Number field 45.45.1822550894130687310163630869707530762635914062521200556148980182659284427677118541704622955169252001171341389227410853361.1

• Label: 45.45.182...361.1
• Degree: $45$
• Signature: $[45, 0]$
• Discriminant: $1.823\times 10^{120}$
• Root discriminant: \(470.39\)
• Ramified prime: $541$
• Class number: not computed
• Class group: not computed
• Galois group: $C_{45}$ (as 45T1)

Normalized defining polynomial

x^45 - x^44 - 264*x^43 + 577*x^42 + 30504*x^41 - 98120*x^40 - 2018702*x^39 + 8460944*x^38 + 84492738*x^37 - 444867299*x^36 - 2320200050*x^35 + 15600272993*x^34 + 41082574326*x^33 - 383669845519*x^32 - 401615692215*x^31 + 6813186987549*x^30 - 248705213777*x^29 - 88665690143496*x^28 + 76259461552137*x^27 + 847486273951872*x^26 - 1335888656272773*x^25 - 5868838401907173*x^24 + 13675801545862322*x^23 + 28183961581580635*x^22 - 95456165563516555*x^21 - 81692357941176698*x^20 + 471789373754560507*x^19 + 45987513402865445*x^18 - 1649280428399137801*x^17 + 754474066803116171*x^16 + 3953734685891549836*x^15 - 3735967195252037225*x^14 - 5995511996337817154*x^13 + 9200133845176559614*x^12 + 4440745936331373641*x^11 - 13174072290529526814*x^10 + 1114529216279611047*x^9 + 10493237142444370059*x^8 - 4803592251955912900*x^7 - 3788065733722662878*x^6 + 3204309942238309333*x^5 + 171253707101259988*x^4 - 725658632317293676*x^3 + 133177483664861744*x^2 + 42013292907430397*x - 11589172153099133

Invariants

Degree:		$45$
Signature:		$[45, 0]$
Discriminant:		1822550894130687310163630869707530762635914062521200556148980182659284427677118541704622955169252001171341389227410853361
Root discriminant:		\(470.39\)
Galois root discriminant:		$541^{44/45}\approx 470.3915895630539$
Ramified primes:		\(541\)
Discriminant root field:		\(\Q\)
$\card{ \Gal(K/\Q) }$:		$45$
This field is Galois and abelian over $\Q$.
Conductor:		\(541\)
Dirichlet character group:		$\lbrace$$\chi_{541}(1,·)$, $\chi_{541}(129,·)$, $\chi_{541}(265,·)$, $\chi_{541}(140,·)$, $\chi_{541}(15,·)$, $\chi_{541}(400,·)$, $\chi_{541}(277,·)$, $\chi_{541}(534,·)$, $\chi_{541}(411,·)$, $\chi_{541}(27,·)$, $\chi_{541}(241,·)$, $\chi_{541}(174,·)$, $\chi_{541}(48,·)$, $\chi_{541}(49,·)$, $\chi_{541}(179,·)$, $\chi_{541}(436,·)$, $\chi_{541}(309,·)$, $\chi_{541}(521,·)$, $\chi_{541}(312,·)$, $\chi_{541}(188,·)$, $\chi_{541}(446,·)$, $\chi_{541}(448,·)$, $\chi_{541}(194,·)$, $\chi_{541}(198,·)$, $\chi_{541}(205,·)$, $\chi_{541}(207,·)$, $\chi_{541}(214,·)$, $\chi_{541}(477,·)$, $\chi_{541}(352,·)$, $\chi_{541}(225,·)$, $\chi_{541}(228,·)$, $\chi_{541}(102,·)$, $\chi_{541}(307,·)$, $\chi_{541}(124,·)$, $\chi_{541}(237,·)$, $\chi_{541}(110,·)$, $\chi_{541}(368,·)$, $\chi_{541}(369,·)$, $\chi_{541}(370,·)$, $\chi_{541}(115,·)$, $\chi_{541}(505,·)$, $\chi_{541}(122,·)$, $\chi_{541}(252,·)$, $\chi_{541}(125,·)$, $\chi_{541}(405,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $a^{40}$, $a^{41}$, $a^{42}$, $\frac{1}{770977}a^{43}+\frac{293292}{770977}a^{42}-\frac{83506}{770977}a^{41}+\frac{79994}{770977}a^{40}+\frac{5316}{770977}a^{39}+\frac{330939}{770977}a^{38}-\frac{78394}{770977}a^{37}+\frac{381110}{770977}a^{36}+\frac{266830}{770977}a^{35}-\frac{336792}{770977}a^{34}-\frac{228610}{770977}a^{33}+\frac{241831}{770977}a^{32}-\frac{112139}{770977}a^{31}-\frac{53412}{770977}a^{30}-\frac{174350}{770977}a^{29}+\frac{27724}{770977}a^{28}-\frac{328825}{770977}a^{27}-\frac{253388}{770977}a^{26}+\frac{158669}{770977}a^{25}-\frac{293386}{770977}a^{24}-\frac{207655}{770977}a^{23}-\frac{62720}{770977}a^{22}+\frac{346040}{770977}a^{21}-\frac{365658}{770977}a^{20}-\frac{176602}{770977}a^{19}-\frac{36212}{770977}a^{18}-\frac{72988}{770977}a^{17}+\frac{273794}{770977}a^{16}-\frac{220528}{770977}a^{15}+\frac{92685}{770977}a^{14}-\frac{113494}{770977}a^{13}+\frac{3938}{770977}a^{12}-\frac{341521}{770977}a^{11}+\frac{36758}{770977}a^{10}-\frac{332276}{770977}a^{9}-\frac{163732}{770977}a^{8}+\frac{111589}{770977}a^{7}+\frac{267872}{770977}a^{6}-\frac{40710}{770977}a^{5}+\frac{52653}{770977}a^{4}-\frac{269383}{770977}a^{3}+\frac{316387}{770977}a^{2}+\frac{44660}{770977}a+\frac{252673}{770977}$, $\frac{1}{49\!\cdots\!77}a^{44}-\frac{21\!\cdots\!66}{49\!\cdots\!77}a^{43}+\frac{22\!\cdots\!87}{49\!\cdots\!77}a^{42}-\frac{19\!\cdots\!02}{49\!\cdots\!77}a^{41}+\frac{73\!\cdots\!14}{49\!\cdots\!77}a^{40}+\frac{17\!\cdots\!29}{49\!\cdots\!77}a^{39}-\frac{19\!\cdots\!96}{49\!\cdots\!77}a^{38}+\frac{15\!\cdots\!19}{49\!\cdots\!77}a^{37}+\frac{13\!\cdots\!88}{49\!\cdots\!77}a^{36}-\frac{74\!\cdots\!31}{49\!\cdots\!77}a^{35}+\frac{16\!\cdots\!06}{49\!\cdots\!77}a^{34}-\frac{11\!\cdots\!12}{49\!\cdots\!77}a^{33}+\frac{16\!\cdots\!12}{49\!\cdots\!77}a^{32}-\frac{14\!\cdots\!84}{49\!\cdots\!77}a^{31}-\frac{25\!\cdots\!53}{49\!\cdots\!77}a^{30}-\frac{13\!\cdots\!30}{49\!\cdots\!77}a^{29}-\frac{86\!\cdots\!44}{49\!\cdots\!77}a^{28}-\frac{24\!\cdots\!23}{49\!\cdots\!77}a^{27}+\frac{22\!\cdots\!60}{49\!\cdots\!77}a^{26}+\frac{22\!\cdots\!03}{49\!\cdots\!77}a^{25}-\frac{17\!\cdots\!65}{49\!\cdots\!77}a^{24}-\frac{83\!\cdots\!80}{49\!\cdots\!77}a^{23}+\frac{20\!\cdots\!62}{49\!\cdots\!77}a^{22}-\frac{16\!\cdots\!84}{49\!\cdots\!77}a^{21}-\frac{24\!\cdots\!64}{49\!\cdots\!77}a^{20}-\frac{16\!\cdots\!88}{49\!\cdots\!77}a^{19}-\frac{11\!\cdots\!21}{49\!\cdots\!77}a^{18}+\frac{91\!\cdots\!17}{49\!\cdots\!77}a^{17}+\frac{23\!\cdots\!37}{49\!\cdots\!77}a^{16}+\frac{11\!\cdots\!13}{49\!\cdots\!77}a^{15}-\frac{11\!\cdots\!80}{49\!\cdots\!77}a^{14}-\frac{11\!\cdots\!01}{49\!\cdots\!77}a^{13}-\frac{10\!\cdots\!26}{49\!\cdots\!77}a^{12}-\frac{32\!\cdots\!07}{49\!\cdots\!77}a^{11}+\frac{20\!\cdots\!95}{49\!\cdots\!77}a^{10}-\frac{24\!\cdots\!90}{49\!\cdots\!77}a^{9}-\frac{12\!\cdots\!73}{49\!\cdots\!77}a^{8}+\frac{14\!\cdots\!43}{49\!\cdots\!77}a^{7}-\frac{22\!\cdots\!90}{49\!\cdots\!77}a^{6}-\frac{26\!\cdots\!06}{49\!\cdots\!77}a^{5}+\frac{44\!\cdots\!66}{49\!\cdots\!77}a^{4}+\frac{68\!\cdots\!65}{49\!\cdots\!77}a^{3}-\frac{98\!\cdots\!30}{49\!\cdots\!77}a^{2}+\frac{15\!\cdots\!98}{49\!\cdots\!77}a-\frac{11\!\cdots\!15}{49\!\cdots\!77}$

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 $C_{45}$ is not computed

Intermediate fields

3.3.292681.1, 5.5.85662167761.1, 9.9.7338006985513707753121.1, 15.15.183976228451617878805132559872306799161.1

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	$45$	$45$	$45$	$15^{3}$	${\href{/padicField/11.9.0.1}{9} }^{5}$	$45$	${\href{/padicField/17.5.0.1}{5} }^{9}$	$45$	$45$	${\href{/padicField/29.5.0.1}{5} }^{9}$	${\href{/padicField/31.9.0.1}{9} }^{5}$	$45$	${\href{/padicField/41.9.0.1}{9} }^{5}$	$45$	$45$	$45$	$45$

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
\(541\)		Deg $45$	$45$	$1$	$44$