Number field 46.0.31332543366972618236508698754187134069701602945566219926588283810613916634762444800000000000000000000000.1

• Label: 46.0.313...000.1
• Degree: $46$
• Signature: $[0, 23]$
• Discriminant: $-3.133\times 10^{103}$
• Root discriminant: \(177.79\)
• Ramified primes: $2,5,47$
• Class number: not computed
• Class group: not computed
• Galois group: $C_{46}$ (as 46T1)

Normalized defining polynomial

x^46 - 2*x^45 + 72*x^44 - 134*x^43 + 2897*x^42 - 5084*x^41 + 82681*x^40 - 137302*x^39 + 1842969*x^38 - 2901188*x^37 + 33715618*x^36 - 50329496*x^35 + 521486533*x^34 - 737794426*x^33 + 6950305674*x^32 - 9307575658*x^31 + 80830810890*x^30 - 102256954530*x^29 + 827118937400*x^28 - 985841461150*x^27 + 7485767552466*x^26 - 8377390757582*x^25 + 60086694646762*x^24 - 62865657320214*x^23 + 428031087265258*x^22 - 416431202055206*x^21 + 2702388396162372*x^20 - 2428526763878674*x^19 + 15070583707733498*x^18 - 12405066876040346*x^17 + 73818176925261607*x^16 - 55061310307265284*x^15 + 314907625765820988*x^14 - 209899452928866436*x^13 + 1155981973200414012*x^12 - 675750258479061684*x^11 + 3589387159241937348*x^10 - 1792691375628199716*x^9 + 9196110333365905782*x^8 - 3775417247902685064*x^7 + 18723655041560322848*x^6 - 5938123951190841976*x^5 + 28494792795353584212*x^4 - 6222180685713906064*x^3 + 28908055782216940872*x^2 - 3269673707415701984*x + 14713531683370658881

Invariants

Degree:		$46$
Signature:		$[0, 23]$
Discriminant:		-31332543366972618236508698754187134069701602945566219926588283810613916634762444800000000000000000000000
Root discriminant:		\(177.79\)
Galois root discriminant:		$2\cdot 5^{1/2}47^{22/23}\approx 177.79230028769678$
Ramified primes:		\(2\), \(5\), \(47\)
Discriminant root field:		\(\Q(\sqrt{-5}) \)
$\card{ \Gal(K/\Q) }$:		$46$
This field is Galois and abelian over $\Q$.
Conductor:		\(940=2^{2}\cdot 5\cdot 47\)
Dirichlet character group:		$\lbrace$$\chi_{940}(1,·)$, $\chi_{940}(259,·)$, $\chi_{940}(901,·)$, $\chi_{940}(519,·)$, $\chi_{940}(521,·)$, $\chi_{940}(779,·)$, $\chi_{940}(401,·)$, $\chi_{940}(899,·)$, $\chi_{940}(661,·)$, $\chi_{940}(921,·)$, $\chi_{940}(541,·)$, $\chi_{940}(159,·)$, $\chi_{940}(801,·)$, $\chi_{940}(679,·)$, $\chi_{940}(299,·)$, $\chi_{940}(639,·)$, $\chi_{940}(559,·)$, $\chi_{940}(439,·)$, $\chi_{940}(441,·)$, $\chi_{940}(59,·)$, $\chi_{940}(61,·)$, $\chi_{940}(319,·)$, $\chi_{940}(581,·)$, $\chi_{940}(841,·)$, $\chi_{940}(459,·)$, $\chi_{940}(79,·)$, $\chi_{940}(81,·)$, $\chi_{940}(341,·)$, $\chi_{940}(121,·)$, $\chi_{940}(601,·)$, $\chi_{940}(719,·)$, $\chi_{940}(101,·)$, $\chi_{940}(479,·)$, $\chi_{940}(739,·)$, $\chi_{940}(741,·)$, $\chi_{940}(721,·)$, $\chi_{940}(379,·)$, $\chi_{940}(361,·)$, $\chi_{940}(619,·)$, $\chi_{940}(239,·)$, $\chi_{940}(241,·)$, $\chi_{940}(659,·)$, $\chi_{940}(759,·)$, $\chi_{940}(761,·)$, $\chi_{940}(119,·)$, $\chi_{940}(21,·)$$\rbrace$
This is a CM field.
Reflex fields:		unavailable$^{4194304}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $a^{40}$, $a^{41}$, $a^{42}$, $a^{43}$, $\frac{1}{79523}a^{44}+\frac{10961}{79523}a^{43}-\frac{29855}{79523}a^{42}+\frac{3226}{79523}a^{41}+\frac{2478}{79523}a^{40}+\frac{31640}{79523}a^{39}+\frac{38124}{79523}a^{38}+\frac{5898}{79523}a^{37}+\frac{37752}{79523}a^{36}+\frac{23175}{79523}a^{35}+\frac{1780}{79523}a^{34}+\frac{7668}{79523}a^{33}+\frac{29019}{79523}a^{32}+\frac{17195}{79523}a^{31}-\frac{10857}{79523}a^{30}+\frac{18763}{79523}a^{29}+\frac{22279}{79523}a^{28}-\frac{17057}{79523}a^{27}-\frac{18893}{79523}a^{26}+\frac{17299}{79523}a^{25}+\frac{5294}{79523}a^{24}+\frac{6139}{79523}a^{23}-\frac{21231}{79523}a^{22}-\frac{14114}{79523}a^{21}+\frac{25821}{79523}a^{20}+\frac{11604}{79523}a^{19}-\frac{35238}{79523}a^{18}+\frac{27579}{79523}a^{17}+\frac{14415}{79523}a^{16}+\frac{13124}{79523}a^{15}-\frac{26994}{79523}a^{14}+\frac{18031}{79523}a^{13}-\frac{19859}{79523}a^{12}+\frac{35489}{79523}a^{11}-\frac{10860}{79523}a^{10}+\frac{35098}{79523}a^{9}-\frac{2363}{79523}a^{8}+\frac{24232}{79523}a^{7}+\frac{27635}{79523}a^{6}-\frac{18521}{79523}a^{5}-\frac{26429}{79523}a^{4}-\frac{27518}{79523}a^{3}-\frac{20801}{79523}a^{2}-\frac{22474}{79523}a+\frac{11979}{79523}$, $\frac{1}{10\!\cdots\!83}a^{45}-\frac{40\!\cdots\!40}{10\!\cdots\!83}a^{44}+\frac{45\!\cdots\!65}{10\!\cdots\!83}a^{43}+\frac{67\!\cdots\!67}{10\!\cdots\!83}a^{42}+\frac{49\!\cdots\!81}{10\!\cdots\!83}a^{41}-\frac{45\!\cdots\!76}{10\!\cdots\!83}a^{40}+\frac{22\!\cdots\!34}{10\!\cdots\!83}a^{39}-\frac{45\!\cdots\!01}{10\!\cdots\!83}a^{38}-\frac{40\!\cdots\!39}{10\!\cdots\!83}a^{37}+\frac{67\!\cdots\!18}{10\!\cdots\!83}a^{36}-\frac{69\!\cdots\!42}{10\!\cdots\!83}a^{35}-\frac{33\!\cdots\!98}{10\!\cdots\!83}a^{34}-\frac{45\!\cdots\!84}{10\!\cdots\!83}a^{33}+\frac{20\!\cdots\!02}{10\!\cdots\!83}a^{32}-\frac{33\!\cdots\!78}{10\!\cdots\!83}a^{31}-\frac{16\!\cdots\!03}{10\!\cdots\!83}a^{30}-\frac{29\!\cdots\!18}{10\!\cdots\!83}a^{29}+\frac{46\!\cdots\!56}{10\!\cdots\!83}a^{28}-\frac{66\!\cdots\!42}{10\!\cdots\!83}a^{27}-\frac{73\!\cdots\!63}{10\!\cdots\!83}a^{26}+\frac{16\!\cdots\!16}{10\!\cdots\!83}a^{25}-\frac{41\!\cdots\!38}{10\!\cdots\!83}a^{24}-\frac{50\!\cdots\!41}{10\!\cdots\!83}a^{23}-\frac{45\!\cdots\!04}{10\!\cdots\!83}a^{22}-\frac{12\!\cdots\!01}{10\!\cdots\!83}a^{21}+\frac{34\!\cdots\!40}{10\!\cdots\!83}a^{20}+\frac{32\!\cdots\!55}{10\!\cdots\!83}a^{19}-\frac{43\!\cdots\!85}{10\!\cdots\!83}a^{18}+\frac{92\!\cdots\!11}{10\!\cdots\!83}a^{17}-\frac{40\!\cdots\!86}{10\!\cdots\!83}a^{16}+\frac{51\!\cdots\!32}{10\!\cdots\!83}a^{15}-\frac{50\!\cdots\!60}{10\!\cdots\!83}a^{14}-\frac{35\!\cdots\!76}{37\!\cdots\!43}a^{13}+\frac{12\!\cdots\!45}{10\!\cdots\!83}a^{12}+\frac{44\!\cdots\!52}{10\!\cdots\!83}a^{11}+\frac{30\!\cdots\!82}{37\!\cdots\!01}a^{10}-\frac{39\!\cdots\!56}{10\!\cdots\!83}a^{9}-\frac{82\!\cdots\!74}{10\!\cdots\!83}a^{8}+\frac{26\!\cdots\!98}{10\!\cdots\!83}a^{7}+\frac{10\!\cdots\!82}{10\!\cdots\!83}a^{6}+\frac{29\!\cdots\!42}{10\!\cdots\!83}a^{5}-\frac{38\!\cdots\!39}{10\!\cdots\!83}a^{4}-\frac{43\!\cdots\!41}{10\!\cdots\!83}a^{3}+\frac{16\!\cdots\!69}{10\!\cdots\!83}a^{2}+\frac{65\!\cdots\!37}{10\!\cdots\!83}a+\frac{40\!\cdots\!10}{18\!\cdots\!41}$

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 $C_{46}$ is not computed

Intermediate fields

\(\Q(\sqrt{-5}) \), \(\Q(\zeta_{47})^+\)

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	R	$23^{2}$	R	$23^{2}$	$46$	$46$	$46$	$46$	$23^{2}$	$23^{2}$	$46$	$46$	$23^{2}$	$23^{2}$	R	$46$	$46$

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
\(2\)		Deg $46$	$2$	$23$	$46$
\(5\)		Deg $46$	$2$	$23$	$23$
\(47\)	47.23.22.1	$x^{23} + 47$	$23$	$1$	$22$	$C_{23}$	$[\ ]_{23}$
	47.23.22.1	$x^{23} + 47$	$23$	$1$	$22$	$C_{23}$	$[\ ]_{23}$