Number field 41.41.1053679516786758709720897165633610283654902079788240782212064028656008150674955537661019599453337948495954124897145643786946499201.1

• Label: 41.41.105...201.1
• Degree: $41$
• Signature: $[41, 0]$
• Discriminant: $1.054\times 10^{129}$
• Root discriminant: \(1402.48\)
• Ramified prime: $41$
• Class number: not computed
• Class group: not computed
• Galois group: $C_{41}$ (as 41T1)

Normalized defining polynomial

x^41 - 820*x^39 - 533*x^38 + 303031*x^37 + 376913*x^36 - 66737627*x^35 - 118821690*x^34 + 9764169434*x^33 + 22080920476*x^32 - 1002016650746*x^31 - 2696024483356*x^30 + 74276881167057*x^29 + 228322554454083*x^28 - 4038255389081271*x^27 - 13810013721005555*x^26 + 161957213612863355*x^25 + 605630162979885300*x^24 - 4783062094323192024*x^23 - 19362795070723667202*x^22 + 103087839028351278942*x^21 + 450266165503910224663*x^20 - 1593603545898162906857*x^19 - 7540369918979718461657*x^18 + 17171487184711671438215*x^17 + 89253263738667697145313*x^16 - 123172673161638273736870*x^15 - 724697045982659590514328*x^14 + 544476915889889670403041*x^13 + 3855061119747652830292013*x^12 - 1279320329561947245453521*x^11 - 12510633939200726076347653*x^10 + 1043523355082184352460715*x^9 + 22013783610841586394614893*x^8 + 598853130650094992883999*x^7 - 16988637915887608842729071*x^6 - 1349373804383563050067379*x^5 + 3896754513924753580219610*x^4 + 333868055127333108898601*x^3 - 256246929318582500140348*x^2 - 10590850429623318361311*x + 2727257042363914863401

Invariants

Degree:		$41$
Signature:		$[41, 0]$
Discriminant:		1053679516786758709720897165633610283654902079788240782212064028656008150674955537661019599453337948495954124897145643786946499201
Root discriminant:		\(1402.48\)
Galois root discriminant:		$41^{80/41}\approx 1402.475651548207$
Ramified primes:		\(41\)
Discriminant root field:		\(\Q\)
$\card{ \Gal(K/\Q) }$:		$41$
This field is Galois and abelian over $\Q$.
Conductor:		\(1681=41^{2}\)
Dirichlet character group:		$\lbrace$$\chi_{1681}(1,·)$, $\chi_{1681}(1026,·)$, $\chi_{1681}(903,·)$, $\chi_{1681}(780,·)$, $\chi_{1681}(657,·)$, $\chi_{1681}(534,·)$, $\chi_{1681}(1559,·)$, $\chi_{1681}(411,·)$, $\chi_{1681}(1436,·)$, $\chi_{1681}(288,·)$, $\chi_{1681}(1313,·)$, $\chi_{1681}(165,·)$, $\chi_{1681}(1190,·)$, $\chi_{1681}(42,·)$, $\chi_{1681}(1067,·)$, $\chi_{1681}(944,·)$, $\chi_{1681}(821,·)$, $\chi_{1681}(698,·)$, $\chi_{1681}(575,·)$, $\chi_{1681}(1600,·)$, $\chi_{1681}(452,·)$, $\chi_{1681}(1477,·)$, $\chi_{1681}(329,·)$, $\chi_{1681}(1354,·)$, $\chi_{1681}(206,·)$, $\chi_{1681}(1231,·)$, $\chi_{1681}(83,·)$, $\chi_{1681}(1108,·)$, $\chi_{1681}(985,·)$, $\chi_{1681}(862,·)$, $\chi_{1681}(739,·)$, $\chi_{1681}(616,·)$, $\chi_{1681}(1641,·)$, $\chi_{1681}(493,·)$, $\chi_{1681}(1518,·)$, $\chi_{1681}(370,·)$, $\chi_{1681}(1395,·)$, $\chi_{1681}(247,·)$, $\chi_{1681}(1272,·)$, $\chi_{1681}(124,·)$, $\chi_{1681}(1149,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $\frac{1}{226315634441027}a^{39}+\frac{68829831999505}{226315634441027}a^{38}+\frac{85277659288374}{226315634441027}a^{37}+\frac{38071080404819}{226315634441027}a^{36}-\frac{66783777056075}{226315634441027}a^{35}-\frac{84838219747661}{226315634441027}a^{34}-\frac{104321769666912}{226315634441027}a^{33}+\frac{105923167280333}{226315634441027}a^{32}-\frac{32881735672093}{226315634441027}a^{31}+\frac{111120921179873}{226315634441027}a^{30}-\frac{104656276531248}{226315634441027}a^{29}+\frac{6451479135422}{226315634441027}a^{28}+\frac{22022404970817}{226315634441027}a^{27}-\frac{86986215702786}{226315634441027}a^{26}+\frac{73250616794429}{226315634441027}a^{25}-\frac{108712554878332}{226315634441027}a^{24}-\frac{13062077579154}{226315634441027}a^{23}-\frac{87899870433838}{226315634441027}a^{22}+\frac{68871135207400}{226315634441027}a^{21}+\frac{65758772582085}{226315634441027}a^{20}+\frac{100830015047028}{226315634441027}a^{19}+\frac{84937298981613}{226315634441027}a^{18}-\frac{21490156759819}{226315634441027}a^{17}-\frac{110536611130363}{226315634441027}a^{16}-\frac{97143126286265}{226315634441027}a^{15}-\frac{34189055557684}{226315634441027}a^{14}-\frac{44686178341650}{226315634441027}a^{13}+\frac{105254690102437}{226315634441027}a^{12}-\frac{49641316303647}{226315634441027}a^{11}+\frac{8096856417015}{226315634441027}a^{10}-\frac{29077702905279}{226315634441027}a^{9}-\frac{112928804397919}{226315634441027}a^{8}-\frac{82851895116003}{226315634441027}a^{7}+\frac{103782156306896}{226315634441027}a^{6}+\frac{106493691366112}{226315634441027}a^{5}-\frac{90006018752259}{226315634441027}a^{4}-\frac{87234973543257}{226315634441027}a^{3}+\frac{65333588526996}{226315634441027}a^{2}+\frac{52105551412211}{226315634441027}a+\frac{81780235767749}{226315634441027}$, $\frac{1}{22\!\cdots\!53}a^{40}+\frac{66\!\cdots\!41}{22\!\cdots\!53}a^{39}-\frac{51\!\cdots\!08}{22\!\cdots\!53}a^{38}+\frac{74\!\cdots\!00}{22\!\cdots\!53}a^{37}+\frac{23\!\cdots\!40}{22\!\cdots\!53}a^{36}+\frac{38\!\cdots\!43}{22\!\cdots\!53}a^{35}-\frac{30\!\cdots\!75}{22\!\cdots\!53}a^{34}-\frac{10\!\cdots\!78}{22\!\cdots\!53}a^{33}+\frac{46\!\cdots\!92}{22\!\cdots\!53}a^{32}-\frac{21\!\cdots\!81}{22\!\cdots\!53}a^{31}-\frac{65\!\cdots\!95}{22\!\cdots\!53}a^{30}-\frac{59\!\cdots\!62}{22\!\cdots\!53}a^{29}+\frac{60\!\cdots\!45}{22\!\cdots\!53}a^{28}-\frac{80\!\cdots\!12}{22\!\cdots\!53}a^{27}+\frac{73\!\cdots\!94}{22\!\cdots\!53}a^{26}-\frac{51\!\cdots\!12}{22\!\cdots\!53}a^{25}+\frac{10\!\cdots\!59}{22\!\cdots\!53}a^{24}-\frac{60\!\cdots\!49}{22\!\cdots\!53}a^{23}+\frac{42\!\cdots\!46}{22\!\cdots\!53}a^{22}-\frac{91\!\cdots\!17}{22\!\cdots\!53}a^{21}+\frac{48\!\cdots\!87}{22\!\cdots\!53}a^{20}+\frac{29\!\cdots\!86}{22\!\cdots\!53}a^{19}+\frac{81\!\cdots\!08}{22\!\cdots\!53}a^{18}+\frac{10\!\cdots\!19}{22\!\cdots\!53}a^{17}-\frac{92\!\cdots\!05}{22\!\cdots\!53}a^{16}-\frac{62\!\cdots\!24}{22\!\cdots\!53}a^{15}-\frac{94\!\cdots\!42}{22\!\cdots\!53}a^{14}-\frac{69\!\cdots\!94}{22\!\cdots\!53}a^{13}-\frac{48\!\cdots\!75}{22\!\cdots\!53}a^{12}+\frac{37\!\cdots\!07}{22\!\cdots\!53}a^{11}+\frac{20\!\cdots\!39}{22\!\cdots\!53}a^{10}+\frac{79\!\cdots\!07}{22\!\cdots\!53}a^{9}-\frac{11\!\cdots\!32}{22\!\cdots\!53}a^{8}-\frac{10\!\cdots\!22}{22\!\cdots\!53}a^{7}-\frac{10\!\cdots\!10}{22\!\cdots\!53}a^{6}+\frac{43\!\cdots\!76}{22\!\cdots\!53}a^{5}+\frac{73\!\cdots\!45}{22\!\cdots\!53}a^{4}+\frac{17\!\cdots\!23}{22\!\cdots\!53}a^{3}-\frac{72\!\cdots\!55}{22\!\cdots\!53}a^{2}+\frac{83\!\cdots\!81}{22\!\cdots\!53}a-\frac{23\!\cdots\!61}{22\!\cdots\!53}$

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

Class group and class number

not computed

Unit group

Rank:		$40$
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

$C_{41}$ (as 41T1):

A cyclic group of order 41

The 41 conjugacy class representatives for $C_{41}$

Character table for $C_{41}$

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	$41$	$41$	$41$	$41$	$41$	$41$	$41$	$41$	$41$	$41$	$41$	$41$	R	$41$	$41$	$41$	$41$

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
\(41\)		Deg $41$	$41$	$1$	$80$