Global number field 23.23.140063703503689367173618364344202364099995564521.1

• Label: 23.23.1400637035...4521.1
• Degree: $23$
• Signature: $[23, 0]$
• Discriminant: $139^{22}$
• Root discriminant: $112.16$
• Ramified prime: $139$
• Class number: $1$ (GRH)
• Class group: Trivial (GRH)
• Galois group: $C_{23}$ (as 23T1)

Normalized defining polynomial

\( x^{23} - x^{22} - 66 x^{21} + 147 x^{20} + 1630 x^{19} - 5648 x^{18} - 16457 x^{17} + 92686 x^{16} + 18441 x^{15} - 709360 x^{14} + 832638 x^{13} + 2239299 x^{12} - 5679764 x^{11} - 156443 x^{10} + 12673530 x^{9} - 11318727 x^{8} - 6468097 x^{7} + 14166332 x^{6} - 3186420 x^{5} - 5386949 x^{4} + 2918745 x^{3} + 436718 x^{2} - 516219 x + 63941 \)

Invariants

Degree:		$23$
Signature:		$[23, 0]$
Discriminant:		\(140063703503689367173618364344202364099995564521=139^{22}\)
Root discriminant:		$112.16$
Ramified primes:		$139$
This field is Galois and abelian over $\Q$.
Conductor:		\(139\)
Dirichlet character group:		$\lbrace$$\chi_{139}(64,·)$, $\chi_{139}(1,·)$, $\chi_{139}(131,·)$, $\chi_{139}(6,·)$, $\chi_{139}(129,·)$, $\chi_{139}(77,·)$, $\chi_{139}(79,·)$, $\chi_{139}(80,·)$, $\chi_{139}(100,·)$, $\chi_{139}(91,·)$, $\chi_{139}(116,·)$, $\chi_{139}(34,·)$, $\chi_{139}(36,·)$, $\chi_{139}(65,·)$, $\chi_{139}(106,·)$, $\chi_{139}(44,·)$, $\chi_{139}(45,·)$, $\chi_{139}(112,·)$, $\chi_{139}(52,·)$, $\chi_{139}(55,·)$, $\chi_{139}(57,·)$, $\chi_{139}(125,·)$, $\chi_{139}(63,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{43} a^{19} - \frac{12}{43} a^{18} + \frac{19}{43} a^{17} - \frac{9}{43} a^{16} + \frac{11}{43} a^{14} - \frac{20}{43} a^{13} + \frac{18}{43} a^{10} + \frac{2}{43} a^{8} + \frac{2}{43} a^{7} + \frac{11}{43} a^{6} + \frac{21}{43} a^{5} - \frac{20}{43} a^{4} + \frac{21}{43} a^{3} - \frac{2}{43} a$, $\frac{1}{43} a^{20} + \frac{4}{43} a^{18} + \frac{4}{43} a^{17} + \frac{21}{43} a^{16} + \frac{11}{43} a^{15} - \frac{17}{43} a^{14} + \frac{18}{43} a^{13} + \frac{18}{43} a^{11} + \frac{1}{43} a^{10} + \frac{2}{43} a^{9} - \frac{17}{43} a^{8} - \frac{8}{43} a^{7} - \frac{19}{43} a^{6} + \frac{17}{43} a^{5} - \frac{4}{43} a^{4} - \frac{6}{43} a^{3} - \frac{2}{43} a^{2} + \frac{19}{43} a$, $\frac{1}{754951} a^{21} - \frac{5533}{754951} a^{20} + \frac{7720}{754951} a^{19} + \frac{366192}{754951} a^{18} + \frac{363702}{754951} a^{17} + \frac{321430}{754951} a^{16} - \frac{74081}{754951} a^{15} - \frac{87043}{754951} a^{14} - \frac{298075}{754951} a^{13} + \frac{72645}{754951} a^{12} - \frac{85747}{754951} a^{11} - \frac{242076}{754951} a^{10} - \frac{312169}{754951} a^{9} + \frac{332913}{754951} a^{8} + \frac{288609}{754951} a^{7} + \frac{242824}{754951} a^{6} + \frac{361274}{754951} a^{5} + \frac{1930}{7783} a^{4} - \frac{318747}{754951} a^{3} + \frac{161757}{754951} a^{2} - \frac{52318}{754951} a + \frac{7827}{17557}$, $\frac{1}{1424880116117105488551658527949} a^{22} + \frac{760445898851806163691671}{1424880116117105488551658527949} a^{21} - \frac{11731147140587931675190947439}{1424880116117105488551658527949} a^{20} + \frac{16517348045468643320241954208}{1424880116117105488551658527949} a^{19} - \frac{392266998589102181978954831679}{1424880116117105488551658527949} a^{18} - \frac{453160600122698737131456859472}{1424880116117105488551658527949} a^{17} - \frac{591812284922908894325999876630}{1424880116117105488551658527949} a^{16} - \frac{304118660702128975433196696289}{1424880116117105488551658527949} a^{15} + \frac{591299700161782929479439974537}{1424880116117105488551658527949} a^{14} - \frac{224213291241531700816956649405}{1424880116117105488551658527949} a^{13} - \frac{703611615285089579928594527900}{1424880116117105488551658527949} a^{12} + \frac{652026345437595812768557348917}{1424880116117105488551658527949} a^{11} - \frac{4471401494433208405902384462}{14689485733166035964450087917} a^{10} - \frac{115845137743880685202933417833}{1424880116117105488551658527949} a^{9} - \frac{142357881441246291245340579092}{1424880116117105488551658527949} a^{8} + \frac{270211501011554981974562726251}{1424880116117105488551658527949} a^{7} - \frac{555497250160671734426512263033}{1424880116117105488551658527949} a^{6} - \frac{79940787894248029944425140770}{1424880116117105488551658527949} a^{5} + \frac{173163381751297406804763935603}{1424880116117105488551658527949} a^{4} - \frac{278515365747601056858491545903}{1424880116117105488551658527949} a^{3} + \frac{602682006965624574094796952891}{1424880116117105488551658527949} a^{2} - \frac{239609006807047678926723835309}{1424880116117105488551658527949} a + \frac{1212150684671470448218646809}{33136746886444313687247872743}$

Class group and class number

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

Unit group

Rank:		$22$
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:		\( 22066241000437346.10236048207 \) (assuming GRH)

Galois group

$C_{23}$ (as 23T1):

A cyclic group of order 23

The 23 conjugacy class representatives for $C_{23}$

Character table for $C_{23}$ is not computed

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Frobenius cycle types

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59
Cycle type	$23$	$23$	$23$	$23$	$23$	$23$	$23$	$23$	$23$	$23$	$23$	$23$	$23$	${\href{/LocalNumberField/43.1.0.1}{1} }^{23}$	$23$	$23$	$23$

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
139	Data not computed