Global number field 33.33.36584611296554742180833097810429342639777502523008874222975105176339833601.1

• Label: 33.33.3658461129...3601.1
• Degree: $33$
• Signature: $[33, 0]$
• Discriminant: $199^{32}$
• Root discriminant: $169.51$
• Ramified prime: $199$
• Class number: $1$ (GRH)
• Class group: Trivial (GRH)
• Galois group: $C_{33}$ (as 33T1)

Normalized defining polynomial

\( x^{33} - x^{32} - 96 x^{31} + 217 x^{30} + 3795 x^{29} - 13403 x^{28} - 74197 x^{27} + 394231 x^{26} + 599821 x^{25} - 6350170 x^{24} + 2832807 x^{23} + 56803105 x^{22} - 104532088 x^{21} - 244229488 x^{20} + 932758015 x^{19} + 5618002 x^{18} - 3890173018 x^{17} + 4529747891 x^{16} + 6495127532 x^{15} - 18110944809 x^{14} + 4574986912 x^{13} + 26694143816 x^{12} - 29645825157 x^{11} - 6037403432 x^{10} + 30417132332 x^{9} - 15468969217 x^{8} - 6737165737 x^{7} + 8515927088 x^{6} - 1017730658 x^{5} - 1409177433 x^{4} + 395068072 x^{3} + 75602260 x^{2} - 21210003 x - 2947097 \)

Invariants

Degree:		$33$
Signature:		$[33, 0]$
Discriminant:		\(36584611296554742180833097810429342639777502523008874222975105176339833601=199^{32}\)
Root discriminant:		$169.51$
Ramified primes:		$199$
This field is Galois and abelian over $\Q$.
Conductor:		\(199\)
Dirichlet character group:		$\lbrace$$\chi_{199}(1,·)$, $\chi_{199}(132,·)$, $\chi_{199}(5,·)$, $\chi_{199}(8,·)$, $\chi_{199}(139,·)$, $\chi_{199}(140,·)$, $\chi_{199}(144,·)$, $\chi_{199}(18,·)$, $\chi_{199}(25,·)$, $\chi_{199}(28,·)$, $\chi_{199}(157,·)$, $\chi_{199}(40,·)$, $\chi_{199}(172,·)$, $\chi_{199}(52,·)$, $\chi_{199}(182,·)$, $\chi_{199}(187,·)$, $\chi_{199}(188,·)$, $\chi_{199}(61,·)$, $\chi_{199}(62,·)$, $\chi_{199}(63,·)$, $\chi_{199}(64,·)$, $\chi_{199}(90,·)$, $\chi_{199}(92,·)$, $\chi_{199}(98,·)$, $\chi_{199}(103,·)$, $\chi_{199}(106,·)$, $\chi_{199}(111,·)$, $\chi_{199}(114,·)$, $\chi_{199}(116,·)$, $\chi_{199}(117,·)$, $\chi_{199}(121,·)$, $\chi_{199}(123,·)$, $\chi_{199}(125,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $\frac{1}{107} a^{28} - \frac{11}{107} a^{27} - \frac{43}{107} a^{26} - \frac{11}{107} a^{25} - \frac{25}{107} a^{24} + \frac{30}{107} a^{23} - \frac{12}{107} a^{22} - \frac{50}{107} a^{21} - \frac{37}{107} a^{19} + \frac{31}{107} a^{18} + \frac{33}{107} a^{16} + \frac{39}{107} a^{15} + \frac{9}{107} a^{14} + \frac{28}{107} a^{13} - \frac{34}{107} a^{12} - \frac{4}{107} a^{11} + \frac{13}{107} a^{10} - \frac{51}{107} a^{9} + \frac{4}{107} a^{8} + \frac{27}{107} a^{7} + \frac{45}{107} a^{6} - \frac{45}{107} a^{5} - \frac{14}{107} a^{4} + \frac{48}{107} a^{3} - \frac{42}{107} a^{2} + \frac{37}{107} a + \frac{34}{107}$, $\frac{1}{107} a^{29} + \frac{50}{107} a^{27} + \frac{51}{107} a^{26} - \frac{39}{107} a^{25} - \frac{31}{107} a^{24} - \frac{3}{107} a^{23} + \frac{32}{107} a^{22} - \frac{15}{107} a^{21} - \frac{37}{107} a^{20} + \frac{52}{107} a^{19} + \frac{20}{107} a^{18} + \frac{33}{107} a^{17} - \frac{26}{107} a^{16} + \frac{10}{107} a^{15} + \frac{20}{107} a^{14} - \frac{47}{107} a^{13} + \frac{50}{107} a^{12} - \frac{31}{107} a^{11} - \frac{15}{107} a^{10} - \frac{22}{107} a^{9} - \frac{36}{107} a^{8} + \frac{21}{107} a^{7} + \frac{22}{107} a^{6} + \frac{26}{107} a^{5} + \frac{1}{107} a^{4} - \frac{49}{107} a^{3} + \frac{3}{107} a^{2} + \frac{13}{107} a + \frac{53}{107}$, $\frac{1}{107} a^{30} - \frac{41}{107} a^{27} - \frac{29}{107} a^{26} - \frac{16}{107} a^{25} - \frac{37}{107} a^{24} + \frac{30}{107} a^{23} + \frac{50}{107} a^{22} + \frac{2}{107} a^{21} + \frac{52}{107} a^{20} + \frac{51}{107} a^{19} - \frac{19}{107} a^{18} - \frac{26}{107} a^{17} - \frac{35}{107} a^{16} - \frac{4}{107} a^{15} + \frac{38}{107} a^{14} + \frac{41}{107} a^{13} - \frac{43}{107} a^{12} - \frac{29}{107} a^{11} - \frac{30}{107} a^{10} + \frac{53}{107} a^{9} + \frac{35}{107} a^{8} - \frac{44}{107} a^{7} + \frac{23}{107} a^{6} + \frac{4}{107} a^{5} + \frac{9}{107} a^{4} - \frac{43}{107} a^{3} - \frac{27}{107} a^{2} + \frac{22}{107} a + \frac{12}{107}$, $\frac{1}{85279} a^{31} + \frac{200}{85279} a^{30} - \frac{310}{85279} a^{29} - \frac{350}{85279} a^{28} - \frac{1}{797} a^{27} + \frac{38955}{85279} a^{26} + \frac{35792}{85279} a^{25} - \frac{19460}{85279} a^{24} + \frac{21143}{85279} a^{23} - \frac{7552}{85279} a^{22} + \frac{18198}{85279} a^{21} - \frac{4936}{85279} a^{20} + \frac{40911}{85279} a^{19} - \frac{16288}{85279} a^{18} + \frac{34825}{85279} a^{17} - \frac{26368}{85279} a^{16} - \frac{23938}{85279} a^{15} + \frac{265}{85279} a^{14} - \frac{11498}{85279} a^{13} + \frac{1678}{85279} a^{12} + \frac{1913}{85279} a^{11} + \frac{30745}{85279} a^{10} - \frac{4236}{85279} a^{9} - \frac{18537}{85279} a^{8} + \frac{22380}{85279} a^{7} + \frac{33206}{85279} a^{6} - \frac{9075}{85279} a^{5} - \frac{41842}{85279} a^{4} - \frac{39299}{85279} a^{3} - \frac{40303}{85279} a^{2} + \frac{25329}{85279} a - \frac{5597}{85279}$, $\frac{1}{102671498988363166441560189308232262476210556522544195280794021105929195422227} a^{32} - \frac{438897633968529796139850794391781373725155282827429592545742453754653758}{102671498988363166441560189308232262476210556522544195280794021105929195422227} a^{31} - \frac{110402546553769652630517426987523221265406509511904028477833912681295184433}{102671498988363166441560189308232262476210556522544195280794021105929195422227} a^{30} + \frac{320936223084897199657985786918830123142659206716931515171038855868669434494}{102671498988363166441560189308232262476210556522544195280794021105929195422227} a^{29} - \frac{374908578497306880128163102914250800633485143768155075842994856790433045172}{102671498988363166441560189308232262476210556522544195280794021105929195422227} a^{28} - \frac{30743493205113985994167010070949657887864805163144174475109645892975829004212}{102671498988363166441560189308232262476210556522544195280794021105929195422227} a^{27} + \frac{25356037735169413811459530807110345442706848648638267190150784388291670032583}{102671498988363166441560189308232262476210556522544195280794021105929195422227} a^{26} + \frac{23470517440383994415605927999524298738720707322662561013555149009748330766351}{102671498988363166441560189308232262476210556522544195280794021105929195422227} a^{25} + \frac{12766944573729428846774254960177509984844734718343700669965983248748774722568}{102671498988363166441560189308232262476210556522544195280794021105929195422227} a^{24} + \frac{14975292975365784641082475344702493089707650002408493449000817364947429348061}{102671498988363166441560189308232262476210556522544195280794021105929195422227} a^{23} + \frac{14924745911544692110799056676101361776099767828562346927514756440921114882622}{102671498988363166441560189308232262476210556522544195280794021105929195422227} a^{22} + \frac{8216190614053508038159652735556314707653996180651639837177153203563934013960}{102671498988363166441560189308232262476210556522544195280794021105929195422227} a^{21} - \frac{22970992314794669892396731847871379849059884104718625186556050762178277931014}{102671498988363166441560189308232262476210556522544195280794021105929195422227} a^{20} + \frac{49955306356803724194778993886028674938590021815018447937533143805121191780330}{102671498988363166441560189308232262476210556522544195280794021105929195422227} a^{19} - \frac{5180635917730927010684352507506494639542192931970873600630632495259793960619}{102671498988363166441560189308232262476210556522544195280794021105929195422227} a^{18} + \frac{48517887467994969535184665359906021871428962537739738059228752747612860407782}{102671498988363166441560189308232262476210556522544195280794021105929195422227} a^{17} + \frac{32812338187970572656159713612624074377771435314801166903073838250896951444758}{102671498988363166441560189308232262476210556522544195280794021105929195422227} a^{16} - \frac{2616901621921753030397829196403387332080446536734032200863868763926503885258}{102671498988363166441560189308232262476210556522544195280794021105929195422227} a^{15} - \frac{21532901663148743214811689407860864906719378229739621308361510383169945793789}{102671498988363166441560189308232262476210556522544195280794021105929195422227} a^{14} + \frac{12054516188201319321522699841002176634071112345030785941738168900616811360095}{102671498988363166441560189308232262476210556522544195280794021105929195422227} a^{13} + \frac{31264541748829993117378410044826614269958018386165198311503746225434683453463}{102671498988363166441560189308232262476210556522544195280794021105929195422227} a^{12} - \frac{47169536983914598731633922943580927997368856169749918614618794685152540242283}{102671498988363166441560189308232262476210556522544195280794021105929195422227} a^{11} - \frac{43168804543004074938266684638479210697079315118805737814490130352217998791075}{102671498988363166441560189308232262476210556522544195280794021105929195422227} a^{10} + \frac{18614299057042667178075574727084464563060072240468685207631630064232530572928}{102671498988363166441560189308232262476210556522544195280794021105929195422227} a^{9} + \frac{33903009564876630326881250934235490396854251770235014190930531384625129631744}{102671498988363166441560189308232262476210556522544195280794021105929195422227} a^{8} + \frac{37454674054966490087073218660817366298874310763514168341148253768557006921844}{102671498988363166441560189308232262476210556522544195280794021105929195422227} a^{7} - \frac{2141071672468326349367177597543086434747190323340729965850954481465294789876}{102671498988363166441560189308232262476210556522544195280794021105929195422227} a^{6} + \frac{17367023243537628972790939335411424812139394598424997369813227291411361484558}{102671498988363166441560189308232262476210556522544195280794021105929195422227} a^{5} - \frac{47586298302744340489101125823909598017395052331123812728399255864118785771119}{102671498988363166441560189308232262476210556522544195280794021105929195422227} a^{4} - \frac{49768997760892950697834957884665557209186192196605180706922785696559925948084}{102671498988363166441560189308232262476210556522544195280794021105929195422227} a^{3} - \frac{25129704869123703134655166045989838573994498991929010347482234215437846523494}{102671498988363166441560189308232262476210556522544195280794021105929195422227} a^{2} + \frac{38404255889277382429064943474916715592440230230959646726738884900693020155634}{102671498988363166441560189308232262476210556522544195280794021105929195422227} a + \frac{36884515875702917506075331585791530761741418441044738793672822864730304131594}{102671498988363166441560189308232262476210556522544195280794021105929195422227}$

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

Unit group

Rank:		$32$
Torsion generator:		\( -1 \) (order $2$)
Fundamental units:		Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
Regulator:		\( 194708110268326050000000000 \) (assuming GRH)

Galois group

$C_{33}$ (as 33T1):

A cyclic group of order 33

The 33 conjugacy class representatives for $C_{33}$

Character table for $C_{33}$ is not computed

Intermediate fields

3.3.39601.1, 11.11.97393677359695041798001.1

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	$33$	$33$	${\href{/LocalNumberField/5.11.0.1}{11} }^{3}$	$33$	${\href{/LocalNumberField/11.11.0.1}{11} }^{3}$	$33$	${\href{/LocalNumberField/17.11.0.1}{11} }^{3}$	${\href{/LocalNumberField/19.3.0.1}{3} }^{11}$	$33$	$33$	$33$	${\href{/LocalNumberField/37.3.0.1}{3} }^{11}$	$33$	${\href{/LocalNumberField/43.3.0.1}{3} }^{11}$	$33$	$33$	${\href{/LocalNumberField/59.11.0.1}{11} }^{3}$

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
199	Data not computed