516156476561948521682864225382107503223044525494007} a^{16} - \frac{472009836497952718848859418984239082707410612303350750342154}{1618253869516156476561948521682864225382107503223044525494007} a^{15} + \frac{741183413497472242841584718357693520078150582607493862857505}{1618253869516156476561948521682864225382107503223044525494007} a^{14} - \frac{317163385944709097486606051658141879231687254546258849474617}{1618253869516156476561948521682864225382107503223044525494007} a^{13} + \frac{556075055478824657192074735922048746024460434208367442519121}{1618253869516156476561948521682864225382107503223044525494007} a^{12} - \frac{556799253243380780296298705373693105649646338407711779597784}{1618253869516156476561948521682864225382107503223044525494007} a^{11} + \frac{343295047102995384555266164144864052483343082720578941101204}{1618253869516156476561948521682864225382107503223044525494007} a^{10} - \frac{634412412936372089352380978375419616199245791386073773561133}{1618253869516156476561948521682864225382107503223044525494007} a^{9} - \frac{549399958919221634222042510790861927677090681058756036384545}{1618253869516156476561948521682864225382107503223044525494007} a^{8} - \frac{575381015467493060624325763787975406168169474205284404354120}{1618253869516156476561948521682864225382107503223044525494007} a^{7} - \frac{30759813490010939914895280204606628263485226879246345914308}{1618253869516156476561948521682864225382107503223044525494007} a^{6} + \frac{641832169536837059278611253167580679308344215783733085919564}{1618253869516156476561948521682864225382107503223044525494007} a^{5} - \frac{132877406557510596807682353763874745205892467787014148710901}{1618253869516156476561948521682864225382107503223044525494007} a^{4} + \frac{340670943646344040360652888590662042852621743265886380231791}{1618253869516156476561948521682864225382107503223044525494007} a^{3} + \frac{643045451093722839362884968038986140669676991535174683110063}{1618253869516156476561948521682864225382107503223044525494007} a^{2} + \frac{108677305155246624520593265696408669094918239646883880875750}{1618253869516156476561948521682864225382107503223044525494007} a - \frac{627203127413407545883763954630939798982057884860588102363546}{1618253869516156476561948521682864225382107503223044525494007}$

Class group and class number

$C_{228253}$, which has order $228253$ (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:		\( 22305.8950792 \) (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

\(\Q(\sqrt{-43}) \), 3.3.361.1, 6.0.10361431747.1, \(\Q(\zeta_{19})^+\)

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	$18$	$18$	$18$	${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$	${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$	${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$	${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$	R	${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$	$18$	${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$	${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$	${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$	R	${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$	${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$	${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
19	Data not computed
43	Data not computed