Number field 47.47.60558628944427886416035618894711378994697503545758179730765967261479053047453845062877188530544503628007178201769.1

• Label: 47.47.605...769.1
• Degree: $47$
• Signature: $[47, 0]$
• Discriminant: $6.056\times 10^{112}$
• Root discriminant: \(250.97\)
• Ramified prime: $283$
• Class number: not computed
• Class group: not computed
• Galois group: $C_{47}$ (as 47T1)

Normalized defining polynomial

x^47 - x^46 - 138*x^45 + 315*x^44 + 8338*x^43 - 29804*x^42 - 276833*x^41 + 1433626*x^40 + 5033859*x^39 - 41190458*x^38 - 30657314*x^37 + 748097961*x^36 - 742659788*x^35 - 8506344013*x^34 + 21519259357*x^33 + 52948548811*x^32 - 268879641855*x^31 - 30332528938*x^30 + 1920252236103*x^29 - 2430736233424*x^28 - 7367030656288*x^27 + 21401598866455*x^26 + 5373046913681*x^25 - 89240242581627*x^24 + 85735098102709*x^23 + 174332690558567*x^22 - 418760640969237*x^21 + 24016008538438*x^20 + 845143941649693*x^19 - 850093833789498*x^18 - 563502754038610*x^17 + 1652337279635119*x^16 - 688139958495907*x^15 - 1132046847057208*x^14 + 1399925903803443*x^13 - 129366579867529*x^12 - 739733924950208*x^11 + 464611724348759*x^10 + 61382685657965*x^9 - 169596960618209*x^8 + 46703998783537*x^7 + 17972540150505*x^6 - 11555848466611*x^5 + 591153994800*x^4 + 757301830397*x^3 - 134933083169*x^2 - 1756074840*x + 132954859

Invariants

Degree:		$47$
Signature:		$[47, 0]$
Discriminant:		60558628944427886416035618894711378994697503545758179730765967261479053047453845062877188530544503628007178201769
Root discriminant:		\(250.97\)
Galois root discriminant:		$283^{46/47}\approx 250.96939670094406$
Ramified primes:		\(283\)
Discriminant root field:		\(\Q\)
$\card{ \Gal(K/\Q) }$:		$47$
This field is Galois and abelian over $\Q$.
Conductor:		\(283\)
Dirichlet character group:		$\lbrace$$\chi_{283}(256,·)$, $\chi_{283}(1,·)$, $\chi_{283}(4,·)$, $\chi_{283}(262,·)$, $\chi_{283}(264,·)$, $\chi_{283}(141,·)$, $\chi_{283}(15,·)$, $\chi_{283}(16,·)$, $\chi_{283}(275,·)$, $\chi_{283}(151,·)$, $\chi_{283}(152,·)$, $\chi_{283}(281,·)$, $\chi_{283}(155,·)$, $\chi_{283}(29,·)$, $\chi_{283}(158,·)$, $\chi_{283}(161,·)$, $\chi_{283}(163,·)$, $\chi_{283}(134,·)$, $\chi_{283}(38,·)$, $\chi_{283}(168,·)$, $\chi_{283}(42,·)$, $\chi_{283}(71,·)$, $\chi_{283}(175,·)$, $\chi_{283}(51,·)$, $\chi_{283}(181,·)$, $\chi_{283}(54,·)$, $\chi_{283}(116,·)$, $\chi_{283}(60,·)$, $\chi_{283}(61,·)$, $\chi_{283}(64,·)$, $\chi_{283}(66,·)$, $\chi_{283}(199,·)$, $\chi_{283}(204,·)$, $\chi_{283}(78,·)$, $\chi_{283}(207,·)$, $\chi_{283}(86,·)$, $\chi_{283}(216,·)$, $\chi_{283}(225,·)$, $\chi_{283}(230,·)$, $\chi_{283}(106,·)$, $\chi_{283}(111,·)$, $\chi_{283}(240,·)$, $\chi_{283}(244,·)$, $\chi_{283}(250,·)$, $\chi_{283}(251,·)$, $\chi_{283}(253,·)$, $\chi_{283}(127,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $a^{40}$, $a^{41}$, $a^{42}$, $a^{43}$, $\frac{1}{521}a^{44}-\frac{18}{521}a^{43}-\frac{205}{521}a^{42}-\frac{101}{521}a^{41}-\frac{103}{521}a^{40}+\frac{100}{521}a^{39}-\frac{36}{521}a^{38}-\frac{97}{521}a^{37}+\frac{223}{521}a^{36}+\frac{137}{521}a^{35}-\frac{16}{521}a^{34}-\frac{221}{521}a^{33}-\frac{135}{521}a^{32}-\frac{138}{521}a^{31}-\frac{192}{521}a^{30}+\frac{66}{521}a^{29}+\frac{208}{521}a^{28}-\frac{205}{521}a^{27}-\frac{87}{521}a^{26}+\frac{90}{521}a^{25}-\frac{164}{521}a^{24}+\frac{57}{521}a^{23}+\frac{82}{521}a^{22}+\frac{252}{521}a^{21}+\frac{249}{521}a^{20}+\frac{196}{521}a^{19}-\frac{239}{521}a^{18}-\frac{175}{521}a^{17}+\frac{13}{521}a^{16}+\frac{93}{521}a^{15}+\frac{208}{521}a^{14}+\frac{204}{521}a^{13}+\frac{229}{521}a^{12}+\frac{184}{521}a^{11}-\frac{212}{521}a^{10}+\frac{64}{521}a^{9}-\frac{89}{521}a^{8}-\frac{120}{521}a^{7}+\frac{193}{521}a^{6}+\frac{60}{521}a^{5}-\frac{124}{521}a^{4}-\frac{224}{521}a^{3}-\frac{111}{521}a^{2}-\frac{213}{521}a+\frac{114}{521}$, $\frac{1}{1755160951}a^{45}-\frac{221972}{1755160951}a^{44}-\frac{767272072}{1755160951}a^{43}-\frac{380323251}{1755160951}a^{42}-\frac{405534754}{1755160951}a^{41}-\frac{612802923}{1755160951}a^{40}+\frac{822122576}{1755160951}a^{39}+\frac{822573226}{1755160951}a^{38}-\frac{780080318}{1755160951}a^{37}+\frac{331184507}{1755160951}a^{36}-\frac{617826878}{1755160951}a^{35}-\frac{855917649}{1755160951}a^{34}+\frac{626061283}{1755160951}a^{33}+\frac{50085193}{1755160951}a^{32}+\frac{293941818}{1755160951}a^{31}-\frac{227595685}{1755160951}a^{30}+\frac{747046992}{1755160951}a^{29}+\frac{238672920}{1755160951}a^{28}-\frac{358112486}{1755160951}a^{27}+\frac{702536463}{1755160951}a^{26}+\frac{447900732}{1755160951}a^{25}-\frac{106511871}{1755160951}a^{24}+\frac{6812222}{1755160951}a^{23}+\frac{315011305}{1755160951}a^{22}+\frac{23191590}{1755160951}a^{21}-\frac{542054885}{1755160951}a^{20}-\frac{798428576}{1755160951}a^{19}+\frac{799868029}{1755160951}a^{18}-\frac{779269749}{1755160951}a^{17}-\frac{223599665}{1755160951}a^{16}+\frac{517602544}{1755160951}a^{15}+\frac{512096213}{1755160951}a^{14}-\frac{787059952}{1755160951}a^{13}+\frac{35918176}{1755160951}a^{12}-\frac{230887523}{1755160951}a^{11}-\frac{538690358}{1755160951}a^{10}+\frac{612784490}{1755160951}a^{9}-\frac{706604095}{1755160951}a^{8}+\frac{239437123}{1755160951}a^{7}+\frac{396669681}{1755160951}a^{6}+\frac{293021258}{1755160951}a^{5}-\frac{802565346}{1755160951}a^{4}-\frac{685475414}{1755160951}a^{3}-\frac{347636054}{1755160951}a^{2}-\frac{665668941}{1755160951}a-\frac{315747231}{1755160951}$, $\frac{1}{22\!\cdots\!61}a^{46}+\frac{61\!\cdots\!45}{22\!\cdots\!61}a^{45}-\frac{53\!\cdots\!38}{22\!\cdots\!61}a^{44}-\frac{32\!\cdots\!68}{22\!\cdots\!61}a^{43}+\frac{10\!\cdots\!20}{22\!\cdots\!61}a^{42}+\frac{11\!\cdots\!27}{22\!\cdots\!61}a^{41}-\frac{10\!\cdots\!78}{22\!\cdots\!61}a^{40}-\frac{18\!\cdots\!15}{22\!\cdots\!61}a^{39}+\frac{45\!\cdots\!22}{22\!\cdots\!61}a^{38}+\frac{69\!\cdots\!48}{22\!\cdots\!61}a^{37}-\frac{45\!\cdots\!35}{22\!\cdots\!61}a^{36}+\frac{22\!\cdots\!02}{22\!\cdots\!61}a^{35}-\frac{89\!\cdots\!35}{22\!\cdots\!61}a^{34}-\frac{83\!\cdots\!04}{22\!\cdots\!61}a^{33}-\frac{51\!\cdots\!42}{22\!\cdots\!61}a^{32}+\frac{84\!\cdots\!29}{22\!\cdots\!61}a^{31}-\frac{92\!\cdots\!39}{22\!\cdots\!61}a^{30}+\frac{44\!\cdots\!96}{22\!\cdots\!61}a^{29}+\frac{33\!\cdots\!68}{22\!\cdots\!61}a^{28}-\frac{90\!\cdots\!58}{22\!\cdots\!61}a^{27}-\frac{92\!\cdots\!52}{22\!\cdots\!61}a^{26}+\frac{41\!\cdots\!71}{22\!\cdots\!61}a^{25}-\frac{22\!\cdots\!30}{22\!\cdots\!61}a^{24}+\frac{30\!\cdots\!66}{22\!\cdots\!61}a^{23}-\frac{10\!\cdots\!42}{22\!\cdots\!61}a^{22}-\frac{19\!\cdots\!29}{22\!\cdots\!61}a^{21}-\frac{25\!\cdots\!92}{22\!\cdots\!61}a^{20}-\frac{98\!\cdots\!57}{22\!\cdots\!61}a^{19}+\frac{11\!\cdots\!02}{22\!\cdots\!61}a^{18}-\frac{68\!\cdots\!20}{22\!\cdots\!61}a^{17}-\frac{36\!\cdots\!11}{22\!\cdots\!61}a^{16}+\frac{58\!\cdots\!31}{22\!\cdots\!61}a^{15}+\frac{77\!\cdots\!49}{22\!\cdots\!61}a^{14}+\frac{58\!\cdots\!19}{22\!\cdots\!61}a^{13}+\frac{16\!\cdots\!63}{22\!\cdots\!61}a^{12}-\frac{60\!\cdots\!95}{22\!\cdots\!61}a^{11}-\frac{15\!\cdots\!74}{22\!\cdots\!61}a^{10}-\frac{61\!\cdots\!18}{22\!\cdots\!61}a^{9}-\frac{90\!\cdots\!21}{22\!\cdots\!61}a^{8}+\frac{77\!\cdots\!96}{22\!\cdots\!61}a^{7}+\frac{87\!\cdots\!10}{22\!\cdots\!61}a^{6}-\frac{30\!\cdots\!10}{22\!\cdots\!61}a^{5}-\frac{31\!\cdots\!75}{22\!\cdots\!61}a^{4}-\frac{10\!\cdots\!24}{22\!\cdots\!61}a^{3}-\frac{11\!\cdots\!23}{22\!\cdots\!61}a^{2}+\frac{79\!\cdots\!85}{22\!\cdots\!61}a+\frac{56\!\cdots\!58}{22\!\cdots\!61}$

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 $C_{47}$

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
\(283\)		Deg $47$	$47$	$1$	$46$