Number field 45.45.612823173021751971297676000847719042376493061735873445231828455051262946540529457157884846548736505117837633099361.1

• Label: 45.45.612...361.1
• Degree: $45$
• Signature: $[45, 0]$
• Discriminant: $6.128\times 10^{113}$
• Root discriminant: \(337.76\)
• Ramified primes: $19,31$
• Class number: not computed
• Class group: not computed
• Galois group: $C_{45}$ (as 45T1)

Normalized defining polynomial

x^45 - x^44 - 250*x^43 - 57*x^42 + 27766*x^41 + 34148*x^40 - 1801054*x^39 - 3701713*x^38 + 75995397*x^37 + 210980315*x^36 - 2204269403*x^35 - 7656169786*x^34 + 45208500184*x^33 + 192019854445*x^32 - 660457840142*x^31 - 3478319662546*x^30 + 6740250648134*x^29 + 46682393472693*x^28 - 44214268231149*x^27 - 471014331370572*x^26 + 119457837724165*x^25 + 3599175944415558*x^24 + 920658436866528*x^23 - 20872201046519003*x^22 - 13494994939768731*x^21 + 91678287510298882*x^20 + 87997267245853232*x^19 - 303360026160716188*x^18 - 371303902979085816*x^17 + 749732931610001378*x^16 + 1094855896004641434*x^15 - 1366727950240442050*x^14 - 2306120975093560007*x^13 + 1803267431605382487*x^12 + 3471556977526533885*x^11 - 1666809346078423573*x^10 - 3682047656706158942*x^9 + 1008367868417943528*x^8 + 2668021411897921137*x^7 - 327662726329807390*x^6 - 1248245094003212175*x^5 + 152924933759221*x^4 + 338163088283119511*x^3 + 38015736547359269*x^2 - 40401517332214916*x - 8937046621536643

Invariants

Degree:		$45$
Signature:		$[45, 0]$
Discriminant:		612823173021751971297676000847719042376493061735873445231828455051262946540529457157884846548736505117837633099361
Root discriminant:		\(337.76\)
Galois root discriminant:		$19^{8/9}31^{14/15}\approx 337.75940307002355$
Ramified primes:		\(19\), \(31\)
Discriminant root field:		\(\Q\)
$\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$:		$C_{45}$
This field is Galois and abelian over $\Q$.
Conductor:		\(589=19\cdot 31\)
Dirichlet character group:		$\lbrace$$\chi_{589}(1,·)$, $\chi_{589}(131,·)$, $\chi_{589}(9,·)$, $\chi_{589}(138,·)$, $\chi_{589}(140,·)$, $\chi_{589}(397,·)$, $\chi_{589}(142,·)$, $\chi_{589}(536,·)$, $\chi_{589}(149,·)$, $\chi_{589}(408,·)$, $\chi_{589}(543,·)$, $\chi_{589}(289,·)$, $\chi_{589}(419,·)$, $\chi_{589}(39,·)$, $\chi_{589}(169,·)$, $\chi_{589}(175,·)$, $\chi_{589}(562,·)$, $\chi_{589}(438,·)$, $\chi_{589}(311,·)$, $\chi_{589}(159,·)$, $\chi_{589}(64,·)$, $\chi_{589}(196,·)$, $\chi_{589}(453,·)$, $\chi_{589}(586,·)$, $\chi_{589}(80,·)$, $\chi_{589}(81,·)$, $\chi_{589}(82,·)$, $\chi_{589}(163,·)$, $\chi_{589}(214,·)$, $\chi_{589}(343,·)$, $\chi_{589}(472,·)$, $\chi_{589}(346,·)$, $\chi_{589}(349,·)$, $\chi_{589}(351,·)$, $\chi_{589}(443,·)$, $\chi_{589}(100,·)$, $\chi_{589}(237,·)$, $\chi_{589}(366,·)$, $\chi_{589}(125,·)$, $\chi_{589}(112,·)$, $\chi_{589}(467,·)$, $\chi_{589}(245,·)$, $\chi_{589}(576,·)$, $\chi_{589}(253,·)$, $\chi_{589}(510,·)$$\rbrace$
This is not a CM field.
This field has no CM subfields.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $a^{40}$, $a^{41}$, $\frac{1}{563}a^{42}+\frac{232}{563}a^{41}-\frac{61}{563}a^{40}-\frac{120}{563}a^{39}-\frac{20}{563}a^{38}+\frac{240}{563}a^{37}+\frac{122}{563}a^{36}-\frac{153}{563}a^{35}+\frac{80}{563}a^{34}-\frac{184}{563}a^{33}-\frac{54}{563}a^{32}-\frac{198}{563}a^{31}-\frac{66}{563}a^{30}+\frac{177}{563}a^{29}-\frac{47}{563}a^{28}-\frac{200}{563}a^{27}-\frac{28}{563}a^{26}+\frac{151}{563}a^{25}-\frac{50}{563}a^{24}+\frac{122}{563}a^{23}+\frac{87}{563}a^{22}+\frac{170}{563}a^{21}-\frac{71}{563}a^{20}-\frac{245}{563}a^{19}-\frac{34}{563}a^{18}+\frac{142}{563}a^{17}-\frac{37}{563}a^{16}-\frac{142}{563}a^{15}-\frac{82}{563}a^{14}+\frac{82}{563}a^{13}-\frac{225}{563}a^{12}+\frac{235}{563}a^{11}+\frac{262}{563}a^{10}+\frac{67}{563}a^{9}-\frac{146}{563}a^{8}-\frac{5}{563}a^{7}+\frac{237}{563}a^{6}-\frac{28}{563}a^{5}+\frac{182}{563}a^{4}+\frac{155}{563}a^{3}+\frac{163}{563}a^{2}-\frac{44}{563}a-\frac{196}{563}$, $\frac{1}{563}a^{43}+\frac{163}{563}a^{41}-\frac{43}{563}a^{40}+\frac{233}{563}a^{39}-\frac{187}{563}a^{38}+\frac{179}{563}a^{37}+\frac{256}{563}a^{36}+\frac{107}{563}a^{35}-\frac{165}{563}a^{34}-\frac{154}{563}a^{33}-\frac{56}{563}a^{32}+\frac{267}{563}a^{31}-\frac{275}{563}a^{30}-\frac{12}{563}a^{29}+\frac{7}{563}a^{28}+\frac{206}{563}a^{27}-\frac{109}{563}a^{26}-\frac{176}{563}a^{25}-\frac{101}{563}a^{24}-\frac{67}{563}a^{23}+\frac{254}{563}a^{22}-\frac{101}{563}a^{21}-\frac{100}{563}a^{20}-\frac{57}{563}a^{19}+\frac{148}{563}a^{18}+\frac{236}{563}a^{17}-\frac{3}{563}a^{16}+\frac{208}{563}a^{15}-\frac{36}{563}a^{14}-\frac{107}{563}a^{13}+\frac{76}{563}a^{12}-\frac{210}{563}a^{11}+\frac{87}{563}a^{10}+\frac{74}{563}a^{9}+\frac{87}{563}a^{8}+\frac{271}{563}a^{7}+\frac{162}{563}a^{6}-\frac{78}{563}a^{5}+\frac{156}{563}a^{4}+\frac{235}{563}a^{3}-\frac{139}{563}a^{2}-\frac{122}{563}a-\frac{131}{563}$, $\frac{1}{59\cdots 99}a^{44}-\frac{48\cdots 39}{59\cdots 99}a^{43}+\frac{50\cdots 37}{59\cdots 99}a^{42}+\frac{15\cdots 07}{59\cdots 99}a^{41}+\frac{35\cdots 30}{59\cdots 99}a^{40}+\frac{21\cdots 38}{59\cdots 99}a^{39}+\frac{18\cdots 94}{59\cdots 99}a^{38}+\frac{25\cdots 13}{59\cdots 99}a^{37}-\frac{25\cdots 34}{59\cdots 99}a^{36}-\frac{18\cdots 41}{59\cdots 99}a^{35}+\frac{26\cdots 18}{59\cdots 99}a^{34}+\frac{26\cdots 64}{59\cdots 99}a^{33}-\frac{29\cdots 44}{59\cdots 99}a^{32}+\frac{81\cdots 68}{59\cdots 99}a^{31}-\frac{95\cdots 62}{59\cdots 99}a^{30}-\frac{13\cdots 81}{59\cdots 99}a^{29}+\frac{23\cdots 19}{59\cdots 99}a^{28}+\frac{15\cdots 17}{59\cdots 99}a^{27}-\frac{21\cdots 04}{59\cdots 99}a^{26}+\frac{19\cdots 68}{59\cdots 99}a^{25}-\frac{53\cdots 28}{59\cdots 99}a^{24}+\frac{17\cdots 35}{59\cdots 99}a^{23}+\frac{11\cdots 45}{59\cdots 99}a^{22}+\frac{21\cdots 24}{59\cdots 99}a^{21}-\frac{14\cdots 53}{59\cdots 99}a^{20}-\frac{15\cdots 78}{59\cdots 99}a^{19}+\frac{18\cdots 97}{59\cdots 99}a^{18}+\frac{28\cdots 84}{59\cdots 99}a^{17}+\frac{82\cdots 99}{59\cdots 99}a^{16}-\frac{29\cdots 01}{59\cdots 99}a^{15}-\frac{27\cdots 47}{59\cdots 99}a^{14}-\frac{74\cdots 48}{59\cdots 99}a^{13}+\frac{28\cdots 25}{59\cdots 99}a^{12}+\frac{13\cdots 88}{59\cdots 99}a^{11}+\frac{10\cdots 16}{59\cdots 99}a^{10}+\frac{21\cdots 81}{59\cdots 99}a^{9}+\frac{14\cdots 33}{59\cdots 99}a^{8}-\frac{80\cdots 06}{59\cdots 99}a^{7}+\frac{24\cdots 99}{59\cdots 99}a^{6}+\frac{39\cdots 38}{59\cdots 99}a^{5}+\frac{14\cdots 62}{59\cdots 99}a^{4}-\frac{19\cdots 08}{59\cdots 99}a^{3}+\frac{13\cdots 35}{59\cdots 99}a^{2}-\frac{23\cdots 47}{59\cdots 99}a+\frac{94\cdots 39}{59\cdots 99}$

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

Class group and class number

Ideal class group:		not computed
Narrow class group:		not computed

Unit group

Rank:		$44$
Torsion generator:		\( -1 \)  (order $2$)
Fundamental units:		not computed
Regulator:		not computed

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr = \mathstrut &\frac{2^{45}\cdot(2\pi)^{0}\cdot R \cdot h}{2\cdot\sqrt{612823173021751971297676000847719042376493061735873445231828455051262946540529457157884846548736505117837633099361}}\cr\mathstrut & \text{ some values not computed } \end{aligned}\]

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

3.3.361.1, 5.5.923521.1, 9.9.15072974715383053921.2, 15.15.4829212716211581952447142935561.1

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	$45$	$45$	${\href{/padicField/5.9.0.1}{9} }^{5}$	$15^{3}$	$15^{3}$	$45$	$45$	R	$45$	$45$	R	${\href{/padicField/37.3.0.1}{3} }^{15}$	$45$	$45$	$45$	$45$	$45$

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
\(19\)		Deg $45$	$9$	$5$	$40$
\(31\)	31.1.15.14a1.12	$x^{15} + 775$	$15$	$1$	$14$	$C_{15}$	$$[\ ]_{15}$$
	31.1.15.14a1.12	$x^{15} + 775$	$15$	$1$	$14$	$C_{15}$	$$[\ ]_{15}$$
	31.1.15.14a1.12	$x^{15} + 775$	$15$	$1$	$14$	$C_{15}$	$$[\ ]_{15}$$

Spectrum of ring of integers