Number field 45.45.1589614467476880833110177740032706464094625919803624842629712851600976752423684909407990098956118088052488259210418247473121.1

• Label: 45.45.158...121.1
• Degree: $45$
• Signature: $[45, 0]$
• Discriminant: $1.590\times 10^{123}$
• Root discriminant: \(546.77\)
• Ramified prime: $631$
• Class number: not computed
• Class group: not computed
• Galois group: $C_{45}$ (as 45T1)

Normalized defining polynomial

x^45 - x^44 - 308*x^43 + 243*x^42 + 42031*x^41 - 29629*x^40 - 3398056*x^39 + 2297498*x^38 + 183048901*x^37 - 121032253*x^36 - 6996152064*x^35 + 4446576994*x^34 + 197152571564*x^33 - 115431928666*x^32 - 4201483097868*x^31 + 2121963890250*x^30 + 68878837826074*x^29 - 27147209842906*x^28 - 878308230905331*x^27 + 225497065522490*x^26 + 8763539163452549*x^25 - 871990800564410*x^24 - 68512286719810484*x^23 - 4780528095474003*x^22 + 418372779020858328*x^21 + 101848276044631524*x^20 - 1980431115576277921*x^19 - 822214007057581210*x^18 + 7177072759573430673*x^17 + 4169540206326707286*x^16 - 19552998002023848240*x^15 - 14466898842903568113*x^14 + 39018305246160367761*x^13 + 34927937499502295142*x^12 - 54906348400741785697*x^11 - 58013755547611372583*x^10 + 51353626114208160219*x^9 + 64112838817587499528*x^8 - 28749362416671453687*x^7 - 44330490069503914218*x^6 + 7543106125554272962*x^5 + 17141372833656407962*x^4 - 119032548118113070*x^3 - 2928899438689275293*x^2 - 140090069545874565*x + 115829050127578637

Invariants

Degree:		$45$
Signature:		$[45, 0]$
Discriminant:		1589614467476880833110177740032706464094625919803624842629712851600976752423684909407990098956118088052488259210418247473121
Root discriminant:		\(546.77\)
Galois root discriminant:		$631^{44/45}\approx 546.7722747055357$
Ramified primes:		\(631\)
Discriminant root field:		\(\Q\)
$\card{ \Gal(K/\Q) }$:		$45$
This field is Galois and abelian over $\Q$.
Conductor:		\(631\)
Dirichlet character group:		$\lbrace$$\chi_{631}(128,·)$, $\chi_{631}(1,·)$, $\chi_{631}(2,·)$, $\chi_{631}(4,·)$, $\chi_{631}(8,·)$, $\chi_{631}(393,·)$, $\chi_{631}(256,·)$, $\chi_{631}(16,·)$, $\chi_{631}(512,·)$, $\chi_{631}(279,·)$, $\chi_{631}(281,·)$, $\chi_{631}(155,·)$, $\chi_{631}(158,·)$, $\chi_{631}(543,·)$, $\chi_{631}(32,·)$, $\chi_{631}(43,·)$, $\chi_{631}(172,·)$, $\chi_{631}(242,·)$, $\chi_{631}(558,·)$, $\chi_{631}(47,·)$, $\chi_{631}(562,·)$, $\chi_{631}(310,·)$, $\chi_{631}(57,·)$, $\chi_{631}(188,·)$, $\chi_{631}(64,·)$, $\chi_{631}(455,·)$, $\chi_{631}(456,·)$, $\chi_{631}(587,·)$, $\chi_{631}(79,·)$, $\chi_{631}(337,·)$, $\chi_{631}(339,·)$, $\chi_{631}(86,·)$, $\chi_{631}(344,·)$, $\chi_{631}(228,·)$, $\chi_{631}(94,·)$, $\chi_{631}(609,·)$, $\chi_{631}(355,·)$, $\chi_{631}(484,·)$, $\chi_{631}(485,·)$, $\chi_{631}(316,·)$, $\chi_{631}(620,·)$, $\chi_{631}(493,·)$, $\chi_{631}(114,·)$, $\chi_{631}(376,·)$, $\chi_{631}(121,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $a^{40}$, $a^{41}$, $\frac{1}{132730249}a^{42}-\frac{42528338}{132730249}a^{41}-\frac{12696122}{132730249}a^{40}+\frac{56825354}{132730249}a^{39}-\frac{39123238}{132730249}a^{38}-\frac{50872548}{132730249}a^{37}-\frac{64842237}{132730249}a^{36}-\frac{47099584}{132730249}a^{35}-\frac{20636773}{132730249}a^{34}+\frac{30218358}{132730249}a^{33}-\frac{10890018}{132730249}a^{32}-\frac{48928993}{132730249}a^{31}+\frac{56667538}{132730249}a^{30}+\frac{26958113}{132730249}a^{29}-\frac{55196071}{132730249}a^{28}+\frac{10838373}{132730249}a^{27}-\frac{30945685}{132730249}a^{26}-\frac{15708082}{132730249}a^{25}-\frac{19889178}{132730249}a^{24}+\frac{54018503}{132730249}a^{23}+\frac{27685572}{132730249}a^{22}+\frac{2049416}{132730249}a^{21}+\frac{36934524}{132730249}a^{20}+\frac{60352851}{132730249}a^{19}-\frac{15378953}{132730249}a^{18}+\frac{29310757}{132730249}a^{17}+\frac{56130862}{132730249}a^{16}+\frac{18969348}{132730249}a^{15}+\frac{56166267}{132730249}a^{14}+\frac{49736682}{132730249}a^{13}+\frac{45799638}{132730249}a^{12}+\frac{28260367}{132730249}a^{11}-\frac{23002832}{132730249}a^{10}-\frac{36477398}{132730249}a^{9}+\frac{52772561}{132730249}a^{8}-\frac{40917792}{132730249}a^{7}+\frac{46983586}{132730249}a^{6}+\frac{41628748}{132730249}a^{5}+\frac{45294637}{132730249}a^{4}+\frac{62633709}{132730249}a^{3}+\frac{17971998}{132730249}a^{2}-\frac{1160809}{132730249}a+\frac{41765217}{132730249}$, $\frac{1}{87734694589}a^{43}+\frac{109}{87734694589}a^{42}-\frac{2568494555}{87734694589}a^{41}-\frac{14109422321}{87734694589}a^{40}-\frac{3937667910}{87734694589}a^{39}-\frac{8179139952}{87734694589}a^{38}+\frac{18722611459}{87734694589}a^{37}-\frac{23291387619}{87734694589}a^{36}-\frac{28978233154}{87734694589}a^{35}-\frac{24329880522}{87734694589}a^{34}-\frac{33021385400}{87734694589}a^{33}-\frac{32197885846}{87734694589}a^{32}+\frac{10820853119}{87734694589}a^{31}+\frac{8971626776}{87734694589}a^{30}+\frac{43433873832}{87734694589}a^{29}-\frac{30356686403}{87734694589}a^{28}+\frac{17025277919}{87734694589}a^{27}-\frac{37850058630}{87734694589}a^{26}-\frac{43121154825}{87734694589}a^{25}-\frac{34675820045}{87734694589}a^{24}-\frac{30216000649}{87734694589}a^{23}-\frac{16629705968}{87734694589}a^{22}-\frac{11941629025}{87734694589}a^{21}-\frac{25603158690}{87734694589}a^{20}-\frac{3574109214}{87734694589}a^{19}-\frac{8417996031}{87734694589}a^{18}-\frac{5321977432}{87734694589}a^{17}+\frac{29700892225}{87734694589}a^{16}-\frac{30403314929}{87734694589}a^{15}-\frac{29836500863}{87734694589}a^{14}-\frac{39686810948}{87734694589}a^{13}-\frac{38322360126}{87734694589}a^{12}-\frac{34099146298}{87734694589}a^{11}-\frac{24153175518}{87734694589}a^{10}-\frac{42382311094}{87734694589}a^{9}-\frac{33409357279}{87734694589}a^{8}+\frac{26727078284}{87734694589}a^{7}-\frac{37926450890}{87734694589}a^{6}+\frac{39235976824}{87734694589}a^{5}-\frac{7984893018}{87734694589}a^{4}+\frac{41015519743}{87734694589}a^{3}-\frac{35707175979}{87734694589}a^{2}-\frac{36705194817}{87734694589}a+\frac{11501826015}{87734694589}$, $\frac{1}{37\!\cdots\!81}a^{44}-\frac{20\!\cdots\!42}{37\!\cdots\!81}a^{43}+\frac{65\!\cdots\!34}{37\!\cdots\!81}a^{42}-\frac{15\!\cdots\!90}{37\!\cdots\!81}a^{41}+\frac{16\!\cdots\!05}{37\!\cdots\!81}a^{40}+\frac{17\!\cdots\!02}{37\!\cdots\!81}a^{39}+\frac{65\!\cdots\!06}{37\!\cdots\!81}a^{38}+\frac{67\!\cdots\!63}{37\!\cdots\!81}a^{37}-\frac{68\!\cdots\!62}{37\!\cdots\!81}a^{36}+\frac{10\!\cdots\!83}{37\!\cdots\!81}a^{35}+\frac{17\!\cdots\!04}{37\!\cdots\!81}a^{34}+\frac{11\!\cdots\!54}{37\!\cdots\!81}a^{33}+\frac{14\!\cdots\!40}{37\!\cdots\!81}a^{32}-\frac{10\!\cdots\!98}{37\!\cdots\!81}a^{31}+\frac{13\!\cdots\!46}{37\!\cdots\!81}a^{30}-\frac{12\!\cdots\!22}{37\!\cdots\!81}a^{29}+\frac{12\!\cdots\!57}{37\!\cdots\!81}a^{28}-\frac{17\!\cdots\!26}{37\!\cdots\!81}a^{27}+\frac{10\!\cdots\!62}{37\!\cdots\!81}a^{26}+\frac{72\!\cdots\!33}{37\!\cdots\!81}a^{25}-\frac{61\!\cdots\!63}{37\!\cdots\!81}a^{24}-\frac{11\!\cdots\!70}{37\!\cdots\!81}a^{23}-\frac{95\!\cdots\!71}{37\!\cdots\!81}a^{22}+\frac{64\!\cdots\!33}{37\!\cdots\!81}a^{21}-\frac{55\!\cdots\!94}{37\!\cdots\!81}a^{20}-\frac{10\!\cdots\!90}{37\!\cdots\!81}a^{19}-\frac{16\!\cdots\!04}{37\!\cdots\!81}a^{18}+\frac{13\!\cdots\!65}{37\!\cdots\!81}a^{17}-\frac{79\!\cdots\!00}{37\!\cdots\!81}a^{16}-\frac{46\!\cdots\!70}{37\!\cdots\!81}a^{15}+\frac{62\!\cdots\!92}{37\!\cdots\!81}a^{14}-\frac{63\!\cdots\!44}{37\!\cdots\!81}a^{13}+\frac{14\!\cdots\!38}{37\!\cdots\!81}a^{12}+\frac{13\!\cdots\!65}{37\!\cdots\!81}a^{11}-\frac{16\!\cdots\!43}{37\!\cdots\!81}a^{10}+\frac{11\!\cdots\!89}{37\!\cdots\!81}a^{9}+\frac{67\!\cdots\!41}{37\!\cdots\!81}a^{8}-\frac{33\!\cdots\!25}{37\!\cdots\!81}a^{7}+\frac{13\!\cdots\!01}{37\!\cdots\!81}a^{6}+\frac{50\!\cdots\!98}{37\!\cdots\!81}a^{5}+\frac{40\!\cdots\!88}{37\!\cdots\!81}a^{4}-\frac{17\!\cdots\!49}{37\!\cdots\!81}a^{3}-\frac{79\!\cdots\!21}{37\!\cdots\!81}a^{2}-\frac{15\!\cdots\!05}{37\!\cdots\!81}a-\frac{70\!\cdots\!36}{37\!\cdots\!81}$

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

3.3.398161.1, 5.5.158532181921.1, 9.9.25132452704633039250241.1, 15.15.1586393893300992738237335763411468444721.1

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	$45$	$45$	${\href{/padicField/5.5.0.1}{5} }^{9}$	$45$	$45$	$45$	$45$	$45$	$45$	$45$	$45$	${\href{/padicField/37.9.0.1}{9} }^{5}$	${\href{/padicField/41.9.0.1}{9} }^{5}$	${\href{/padicField/43.3.0.1}{3} }^{15}$	$45$	$45$	$45$

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
\(631\)		Deg $45$	$45$	$1$	$44$