Number field 45.45.99292760576690774537904993081140088218976054890294423342092931819201133113087291038341877092167846038415081660922852502562820241.1

• Label: 45.45.992...241.1
• Degree: $45$
• Signature: $[45, 0]$
• Discriminant: $9.929\times 10^{127}$
• Root discriminant: \(698.84\)
• Ramified prime: $811$
• Class number: not computed
• Class group: not computed
• Galois group: $C_{45}$ (as 45T1)

Normalized defining polynomial

x^45 - x^44 - 396*x^43 + 829*x^42 + 69093*x^41 - 213329*x^40 - 6974865*x^39 + 28148607*x^38 + 450159883*x^37 - 2263089158*x^36 - 19342390441*x^35 + 120399126257*x^34 + 554492360509*x^33 - 4434413326456*x^32 - 10046622781109*x^31 + 116018246394393*x^30 + 86886702429318*x^29 - 2185646177028281*x^28 + 693994919431743*x^27 + 29756816334260440*x^26 - 33973753215967284*x^25 - 290582689767894657*x^24 + 543184152125443033*x^23 + 1984489268632473104*x^22 - 5275691290270373630*x^21 - 8880531728704621017*x^20 + 34163102478237698853*x^19 + 20979649586488016898*x^18 - 149269365665132134938*x^17 + 10063198266998551730*x^16 + 427685059470653411785*x^15 - 241034540985719055340*x^14 - 743760671890407982334*x^13 + 747995661858071416101*x^12 + 642020392083923396842*x^11 - 1065427764689162633315*x^10 - 59930805542863649405*x^9 + 666581021324279560724*x^8 - 231175789721089989336*x^7 - 115726943075772621725*x^6 + 74801496316603079673*x^5 - 1106196938800865215*x^4 - 5699180925312701810*x^3 + 720276517742393070*x^2 + 88437835208003944*x - 13311721526413589

Invariants

Degree:		$45$
Signature:		$[45, 0]$
Discriminant:		99292760576690774537904993081140088218976054890294423342092931819201133113087291038341877092167846038415081660922852502562820241
Root discriminant:		\(698.84\)
Galois root discriminant:		$811^{44/45}\approx 698.837089551728$
Ramified primes:		\(811\)
Discriminant root field:		\(\Q\)
$\card{ \Gal(K/\Q) }$:		$45$
This field is Galois and abelian over $\Q$.
Conductor:		\(811\)
Dirichlet character group:		$\lbrace$$\chi_{811}(1,·)$, $\chi_{811}(130,·)$, $\chi_{811}(773,·)$, $\chi_{811}(276,·)$, $\chi_{811}(662,·)$, $\chi_{811}(537,·)$, $\chi_{811}(796,·)$, $\chi_{811}(797,·)$, $\chi_{811}(803,·)$, $\chi_{811}(680,·)$, $\chi_{811}(41,·)$, $\chi_{811}(299,·)$, $\chi_{811}(304,·)$, $\chi_{811}(306,·)$, $\chi_{811}(54,·)$, $\chi_{811}(55,·)$, $\chi_{811}(570,·)$, $\chi_{811}(59,·)$, $\chi_{811}(191,·)$, $\chi_{811}(64,·)$, $\chi_{811}(196,·)$, $\chi_{811}(582,·)$, $\chi_{811}(737,·)$, $\chi_{811}(464,·)$, $\chi_{811}(210,·)$, $\chi_{811}(339,·)$, $\chi_{811}(212,·)$, $\chi_{811}(726,·)$, $\chi_{811}(343,·)$, $\chi_{811}(592,·)$, $\chi_{811}(94,·)$, $\chi_{811}(225,·)$, $\chi_{811}(610,·)$, $\chi_{811}(483,·)$, $\chi_{811}(613,·)$, $\chi_{811}(237,·)$, $\chi_{811}(532,·)$, $\chi_{811}(112,·)$, $\chi_{811}(753,·)$, $\chi_{811}(371,·)$, $\chi_{811}(500,·)$, $\chi_{811}(120,·)$, $\chi_{811}(633,·)$, $\chi_{811}(379,·)$, $\chi_{811}(381,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $\frac{1}{131}a^{35}-\frac{62}{131}a^{34}-\frac{10}{131}a^{33}-\frac{62}{131}a^{32}+\frac{46}{131}a^{31}-\frac{45}{131}a^{30}+\frac{15}{131}a^{29}-\frac{7}{131}a^{28}+\frac{12}{131}a^{27}-\frac{17}{131}a^{26}+\frac{63}{131}a^{25}-\frac{3}{131}a^{24}-\frac{37}{131}a^{23}+\frac{64}{131}a^{22}-\frac{9}{131}a^{21}-\frac{41}{131}a^{20}-\frac{60}{131}a^{19}-\frac{13}{131}a^{18}-\frac{21}{131}a^{17}+\frac{59}{131}a^{16}+\frac{23}{131}a^{15}+\frac{9}{131}a^{14}+\frac{35}{131}a^{13}-\frac{3}{131}a^{12}-\frac{32}{131}a^{11}-\frac{65}{131}a^{10}+\frac{23}{131}a^{9}+\frac{61}{131}a^{8}-\frac{53}{131}a^{7}+\frac{10}{131}a^{6}-\frac{27}{131}a^{5}+\frac{63}{131}a^{4}-\frac{31}{131}a^{3}+\frac{40}{131}a^{2}-\frac{57}{131}a$, $\frac{1}{131}a^{36}-\frac{55}{131}a^{34}-\frac{27}{131}a^{33}+\frac{1}{131}a^{32}+\frac{56}{131}a^{31}-\frac{24}{131}a^{30}+\frac{6}{131}a^{29}-\frac{29}{131}a^{28}-\frac{59}{131}a^{27}+\frac{57}{131}a^{26}-\frac{27}{131}a^{25}+\frac{39}{131}a^{24}-\frac{3}{131}a^{23}+\frac{29}{131}a^{22}+\frac{56}{131}a^{21}+\frac{18}{131}a^{20}-\frac{65}{131}a^{19}-\frac{41}{131}a^{18}-\frac{64}{131}a^{17}+\frac{13}{131}a^{16}-\frac{6}{131}a^{15}-\frac{62}{131}a^{14}-\frac{60}{131}a^{13}+\frac{44}{131}a^{12}+\frac{47}{131}a^{11}+\frac{54}{131}a^{10}+\frac{46}{131}a^{9}+\frac{61}{131}a^{8}-\frac{1}{131}a^{7}-\frac{62}{131}a^{6}-\frac{39}{131}a^{5}-\frac{55}{131}a^{4}-\frac{48}{131}a^{3}+\frac{65}{131}a^{2}+\frac{3}{131}a$, $\frac{1}{131}a^{37}-\frac{31}{131}a^{34}-\frac{25}{131}a^{33}+\frac{52}{131}a^{32}+\frac{17}{131}a^{31}+\frac{20}{131}a^{30}+\frac{10}{131}a^{29}-\frac{51}{131}a^{28}+\frac{62}{131}a^{27}-\frac{45}{131}a^{26}-\frac{33}{131}a^{25}-\frac{37}{131}a^{24}-\frac{41}{131}a^{23}+\frac{39}{131}a^{22}+\frac{47}{131}a^{21}+\frac{38}{131}a^{20}+\frac{65}{131}a^{19}+\frac{7}{131}a^{18}+\frac{37}{131}a^{17}-\frac{36}{131}a^{16}+\frac{24}{131}a^{15}+\frac{42}{131}a^{14}+\frac{4}{131}a^{13}+\frac{13}{131}a^{12}-\frac{3}{131}a^{11}+\frac{8}{131}a^{10}+\frac{16}{131}a^{9}-\frac{52}{131}a^{8}+\frac{36}{131}a^{7}-\frac{13}{131}a^{6}+\frac{32}{131}a^{5}+\frac{11}{131}a^{4}+\frac{63}{131}a^{3}-\frac{24}{131}a^{2}+\frac{9}{131}a$, $\frac{1}{131}a^{38}+\frac{18}{131}a^{34}+\frac{4}{131}a^{33}+\frac{60}{131}a^{32}+\frac{5}{131}a^{31}+\frac{56}{131}a^{30}+\frac{21}{131}a^{29}-\frac{24}{131}a^{28}+\frac{65}{131}a^{27}-\frac{36}{131}a^{26}-\frac{49}{131}a^{25}-\frac{3}{131}a^{24}-\frac{60}{131}a^{23}-\frac{65}{131}a^{22}+\frac{21}{131}a^{21}-\frac{27}{131}a^{20}-\frac{19}{131}a^{19}+\frac{27}{131}a^{18}-\frac{32}{131}a^{17}+\frac{19}{131}a^{16}-\frac{31}{131}a^{15}+\frac{21}{131}a^{14}+\frac{50}{131}a^{13}+\frac{35}{131}a^{12}+\frac{64}{131}a^{11}-\frac{34}{131}a^{10}+\frac{6}{131}a^{9}-\frac{38}{131}a^{8}+\frac{47}{131}a^{7}-\frac{51}{131}a^{6}-\frac{40}{131}a^{5}+\frac{51}{131}a^{4}+\frac{63}{131}a^{3}-\frac{61}{131}a^{2}-\frac{64}{131}a$, $\frac{1}{917}a^{39}-\frac{1}{917}a^{38}-\frac{1}{917}a^{37}+\frac{1}{917}a^{36}+\frac{3}{917}a^{35}-\frac{156}{917}a^{34}+\frac{204}{917}a^{33}-\frac{32}{131}a^{32}+\frac{64}{131}a^{31}+\frac{334}{917}a^{30}+\frac{381}{917}a^{29}-\frac{44}{131}a^{28}-\frac{402}{917}a^{27}+\frac{344}{917}a^{26}+\frac{417}{917}a^{25}-\frac{67}{917}a^{24}-\frac{198}{917}a^{23}+\frac{164}{917}a^{22}-\frac{428}{917}a^{21}-\frac{183}{917}a^{20}+\frac{30}{917}a^{19}-\frac{305}{917}a^{18}+\frac{265}{917}a^{17}+\frac{424}{917}a^{16}+\frac{332}{917}a^{15}-\frac{79}{917}a^{14}-\frac{342}{917}a^{13}+\frac{367}{917}a^{12}+\frac{170}{917}a^{11}-\frac{249}{917}a^{10}+\frac{296}{917}a^{9}-\frac{62}{917}a^{8}+\frac{136}{917}a^{7}+\frac{74}{917}a^{6}-\frac{230}{917}a^{5}+\frac{180}{917}a^{4}-\frac{42}{131}a^{3}+\frac{403}{917}a^{2}+\frac{258}{917}a-\frac{3}{7}$, $\frac{1}{917}a^{40}-\frac{2}{917}a^{38}-\frac{3}{917}a^{36}+\frac{1}{917}a^{35}+\frac{55}{917}a^{34}-\frac{454}{917}a^{33}-\frac{23}{131}a^{32}+\frac{138}{917}a^{31}+\frac{372}{917}a^{30}-\frac{410}{917}a^{29}+\frac{249}{917}a^{28}+\frac{369}{917}a^{27}-\frac{422}{917}a^{26}+\frac{22}{131}a^{25}-\frac{83}{917}a^{24}-\frac{209}{917}a^{23}+\frac{219}{917}a^{22}+\frac{362}{917}a^{21}-\frac{174}{917}a^{20}+\frac{110}{917}a^{19}+\frac{79}{917}a^{18}-\frac{263}{917}a^{17}-\frac{48}{131}a^{16}+\frac{169}{917}a^{15}-\frac{435}{917}a^{14}+\frac{333}{917}a^{13}-\frac{233}{917}a^{12}+\frac{166}{917}a^{11}-\frac{254}{917}a^{10}-\frac{214}{917}a^{9}-\frac{129}{917}a^{8}+\frac{44}{131}a^{7}-\frac{16}{917}a^{6}-\frac{267}{917}a^{5}-\frac{114}{917}a^{4}+\frac{256}{917}a^{3}-\frac{53}{917}a^{2}+\frac{236}{917}a-\frac{3}{7}$, $\frac{1}{917}a^{41}-\frac{2}{917}a^{38}+\frac{2}{917}a^{37}+\frac{3}{917}a^{36}-\frac{2}{917}a^{35}+\frac{172}{917}a^{34}-\frac{215}{917}a^{33}+\frac{292}{917}a^{32}+\frac{323}{917}a^{31}-\frac{435}{917}a^{30}+\frac{136}{917}a^{29}-\frac{163}{917}a^{28}+\frac{286}{917}a^{27}-\frac{236}{917}a^{26}+\frac{219}{917}a^{25}-\frac{59}{131}a^{24}+\frac{33}{917}a^{23}-\frac{318}{917}a^{22}-\frac{134}{917}a^{21}-\frac{158}{917}a^{20}-\frac{211}{917}a^{19}-\frac{5}{917}a^{18}-\frac{58}{917}a^{17}-\frac{201}{917}a^{16}-\frac{135}{917}a^{15}-\frac{14}{131}a^{14}-\frac{49}{131}a^{13}+\frac{263}{917}a^{12}+\frac{247}{917}a^{11}-\frac{229}{917}a^{10}+\frac{43}{917}a^{9}-\frac{355}{917}a^{8}+\frac{179}{917}a^{7}+\frac{11}{131}a^{6}+\frac{62}{131}a^{5}+\frac{56}{131}a^{4}-\frac{81}{917}a^{3}+\frac{188}{917}a^{2}+\frac{109}{917}a+\frac{1}{7}$, $\frac{1}{917}a^{42}+\frac{1}{917}a^{37}+\frac{3}{917}a^{35}+\frac{236}{917}a^{34}-\frac{43}{131}a^{33}-\frac{279}{917}a^{32}-\frac{253}{917}a^{31}+\frac{426}{917}a^{30}-\frac{192}{917}a^{29}-\frac{22}{917}a^{28}-\frac{389}{917}a^{27}+\frac{214}{917}a^{26}+\frac{400}{917}a^{25}+\frac{424}{917}a^{24}+\frac{37}{131}a^{23}-\frac{2}{917}a^{22}-\frac{356}{917}a^{21}+\frac{179}{917}a^{20}-\frac{449}{917}a^{19}-\frac{227}{917}a^{18}+\frac{48}{131}a^{17}-\frac{442}{917}a^{16}+\frac{209}{917}a^{15}-\frac{242}{917}a^{14}-\frac{127}{917}a^{13}-\frac{328}{917}a^{12}+\frac{209}{917}a^{11}-\frac{12}{131}a^{10}-\frac{120}{917}a^{9}+\frac{384}{917}a^{8}+\frac{454}{917}a^{7}-\frac{251}{917}a^{6}+\frac{72}{917}a^{5}+\frac{258}{917}a^{4}+\frac{440}{917}a^{3}+\frac{334}{917}a^{2}-\frac{382}{917}a+\frac{1}{7}$, $\frac{1}{917}a^{43}+\frac{1}{917}a^{38}+\frac{3}{917}a^{36}-\frac{2}{917}a^{35}-\frac{31}{131}a^{34}+\frac{267}{917}a^{33}-\frac{169}{917}a^{32}-\frac{435}{917}a^{31}+\frac{431}{917}a^{30}+\frac{76}{917}a^{29}+\frac{360}{917}a^{28}+\frac{109}{917}a^{27}-\frac{139}{917}a^{26}+\frac{102}{917}a^{25}+\frac{8}{131}a^{24}-\frac{366}{917}a^{23}+\frac{1}{917}a^{22}-\frac{430}{917}a^{21}+\frac{139}{917}a^{20}+\frac{298}{917}a^{19}-\frac{34}{131}a^{18}-\frac{29}{917}a^{17}-\frac{78}{917}a^{16}-\frac{214}{917}a^{15}-\frac{435}{917}a^{14}-\frac{405}{917}a^{13}+\frac{6}{917}a^{12}+\frac{28}{131}a^{11}-\frac{239}{917}a^{10}+\frac{412}{917}a^{9}-\frac{309}{917}a^{8}+\frac{442}{917}a^{7}+\frac{443}{917}a^{6}+\frac{265}{917}a^{5}+\frac{118}{917}a^{4}+\frac{376}{917}a^{3}+\frac{185}{917}a^{2}-\frac{58}{917}a$, $\frac{1}{76\!\cdots\!57}a^{44}+\frac{87\!\cdots\!69}{76\!\cdots\!57}a^{43}-\frac{90\!\cdots\!58}{76\!\cdots\!57}a^{42}-\frac{32\!\cdots\!13}{76\!\cdots\!57}a^{41}-\frac{60\!\cdots\!20}{76\!\cdots\!57}a^{40}-\frac{69\!\cdots\!20}{76\!\cdots\!57}a^{39}-\frac{52\!\cdots\!94}{76\!\cdots\!57}a^{38}+\frac{22\!\cdots\!08}{76\!\cdots\!57}a^{37}-\frac{40\!\cdots\!95}{76\!\cdots\!57}a^{36}-\frac{10\!\cdots\!18}{76\!\cdots\!57}a^{35}-\frac{88\!\cdots\!32}{76\!\cdots\!57}a^{34}+\frac{35\!\cdots\!59}{10\!\cdots\!51}a^{33}-\frac{21\!\cdots\!85}{76\!\cdots\!57}a^{32}+\frac{66\!\cdots\!08}{76\!\cdots\!57}a^{31}+\frac{26\!\cdots\!25}{10\!\cdots\!51}a^{30}-\frac{22\!\cdots\!80}{76\!\cdots\!57}a^{29}-\frac{54\!\cdots\!95}{10\!\cdots\!51}a^{28}-\frac{61\!\cdots\!22}{76\!\cdots\!57}a^{27}-\frac{68\!\cdots\!91}{76\!\cdots\!57}a^{26}+\frac{26\!\cdots\!11}{76\!\cdots\!57}a^{25}+\frac{36\!\cdots\!82}{76\!\cdots\!57}a^{24}+\frac{24\!\cdots\!37}{76\!\cdots\!57}a^{23}-\frac{22\!\cdots\!23}{76\!\cdots\!57}a^{22}-\frac{13\!\cdots\!98}{76\!\cdots\!57}a^{21}-\frac{28\!\cdots\!83}{76\!\cdots\!57}a^{20}+\frac{66\!\cdots\!50}{76\!\cdots\!57}a^{19}+\frac{10\!\cdots\!25}{76\!\cdots\!57}a^{18}-\frac{10\!\cdots\!68}{76\!\cdots\!57}a^{17}+\frac{62\!\cdots\!84}{76\!\cdots\!57}a^{16}-\frac{31\!\cdots\!44}{76\!\cdots\!57}a^{15}-\frac{16\!\cdots\!40}{76\!\cdots\!57}a^{14}+\frac{49\!\cdots\!91}{10\!\cdots\!51}a^{13}+\frac{33\!\cdots\!97}{76\!\cdots\!57}a^{12}+\frac{12\!\cdots\!87}{76\!\cdots\!57}a^{11}+\frac{23\!\cdots\!35}{76\!\cdots\!57}a^{10}+\frac{32\!\cdots\!77}{10\!\cdots\!51}a^{9}-\frac{83\!\cdots\!24}{76\!\cdots\!57}a^{8}+\frac{29\!\cdots\!03}{10\!\cdots\!51}a^{7}-\frac{15\!\cdots\!22}{76\!\cdots\!57}a^{6}+\frac{12\!\cdots\!58}{76\!\cdots\!57}a^{5}-\frac{36\!\cdots\!98}{76\!\cdots\!57}a^{4}+\frac{31\!\cdots\!75}{76\!\cdots\!57}a^{3}-\frac{36\!\cdots\!60}{10\!\cdots\!51}a^{2}-\frac{23\!\cdots\!04}{76\!\cdots\!57}a-\frac{11\!\cdots\!22}{58\!\cdots\!47}$

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

Class group and class number

not computed

Unit group

Rank:		$44$
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

$C_{45}$ (as 45T1):

A cyclic group of order 45

The 45 conjugacy class representatives for $C_{45}$

Character table for $C_{45}$

Intermediate fields

3.3.657721.1, 5.5.432596913841.1, 9.9.187140089864757577373281.1, 15.15.53246609481207900818015621743146524650441.1

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	$15^{3}$	$45$	$45$	${\href{/padicField/7.3.0.1}{3} }^{15}$	$45$	$45$	${\href{/padicField/17.9.0.1}{9} }^{5}$	$15^{3}$	$45$	$15^{3}$	$45$	$15^{3}$	${\href{/padicField/41.5.0.1}{5} }^{9}$	$45$	${\href{/padicField/47.3.0.1}{3} }^{15}$	$15^{3}$	${\href{/padicField/59.5.0.1}{5} }^{9}$

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
\(811\)		Deg $45$	$45$	$1$	$44$