Number field 46.0.2726879904444547310266403727163556673049075149201429809891173702168707134366554981450412880362299.1

• Label: 46.0.272...299.1
• Degree: $46$
• Signature: $[0, 23]$
• Discriminant: $-2.727\times 10^{96}$
• Root discriminant: \(124.86\)
• Ramified prime: $139$
• Class number: not computed
• Class group: not computed
• Galois group: $C_{46}$ (as 46T1)

Normalized defining polynomial

x^46 - x^45 + 2*x^44 + 88*x^43 - 81*x^42 + 155*x^41 + 3247*x^40 - 2737*x^39 + 4990*x^38 + 65894*x^37 - 50535*x^36 + 87297*x^35 + 812047*x^34 - 562441*x^33 + 914794*x^32 + 6340570*x^31 - 3933649*x^30 + 5969903*x^29 + 31840228*x^28 - 17522906*x^27 + 24664884*x^26 + 102351606*x^25 - 49045994*x^24 + 67898048*x^23 + 203991966*x^22 - 79205588*x^21 + 147048808*x^20 + 226793510*x^19 - 44044110*x^18 + 277327722*x^17 + 97270295*x^16 + 52705940*x^15 + 374031026*x^14 - 2906153*x^13 + 88850041*x^12 + 243869890*x^11 + 66181028*x^10 + 138359819*x^9 + 68541629*x^8 + 60480343*x^7 + 77496743*x^6 - 68525839*x^5 + 45005646*x^4 - 25416074*x^3 - 12752729*x^2 + 8659371*x + 2726249

Invariants

Degree:		$46$
Signature:		$[0, 23]$
Discriminant:		-2726879904444547310266403727163556673049075149201429809891173702168707134366554981450412880362299
Root discriminant:		\(124.86\)
Galois root discriminant:		$139^{45/46}\approx 124.86120503438597$
Ramified primes:		\(139\)
Discriminant root field:		\(\Q(\sqrt{-139}) \)
$\card{ \Gal(K/\Q) }$:		$46$
This field is Galois and abelian over $\Q$.
Conductor:		\(139\)
Dirichlet character group:		$\lbrace$$\chi_{139}(1,·)$, $\chi_{139}(131,·)$, $\chi_{139}(133,·)$, $\chi_{139}(6,·)$, $\chi_{139}(129,·)$, $\chi_{139}(8,·)$, $\chi_{139}(10,·)$, $\chi_{139}(14,·)$, $\chi_{139}(23,·)$, $\chi_{139}(27,·)$, $\chi_{139}(33,·)$, $\chi_{139}(34,·)$, $\chi_{139}(36,·)$, $\chi_{139}(39,·)$, $\chi_{139}(44,·)$, $\chi_{139}(45,·)$, $\chi_{139}(48,·)$, $\chi_{139}(52,·)$, $\chi_{139}(55,·)$, $\chi_{139}(57,·)$, $\chi_{139}(59,·)$, $\chi_{139}(60,·)$, $\chi_{139}(138,·)$, $\chi_{139}(62,·)$, $\chi_{139}(63,·)$, $\chi_{139}(64,·)$, $\chi_{139}(65,·)$, $\chi_{139}(74,·)$, $\chi_{139}(75,·)$, $\chi_{139}(76,·)$, $\chi_{139}(77,·)$, $\chi_{139}(79,·)$, $\chi_{139}(80,·)$, $\chi_{139}(82,·)$, $\chi_{139}(84,·)$, $\chi_{139}(87,·)$, $\chi_{139}(91,·)$, $\chi_{139}(94,·)$, $\chi_{139}(95,·)$, $\chi_{139}(100,·)$, $\chi_{139}(103,·)$, $\chi_{139}(105,·)$, $\chi_{139}(106,·)$, $\chi_{139}(112,·)$, $\chi_{139}(116,·)$, $\chi_{139}(125,·)$$\rbrace$
This is a CM field.
Reflex fields:		unavailable$^{4194304}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $a^{40}$, $a^{41}$, $\frac{1}{181}a^{42}-\frac{68}{181}a^{41}-\frac{81}{181}a^{40}-\frac{47}{181}a^{39}-\frac{35}{181}a^{38}-\frac{45}{181}a^{37}+\frac{67}{181}a^{36}-\frac{21}{181}a^{35}-\frac{13}{181}a^{34}+\frac{78}{181}a^{33}-\frac{36}{181}a^{32}-\frac{7}{181}a^{31}-\frac{81}{181}a^{30}-\frac{21}{181}a^{29}-\frac{75}{181}a^{28}-\frac{55}{181}a^{27}-\frac{9}{181}a^{26}-\frac{15}{181}a^{25}-\frac{64}{181}a^{24}+\frac{54}{181}a^{23}-\frac{52}{181}a^{22}+\frac{2}{181}a^{21}+\frac{46}{181}a^{20}+\frac{72}{181}a^{19}+\frac{28}{181}a^{18}+\frac{86}{181}a^{17}-\frac{22}{181}a^{16}-\frac{70}{181}a^{15}+\frac{39}{181}a^{14}+\frac{47}{181}a^{13}+\frac{64}{181}a^{12}+\frac{60}{181}a^{11}-\frac{15}{181}a^{10}+\frac{4}{181}a^{9}-\frac{5}{181}a^{8}-\frac{25}{181}a^{7}-\frac{39}{181}a^{6}+\frac{80}{181}a^{5}+\frac{83}{181}a^{4}-\frac{33}{181}a^{3}+\frac{35}{181}a^{2}+\frac{58}{181}a+\frac{30}{181}$, $\frac{1}{181}a^{43}+\frac{1}{181}a^{41}+\frac{56}{181}a^{40}+\frac{27}{181}a^{39}-\frac{72}{181}a^{38}+\frac{84}{181}a^{37}+\frac{10}{181}a^{36}+\frac{7}{181}a^{35}-\frac{82}{181}a^{34}+\frac{19}{181}a^{33}+\frac{79}{181}a^{32}-\frac{14}{181}a^{31}+\frac{82}{181}a^{30}-\frac{55}{181}a^{29}-\frac{87}{181}a^{28}+\frac{52}{181}a^{27}-\frac{84}{181}a^{26}+\frac{2}{181}a^{25}+\frac{46}{181}a^{24}+\frac{86}{181}a^{22}+\frac{1}{181}a^{21}-\frac{58}{181}a^{20}+\frac{37}{181}a^{19}-\frac{1}{181}a^{18}+\frac{34}{181}a^{17}+\frac{63}{181}a^{16}-\frac{15}{181}a^{15}-\frac{16}{181}a^{14}+\frac{2}{181}a^{13}+\frac{68}{181}a^{12}+\frac{83}{181}a^{11}+\frac{70}{181}a^{10}+\frac{86}{181}a^{9}-\frac{3}{181}a^{8}+\frac{71}{181}a^{7}-\frac{38}{181}a^{6}-\frac{88}{181}a^{5}-\frac{37}{181}a^{3}+\frac{85}{181}a^{2}-\frac{8}{181}a+\frac{49}{181}$, $\frac{1}{193489}a^{44}-\frac{152}{193489}a^{43}-\frac{201}{193489}a^{42}-\frac{2288}{193489}a^{41}-\frac{61084}{193489}a^{40}-\frac{69978}{193489}a^{39}+\frac{4704}{193489}a^{38}+\frac{42306}{193489}a^{37}+\frac{67489}{193489}a^{36}-\frac{63693}{193489}a^{35}+\frac{4068}{193489}a^{34}+\frac{40441}{193489}a^{33}+\frac{52989}{193489}a^{32}+\frac{46521}{193489}a^{31}+\frac{73709}{193489}a^{30}+\frac{28624}{193489}a^{29}-\frac{78726}{193489}a^{28}-\frac{32354}{193489}a^{27}-\frac{26499}{193489}a^{26}-\frac{41935}{193489}a^{25}-\frac{15965}{193489}a^{24}+\frac{84927}{193489}a^{23}+\frac{32366}{193489}a^{22}+\frac{2463}{193489}a^{21}+\frac{48793}{193489}a^{20}+\frac{28882}{193489}a^{19}+\frac{11001}{193489}a^{18}-\frac{42387}{193489}a^{17}-\frac{4604}{193489}a^{16}-\frac{25407}{193489}a^{15}+\frac{55915}{193489}a^{14}+\frac{88010}{193489}a^{13}-\frac{54494}{193489}a^{12}+\frac{54974}{193489}a^{11}+\frac{62342}{193489}a^{10}-\frac{28363}{193489}a^{9}-\frac{31948}{193489}a^{8}-\frac{51573}{193489}a^{7}-\frac{80373}{193489}a^{6}+\frac{88621}{193489}a^{5}+\frac{67724}{193489}a^{4}-\frac{26721}{193489}a^{3}+\frac{4437}{193489}a^{2}+\frac{57786}{193489}a+\frac{14547}{193489}$, $\frac{1}{54\!\cdots\!67}a^{45}+\frac{13\!\cdots\!41}{54\!\cdots\!67}a^{44}+\frac{12\!\cdots\!80}{54\!\cdots\!67}a^{43}+\frac{88\!\cdots\!22}{54\!\cdots\!67}a^{42}-\frac{74\!\cdots\!49}{54\!\cdots\!67}a^{41}-\frac{21\!\cdots\!45}{54\!\cdots\!67}a^{40}-\frac{12\!\cdots\!53}{54\!\cdots\!67}a^{39}-\frac{13\!\cdots\!85}{54\!\cdots\!67}a^{38}-\frac{13\!\cdots\!93}{54\!\cdots\!67}a^{37}-\frac{24\!\cdots\!12}{54\!\cdots\!67}a^{36}-\frac{12\!\cdots\!72}{54\!\cdots\!67}a^{35}+\frac{68\!\cdots\!19}{54\!\cdots\!67}a^{34}-\frac{12\!\cdots\!29}{54\!\cdots\!67}a^{33}-\frac{23\!\cdots\!70}{54\!\cdots\!67}a^{32}-\frac{18\!\cdots\!79}{54\!\cdots\!67}a^{31}-\frac{27\!\cdots\!42}{54\!\cdots\!67}a^{30}-\frac{17\!\cdots\!07}{54\!\cdots\!67}a^{29}-\frac{14\!\cdots\!34}{54\!\cdots\!67}a^{28}-\frac{59\!\cdots\!22}{29\!\cdots\!07}a^{27}+\frac{40\!\cdots\!30}{54\!\cdots\!67}a^{26}-\frac{28\!\cdots\!13}{54\!\cdots\!67}a^{25}-\frac{51\!\cdots\!96}{54\!\cdots\!67}a^{24}+\frac{11\!\cdots\!00}{54\!\cdots\!67}a^{23}-\frac{19\!\cdots\!28}{54\!\cdots\!67}a^{22}+\frac{28\!\cdots\!74}{54\!\cdots\!67}a^{21}-\frac{32\!\cdots\!38}{54\!\cdots\!67}a^{20}+\frac{77\!\cdots\!74}{54\!\cdots\!67}a^{19}+\frac{38\!\cdots\!95}{54\!\cdots\!67}a^{18}+\frac{11\!\cdots\!70}{54\!\cdots\!67}a^{17}-\frac{14\!\cdots\!61}{54\!\cdots\!67}a^{16}-\frac{13\!\cdots\!35}{54\!\cdots\!67}a^{15}-\frac{20\!\cdots\!83}{54\!\cdots\!67}a^{14}-\frac{24\!\cdots\!92}{54\!\cdots\!67}a^{13}-\frac{11\!\cdots\!57}{54\!\cdots\!67}a^{12}-\frac{80\!\cdots\!43}{54\!\cdots\!67}a^{11}-\frac{61\!\cdots\!76}{54\!\cdots\!67}a^{10}+\frac{22\!\cdots\!55}{54\!\cdots\!67}a^{9}-\frac{26\!\cdots\!10}{54\!\cdots\!67}a^{8}-\frac{30\!\cdots\!70}{54\!\cdots\!67}a^{7}-\frac{11\!\cdots\!62}{54\!\cdots\!67}a^{6}+\frac{51\!\cdots\!72}{54\!\cdots\!67}a^{5}+\frac{11\!\cdots\!62}{54\!\cdots\!67}a^{4}-\frac{26\!\cdots\!52}{54\!\cdots\!67}a^{3}-\frac{16\!\cdots\!66}{54\!\cdots\!67}a^{2}-\frac{14\!\cdots\!78}{54\!\cdots\!67}a+\frac{52\!\cdots\!99}{19\!\cdots\!83}$

Monogenic:		Not computed
Index:		$1$
Inessential primes:		None

Class group and class number

not computed

Unit group

Rank:		$22$
Torsion generator:		\( -1 \)  (order $2$)
Fundamental units:		not computed
Regulator:		not computed

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr $ not computed \end{aligned}\]

Galois group

$C_{46}$ (as 46T1):

A cyclic group of order 46

The 46 conjugacy class representatives for $C_{46}$

Character table for $C_{46}$

Intermediate fields

\(\Q(\sqrt{-139}) \), 23.23.140063703503689367173618364344202364099995564521.1

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	$46$	$46$	$23^{2}$	$23^{2}$	$23^{2}$	$23^{2}$	$46$	$46$	$46$	$23^{2}$	$23^{2}$	$23^{2}$	$23^{2}$	${\href{/padicField/43.2.0.1}{2} }^{23}$	$23^{2}$	$46$	$46$

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
\(139\)		Deg $46$	$46$	$1$	$45$