Number field 47.47.4663683520257583455651486936454104910113678905866717884728024389998128903778476103820811059427504675587221679201531780973276735641.1

• Label: 47.47.466...641.1
• Degree: $47$
• Signature: $[47, 0]$
• Discriminant: $4.664\times 10^{129}$
• Root discriminant: \(574.00\)
• Ramified prime: $659$
• Class number: not computed
• Class group: not computed
• Galois group: $C_{47}$ (as 47T1)

Normalized defining polynomial

x^47 - x^46 - 322*x^45 + 257*x^44 + 46168*x^43 - 32856*x^42 - 3928972*x^41 + 2800261*x^40 + 223114373*x^39 - 173143135*x^38 - 9003548066*x^37 + 7929128719*x^36 + 268197678343*x^35 - 271080896060*x^34 - 6041385515832*x^33 + 6979624724342*x^32 + 104494842005772*x^31 - 136657122243604*x^30 - 1400247413062994*x^29 + 2052225750352966*x^28 + 14591458839594017*x^27 - 23782847219941819*x^26 - 118118630838181495*x^25 + 213321125182479282*x^24 + 738128822260196887*x^23 - 1480085655858130569*x^22 - 3514638820507932409*x^21 + 7908202660646656434*x^20 + 12455131093399060106*x^19 - 32248927636656562527*x^18 - 31433221457470396690*x^17 + 98930979031819081241*x^16 + 51111011671312688653*x^15 - 223422112969694652903*x^14 - 36045180999704955067*x^13 + 359687682365174498379*x^12 - 43326797472112521543*x^11 - 392741974421328427418*x^10 + 139313975200941570964*x^9 + 266729246622027266884*x^8 - 153819291119830123270*x^7 - 92300136797476126809*x^6 + 82337017997204992796*x^5 + 4438182302442319866*x^4 - 17703030829530739458*x^3 + 4375073965542285492*x^2 - 7552331260195558*x - 66410957360928749

Invariants

Degree:		$47$
Signature:		$[47, 0]$
Discriminant:		4663683520257583455651486936454104910113678905866717884728024389998128903778476103820811059427504675587221679201531780973276735641
Root discriminant:		\(574.00\)
Galois root discriminant:		$659^{46/47}\approx 573.9963479072553$
Ramified primes:		\(659\)
Discriminant root field:		\(\Q\)
$\card{ \Gal(K/\Q) }$:		$47$
This field is Galois and abelian over $\Q$.
Conductor:		\(659\)
Dirichlet character group:		$\lbrace$$\chi_{659}(1,·)$, $\chi_{659}(514,·)$, $\chi_{659}(262,·)$, $\chi_{659}(139,·)$, $\chi_{659}(14,·)$, $\chi_{659}(15,·)$, $\chi_{659}(274,·)$, $\chi_{659}(531,·)$, $\chi_{659}(156,·)$, $\chi_{659}(541,·)$, $\chi_{659}(299,·)$, $\chi_{659}(44,·)$, $\chi_{659}(173,·)$, $\chi_{659}(302,·)$, $\chi_{659}(304,·)$, $\chi_{659}(436,·)$, $\chi_{659}(568,·)$, $\chi_{659}(185,·)$, $\chi_{659}(445,·)$, $\chi_{659}(576,·)$, $\chi_{659}(194,·)$, $\chi_{659}(323,·)$, $\chi_{659}(196,·)$, $\chi_{659}(325,·)$, $\chi_{659}(225,·)$, $\chi_{659}(73,·)$, $\chi_{659}(461,·)$, $\chi_{659}(207,·)$, $\chi_{659}(80,·)$, $\chi_{659}(210,·)$, $\chi_{659}(596,·)$, $\chi_{659}(85,·)$, $\chi_{659}(57,·)$, $\chi_{659}(606,·)$, $\chi_{659}(523,·)$, $\chi_{659}(609,·)$, $\chi_{659}(612,·)$, $\chi_{659}(613,·)$, $\chi_{659}(232,·)$, $\chi_{659}(618,·)$, $\chi_{659}(363,·)$, $\chi_{659}(108,·)$, $\chi_{659}(616,·)$, $\chi_{659}(628,·)$, $\chi_{659}(373,·)$, $\chi_{659}(635,·)$, $\chi_{659}(469,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $a^{40}$, $a^{41}$, $a^{42}$, $\frac{1}{404933}a^{43}-\frac{78420}{404933}a^{42}-\frac{11277}{404933}a^{41}+\frac{79396}{404933}a^{40}-\frac{117663}{404933}a^{39}+\frac{56793}{404933}a^{38}-\frac{166178}{404933}a^{37}+\frac{11171}{404933}a^{36}-\frac{58985}{404933}a^{35}-\frac{135592}{404933}a^{34}+\frac{11532}{404933}a^{33}+\frac{188454}{404933}a^{32}-\frac{61073}{404933}a^{31}+\frac{72045}{404933}a^{30}+\frac{135960}{404933}a^{29}+\frac{158575}{404933}a^{28}-\frac{10949}{404933}a^{27}+\frac{159469}{404933}a^{26}-\frac{91685}{404933}a^{25}-\frac{190737}{404933}a^{24}+\frac{165373}{404933}a^{23}-\frac{141925}{404933}a^{22}+\frac{196465}{404933}a^{21}-\frac{117115}{404933}a^{20}+\frac{78647}{404933}a^{19}-\frac{40613}{404933}a^{18}+\frac{143833}{404933}a^{17}+\frac{42548}{404933}a^{16}-\frac{3433}{404933}a^{15}-\frac{116922}{404933}a^{14}-\frac{181017}{404933}a^{13}+\frac{64475}{404933}a^{12}-\frac{2257}{404933}a^{11}+\frac{185314}{404933}a^{10}-\frac{180507}{404933}a^{9}-\frac{90068}{404933}a^{8}-\frac{81349}{404933}a^{7}+\frac{58264}{404933}a^{6}+\frac{44101}{404933}a^{5}+\frac{36411}{404933}a^{4}+\frac{138248}{404933}a^{3}+\frac{132321}{404933}a^{2}-\frac{110867}{404933}a+\frac{43396}{404933}$, $\frac{1}{404933}a^{44}+\frac{9794}{404933}a^{42}+\frac{110728}{404933}a^{41}-\frac{133151}{404933}a^{40}+\frac{132604}{404933}a^{39}+\frac{87748}{404933}a^{38}-\frac{113783}{404933}a^{37}+\frac{100756}{404933}a^{36}-\frac{189633}{404933}a^{35}+\frac{22539}{404933}a^{34}-\frac{92428}{404933}a^{33}+\frac{66839}{404933}a^{32}-\frac{130024}{404933}a^{31}-\frac{125289}{404933}a^{30}-\frac{149048}{404933}a^{29}-\frac{51879}{404933}a^{28}-\frac{3151}{404933}a^{27}-\frac{78544}{404933}a^{26}-\frac{138089}{404933}a^{25}-\frac{15013}{404933}a^{24}+\frac{24477}{404933}a^{23}+\frac{21470}{404933}a^{22}+\frac{182334}{404933}a^{21}-\frac{199213}{404933}a^{20}-\frac{77396}{404933}a^{19}+\frac{70418}{404933}a^{18}+\frac{17693}{404933}a^{17}-\frac{37193}{404933}a^{16}-\frac{52337}{404933}a^{15}+\frac{98595}{404933}a^{14}+\frac{42583}{404933}a^{13}+\frac{133805}{404933}a^{12}+\frac{147095}{404933}a^{11}-\frac{92131}{404933}a^{10}+\frac{198806}{404933}a^{9}+\frac{32410}{404933}a^{8}-\frac{15834}{404933}a^{7}-\frac{156991}{404933}a^{6}-\frac{95922}{404933}a^{5}-\frac{98648}{404933}a^{4}-\frac{135661}{404933}a^{3}+\frac{93828}{404933}a^{2}+\frac{169699}{404933}a+\frac{57388}{404933}$, $\frac{1}{157518937}a^{45}+\frac{110}{157518937}a^{44}-\frac{41}{157518937}a^{43}+\frac{41544703}{157518937}a^{42}-\frac{8242068}{157518937}a^{41}-\frac{67709065}{157518937}a^{40}+\frac{14996212}{157518937}a^{39}-\frac{61076393}{157518937}a^{38}+\frac{55664412}{157518937}a^{37}-\frac{24465586}{157518937}a^{36}-\frac{46499384}{157518937}a^{35}+\frac{870081}{157518937}a^{34}+\frac{76114508}{157518937}a^{33}-\frac{13091133}{157518937}a^{32}-\frac{77865094}{157518937}a^{31}-\frac{78245010}{157518937}a^{30}-\frac{76050144}{157518937}a^{29}+\frac{34190384}{157518937}a^{28}+\frac{14933537}{157518937}a^{27}+\frac{15850878}{157518937}a^{26}-\frac{11624222}{157518937}a^{25}+\frac{22107900}{157518937}a^{24}-\frac{16550438}{157518937}a^{23}-\frac{50473865}{157518937}a^{22}+\frac{53573967}{157518937}a^{21}+\frac{40970362}{157518937}a^{20}-\frac{78567766}{157518937}a^{19}-\frac{39043638}{157518937}a^{18}+\frac{43450550}{157518937}a^{17}-\frac{40751328}{157518937}a^{16}+\frac{65358916}{157518937}a^{15}-\frac{55602829}{157518937}a^{14}-\frac{12782390}{157518937}a^{13}+\frac{30266552}{157518937}a^{12}-\frac{22454036}{157518937}a^{11}-\frac{45529465}{157518937}a^{10}-\frac{41613338}{157518937}a^{9}-\frac{29424931}{157518937}a^{8}+\frac{51877165}{157518937}a^{7}-\frac{23080239}{157518937}a^{6}+\frac{52469388}{157518937}a^{5}-\frac{39878597}{157518937}a^{4}+\frac{70302982}{157518937}a^{3}+\frac{34864386}{157518937}a^{2}-\frac{76137668}{157518937}a-\frac{32156233}{157518937}$, $\frac{1}{12\!\cdots\!67}a^{46}+\frac{19\!\cdots\!96}{12\!\cdots\!67}a^{45}+\frac{14\!\cdots\!60}{12\!\cdots\!67}a^{44}+\frac{32\!\cdots\!96}{12\!\cdots\!67}a^{43}+\frac{52\!\cdots\!26}{12\!\cdots\!67}a^{42}+\frac{47\!\cdots\!29}{12\!\cdots\!67}a^{41}-\frac{21\!\cdots\!52}{12\!\cdots\!67}a^{40}-\frac{41\!\cdots\!24}{12\!\cdots\!67}a^{39}+\frac{80\!\cdots\!74}{12\!\cdots\!67}a^{38}-\frac{32\!\cdots\!82}{12\!\cdots\!67}a^{37}-\frac{26\!\cdots\!93}{12\!\cdots\!67}a^{36}+\frac{58\!\cdots\!26}{12\!\cdots\!67}a^{35}+\frac{14\!\cdots\!43}{12\!\cdots\!67}a^{34}+\frac{52\!\cdots\!35}{12\!\cdots\!67}a^{33}+\frac{17\!\cdots\!93}{12\!\cdots\!67}a^{32}+\frac{24\!\cdots\!75}{12\!\cdots\!67}a^{31}+\frac{19\!\cdots\!26}{12\!\cdots\!67}a^{30}+\frac{53\!\cdots\!35}{12\!\cdots\!67}a^{29}-\frac{13\!\cdots\!97}{12\!\cdots\!67}a^{28}+\frac{37\!\cdots\!35}{12\!\cdots\!67}a^{27}+\frac{13\!\cdots\!86}{12\!\cdots\!67}a^{26}+\frac{55\!\cdots\!32}{12\!\cdots\!67}a^{25}+\frac{44\!\cdots\!87}{12\!\cdots\!67}a^{24}-\frac{24\!\cdots\!17}{12\!\cdots\!67}a^{23}+\frac{56\!\cdots\!26}{12\!\cdots\!67}a^{22}-\frac{26\!\cdots\!41}{12\!\cdots\!67}a^{21}-\frac{41\!\cdots\!62}{12\!\cdots\!67}a^{20}+\frac{22\!\cdots\!81}{12\!\cdots\!67}a^{19}+\frac{26\!\cdots\!14}{12\!\cdots\!67}a^{18}-\frac{23\!\cdots\!05}{12\!\cdots\!67}a^{17}+\frac{47\!\cdots\!85}{12\!\cdots\!67}a^{16}+\frac{13\!\cdots\!56}{12\!\cdots\!67}a^{15}+\frac{58\!\cdots\!05}{12\!\cdots\!67}a^{14}+\frac{28\!\cdots\!36}{12\!\cdots\!67}a^{13}-\frac{89\!\cdots\!71}{12\!\cdots\!67}a^{12}+\frac{12\!\cdots\!90}{12\!\cdots\!67}a^{11}+\frac{44\!\cdots\!76}{12\!\cdots\!67}a^{10}-\frac{47\!\cdots\!57}{12\!\cdots\!67}a^{9}-\frac{24\!\cdots\!94}{12\!\cdots\!67}a^{8}-\frac{41\!\cdots\!77}{12\!\cdots\!67}a^{7}-\frac{24\!\cdots\!98}{12\!\cdots\!67}a^{6}-\frac{60\!\cdots\!19}{12\!\cdots\!67}a^{5}+\frac{22\!\cdots\!41}{12\!\cdots\!67}a^{4}-\frac{14\!\cdots\!46}{12\!\cdots\!67}a^{3}-\frac{19\!\cdots\!89}{12\!\cdots\!67}a^{2}+\frac{52\!\cdots\!58}{12\!\cdots\!67}a-\frac{20\!\cdots\!28}{12\!\cdots\!67}$

Monogenic:		No
Index:		$1$
Inessential primes:		None

Class group and class number

not computed

Unit group

Rank:		$46$
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

$C_{47}$ (as 47T1):

A cyclic group of order 47

The 47 conjugacy class representatives for $C_{47}$

Character table for $C_{47}$ 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	$47$	$47$	$47$	$47$	$47$	$47$	$47$	$47$	$47$	$47$	$47$	$47$	$47$	$47$	$47$	$47$	$47$

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
\(659\)		Deg $47$	$47$	$1$	$46$