Number field 46.46.1559562745220946355455174068757250546262830593902488581782176430066187304634955035230281728558464357.1

• Label: 46.46.155...357.1
• Degree: $46$
• Signature: $[46, 0]$
• Discriminant: $1.560\times 10^{99}$
• Root discriminant: \(143.34\)
• Ramified primes: $13,47$
• Class number: not computed
• Class group: not computed
• Galois group: $C_{46}$ (as 46T1)

Normalized defining polynomial

x^46 - 21*x^45 + 96*x^44 + 959*x^43 - 9950*x^42 - 1825*x^41 + 335613*x^40 - 834537*x^39 - 5657221*x^38 + 26988590*x^37 + 44109968*x^36 - 443216122*x^35 + 100002271*x^34 + 4479212294*x^33 - 6237187098*x^32 - 28571535040*x^31 + 72586582734*x^30 + 103336700236*x^29 - 480319603281*x^28 - 66786211526*x^27 + 2018290007358*x^26 - 1355370040843*x^25 - 5285056712013*x^24 + 7582856629140*x^23 + 7178933729591*x^22 - 20645533933596*x^21 + 1046437412802*x^20 + 31269803408789*x^19 - 21465428316192*x^18 - 22251983955427*x^17 + 34519826963183*x^16 - 2181540751435*x^15 - 22787259295206*x^14 + 14052337420836*x^13 + 3424903604271*x^12 - 7356504751997*x^11 + 2524580197417*x^10 + 719437437556*x^9 - 822188957900*x^8 + 216958911989*x^7 + 16546070228*x^6 - 23274902617*x^5 + 5444167373*x^4 - 387991071*x^3 - 44784143*x^2 + 9031748*x - 400721

Invariants

Degree:		$46$
Signature:		$[46, 0]$
Discriminant:		1559562745220946355455174068757250546262830593902488581782176430066187304634955035230281728558464357
Root discriminant:		\(143.34\)
Galois root discriminant:		$13^{1/2}47^{22/23}\approx 143.3407350582306$
Ramified primes:		\(13\), \(47\)
Discriminant root field:		\(\Q(\sqrt{13}) \)
$\card{ \Gal(K/\Q) }$:		$46$
This field is Galois and abelian over $\Q$.
Conductor:		\(611=13\cdot 47\)
Dirichlet character group:		$\lbrace$$\chi_{611}(1,·)$, $\chi_{611}(259,·)$, $\chi_{611}(519,·)$, $\chi_{611}(324,·)$, $\chi_{611}(521,·)$, $\chi_{611}(12,·)$, $\chi_{611}(14,·)$, $\chi_{611}(272,·)$, $\chi_{611}(131,·)$, $\chi_{611}(404,·)$, $\chi_{611}(534,·)$, $\chi_{611}(25,·)$, $\chi_{611}(27,·)$, $\chi_{611}(285,·)$, $\chi_{611}(545,·)$, $\chi_{611}(155,·)$, $\chi_{611}(168,·)$, $\chi_{611}(363,·)$, $\chi_{611}(298,·)$, $\chi_{611}(300,·)$, $\chi_{611}(430,·)$, $\chi_{611}(157,·)$, $\chi_{611}(51,·)$, $\chi_{611}(53,·)$, $\chi_{611}(183,·)$, $\chi_{611}(441,·)$, $\chi_{611}(571,·)$, $\chi_{611}(573,·)$, $\chi_{611}(64,·)$, $\chi_{611}(194,·)$, $\chi_{611}(196,·)$, $\chi_{611}(79,·)$, $\chi_{611}(337,·)$, $\chi_{611}(378,·)$, $\chi_{611}(142,·)$, $\chi_{611}(220,·)$, $\chi_{611}(222,·)$, $\chi_{611}(350,·)$, $\chi_{611}(144,·)$, $\chi_{611}(482,·)$, $\chi_{611}(209,·)$, $\chi_{611}(103,·)$, $\chi_{611}(365,·)$, $\chi_{611}(495,·)$, $\chi_{611}(118,·)$, $\chi_{611}(506,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $a^{40}$, $a^{41}$, $a^{42}$, $\frac{1}{283}a^{43}-\frac{68}{283}a^{42}+\frac{22}{283}a^{41}+\frac{111}{283}a^{40}-\frac{66}{283}a^{39}+\frac{129}{283}a^{38}-\frac{99}{283}a^{37}+\frac{117}{283}a^{36}-\frac{99}{283}a^{35}+\frac{63}{283}a^{34}+\frac{61}{283}a^{33}-\frac{40}{283}a^{32}+\frac{138}{283}a^{31}+\frac{129}{283}a^{30}+\frac{91}{283}a^{29}+\frac{80}{283}a^{28}+\frac{59}{283}a^{27}+\frac{104}{283}a^{26}+\frac{58}{283}a^{25}+\frac{115}{283}a^{24}+\frac{55}{283}a^{23}-\frac{104}{283}a^{22}+\frac{129}{283}a^{21}+\frac{85}{283}a^{20}+\frac{91}{283}a^{19}-\frac{118}{283}a^{18}-\frac{32}{283}a^{17}-\frac{79}{283}a^{16}-\frac{24}{283}a^{15}-\frac{128}{283}a^{14}-\frac{19}{283}a^{13}+\frac{114}{283}a^{12}-\frac{71}{283}a^{11}-\frac{15}{283}a^{10}-\frac{139}{283}a^{9}+\frac{6}{283}a^{8}-\frac{37}{283}a^{7}+\frac{84}{283}a^{6}-\frac{46}{283}a^{5}-\frac{141}{283}a^{4}-\frac{128}{283}a^{3}+\frac{120}{283}a^{2}+\frac{25}{283}a-\frac{89}{283}$, $\frac{1}{1595837}a^{44}+\frac{2259}{1595837}a^{43}+\frac{694748}{1595837}a^{42}-\frac{762886}{1595837}a^{41}+\frac{675939}{1595837}a^{40}-\frac{354100}{1595837}a^{39}-\frac{482977}{1595837}a^{38}-\frac{511558}{1595837}a^{37}+\frac{288291}{1595837}a^{36}+\frac{351821}{1595837}a^{35}+\frac{627762}{1595837}a^{34}-\frac{115906}{1595837}a^{33}+\frac{345142}{1595837}a^{32}-\frac{235972}{1595837}a^{31}-\frac{774843}{1595837}a^{30}-\frac{122952}{1595837}a^{29}+\frac{8495}{1595837}a^{28}-\frac{651890}{1595837}a^{27}+\frac{325268}{1595837}a^{26}-\frac{723541}{1595837}a^{25}-\frac{445783}{1595837}a^{24}+\frac{499743}{1595837}a^{23}+\frac{638817}{1595837}a^{22}+\frac{196690}{1595837}a^{21}+\frac{794450}{1595837}a^{20}-\frac{121452}{1595837}a^{19}+\frac{31305}{1595837}a^{18}-\frac{249437}{1595837}a^{17}-\frac{797684}{1595837}a^{16}+\frac{331168}{1595837}a^{15}+\frac{432831}{1595837}a^{14}+\frac{56083}{1595837}a^{13}+\frac{71918}{1595837}a^{12}+\frac{8247}{1595837}a^{11}+\frac{226448}{1595837}a^{10}-\frac{131290}{1595837}a^{9}+\frac{147218}{1595837}a^{8}-\frac{3096}{1595837}a^{7}+\frac{378240}{1595837}a^{6}-\frac{126993}{1595837}a^{5}-\frac{570483}{1595837}a^{4}-\frac{787326}{1595837}a^{3}-\frac{90050}{1595837}a^{2}-\frac{360754}{1595837}a+\frac{151741}{1595837}$, $\frac{1}{19\!\cdots\!51}a^{45}-\frac{36\!\cdots\!81}{19\!\cdots\!51}a^{44}-\frac{21\!\cdots\!58}{19\!\cdots\!51}a^{43}-\frac{30\!\cdots\!01}{19\!\cdots\!51}a^{42}-\frac{88\!\cdots\!83}{19\!\cdots\!51}a^{41}-\frac{36\!\cdots\!33}{19\!\cdots\!51}a^{40}+\frac{29\!\cdots\!83}{19\!\cdots\!51}a^{39}+\frac{63\!\cdots\!83}{19\!\cdots\!51}a^{38}+\frac{41\!\cdots\!88}{19\!\cdots\!51}a^{37}-\frac{30\!\cdots\!36}{19\!\cdots\!51}a^{36}+\frac{48\!\cdots\!16}{19\!\cdots\!51}a^{35}+\frac{69\!\cdots\!96}{19\!\cdots\!51}a^{34}+\frac{84\!\cdots\!10}{19\!\cdots\!51}a^{33}+\frac{39\!\cdots\!70}{19\!\cdots\!51}a^{32}+\frac{30\!\cdots\!67}{19\!\cdots\!51}a^{31}-\frac{62\!\cdots\!67}{19\!\cdots\!51}a^{30}-\frac{35\!\cdots\!81}{19\!\cdots\!51}a^{29}-\frac{58\!\cdots\!73}{19\!\cdots\!51}a^{28}+\frac{61\!\cdots\!90}{19\!\cdots\!51}a^{27}-\frac{86\!\cdots\!68}{19\!\cdots\!51}a^{26}-\frac{22\!\cdots\!86}{19\!\cdots\!51}a^{25}-\frac{82\!\cdots\!50}{19\!\cdots\!51}a^{24}-\frac{37\!\cdots\!09}{19\!\cdots\!51}a^{23}+\frac{53\!\cdots\!15}{19\!\cdots\!51}a^{22}-\frac{61\!\cdots\!19}{19\!\cdots\!51}a^{21}+\frac{68\!\cdots\!08}{19\!\cdots\!51}a^{20}+\frac{90\!\cdots\!19}{19\!\cdots\!51}a^{19}-\frac{36\!\cdots\!70}{19\!\cdots\!51}a^{18}+\frac{50\!\cdots\!89}{19\!\cdots\!51}a^{17}+\frac{78\!\cdots\!34}{19\!\cdots\!51}a^{16}+\frac{29\!\cdots\!65}{19\!\cdots\!51}a^{15}-\frac{54\!\cdots\!89}{19\!\cdots\!51}a^{14}-\frac{48\!\cdots\!21}{19\!\cdots\!51}a^{13}-\frac{55\!\cdots\!07}{19\!\cdots\!51}a^{12}+\frac{68\!\cdots\!44}{19\!\cdots\!51}a^{11}+\frac{36\!\cdots\!76}{19\!\cdots\!51}a^{10}+\frac{36\!\cdots\!97}{19\!\cdots\!51}a^{9}+\frac{46\!\cdots\!54}{19\!\cdots\!51}a^{8}+\frac{35\!\cdots\!15}{19\!\cdots\!51}a^{7}-\frac{81\!\cdots\!79}{19\!\cdots\!51}a^{6}+\frac{84\!\cdots\!37}{19\!\cdots\!51}a^{5}-\frac{16\!\cdots\!08}{19\!\cdots\!51}a^{4}-\frac{82\!\cdots\!61}{19\!\cdots\!51}a^{3}-\frac{54\!\cdots\!46}{19\!\cdots\!51}a^{2}-\frac{40\!\cdots\!10}{19\!\cdots\!51}a-\frac{43\!\cdots\!37}{19\!\cdots\!51}$

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

Class group and class number

not computed

Unit group

Rank:		$45$
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

\(\Q(\sqrt{13}) \), \(\Q(\zeta_{47})^+\)

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

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
\(13\)		Deg $46$	$2$	$23$	$23$
\(47\)		Deg $46$	$23$	$2$	$44$