Normalized defining polynomial
\( x^{28} - 12 x^{27} + 109 x^{26} - 700 x^{25} + 4035 x^{24} - 19574 x^{23} + 88778 x^{22} - 356475 x^{21} + 1362190 x^{20} - 4712493 x^{19} + 15680341 x^{18} - 47682224 x^{17} + 140523186 x^{16} - 379222674 x^{15} + 998556167 x^{14} - 2397439896 x^{13} + 5658735650 x^{12} - 12041626214 x^{11} + 25449411835 x^{10} - 47468325131 x^{9} + 89362532230 x^{8} - 142895968159 x^{7} + 237288842983 x^{6} - 311702665937 x^{5} + 449511995851 x^{4} - 443009324759 x^{3} + 541930532937 x^{2} - 311321664597 x + 311907554911 \)
Invariants
| Degree: | $28$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 14]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(84119824649142740425307369706365450159735457977329962241=19^{14}\cdot 29^{26}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $99.38$ | magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
| |
| Ramified primes: | $19, 29$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(551=19\cdot 29\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{551}(1,·)$, $\chi_{551}(343,·)$, $\chi_{551}(132,·)$, $\chi_{551}(455,·)$, $\chi_{551}(265,·)$, $\chi_{551}(267,·)$, $\chi_{551}(400,·)$, $\chi_{551}(210,·)$, $\chi_{551}(531,·)$, $\chi_{551}(20,·)$, $\chi_{551}(341,·)$, $\chi_{551}(151,·)$, $\chi_{551}(324,·)$, $\chi_{551}(284,·)$, $\chi_{551}(286,·)$, $\chi_{551}(96,·)$, $\chi_{551}(208,·)$, $\chi_{551}(419,·)$, $\chi_{551}(550,·)$, $\chi_{551}(227,·)$, $\chi_{551}(170,·)$, $\chi_{551}(303,·)$, $\chi_{551}(115,·)$, $\chi_{551}(436,·)$, $\chi_{551}(94,·)$, $\chi_{551}(457,·)$, $\chi_{551}(248,·)$, $\chi_{551}(381,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $\frac{1}{41} a^{26} - \frac{16}{41} a^{25} - \frac{13}{41} a^{24} - \frac{9}{41} a^{23} + \frac{11}{41} a^{22} + \frac{14}{41} a^{21} + \frac{2}{41} a^{20} - \frac{9}{41} a^{19} - \frac{2}{41} a^{18} + \frac{7}{41} a^{17} - \frac{11}{41} a^{16} + \frac{10}{41} a^{15} - \frac{12}{41} a^{14} + \frac{12}{41} a^{13} - \frac{11}{41} a^{12} - \frac{7}{41} a^{11} + \frac{11}{41} a^{10} + \frac{14}{41} a^{9} - \frac{18}{41} a^{8} - \frac{14}{41} a^{7} - \frac{6}{41} a^{6} + \frac{1}{41} a^{5} - \frac{8}{41} a^{4} + \frac{7}{41} a^{2} + \frac{16}{41} a + \frac{8}{41}$, $\frac{1}{82099482226810191492738533382404672826255812410355634554904945505332895251254960377147807045293446593834085943100725535403} a^{27} - \frac{899743915158104831432116063149850164269677787747129310699098308886181235293555010177138916964501841232335190044506249393}{82099482226810191492738533382404672826255812410355634554904945505332895251254960377147807045293446593834085943100725535403} a^{26} + \frac{29534619550947685181272477545109456690769469971848960509509584182298736150538031291498440435115193944780496541126881968429}{82099482226810191492738533382404672826255812410355634554904945505332895251254960377147807045293446593834085943100725535403} a^{25} - \frac{11368932700028006905286204547338032029031938835073711188413023762969359564997116942045602307731817651449226861531425480254}{82099482226810191492738533382404672826255812410355634554904945505332895251254960377147807045293446593834085943100725535403} a^{24} - \frac{26722787565721585664353013783797926161622785960133871282044998497665340584858974358092156777186586009508566131206768340749}{82099482226810191492738533382404672826255812410355634554904945505332895251254960377147807045293446593834085943100725535403} a^{23} - \frac{37961093918619791697223843075739276018445172915206740349134203509482200226941265234868722176295044803536239768687058422577}{82099482226810191492738533382404672826255812410355634554904945505332895251254960377147807045293446593834085943100725535403} a^{22} - \frac{40316997773181807600244325335375524144612705437289058020594215664959835960570284213857853359873783841074928246863694615782}{82099482226810191492738533382404672826255812410355634554904945505332895251254960377147807045293446593834085943100725535403} a^{21} - \frac{38464990360438758036299409207885737232803873412981710741434742653738828829535544794049973636963116466345776993252082052695}{82099482226810191492738533382404672826255812410355634554904945505332895251254960377147807045293446593834085943100725535403} a^{20} - \frac{40891281350092339484540153751293162487993227736534048627759935222457767578070674588054545554530051531376088526450148291563}{82099482226810191492738533382404672826255812410355634554904945505332895251254960377147807045293446593834085943100725535403} a^{19} + \frac{7937318670264750644744367802772126911585097950209667988198303317146905600734020092086536465032829407718572034491397122078}{82099482226810191492738533382404672826255812410355634554904945505332895251254960377147807045293446593834085943100725535403} a^{18} - \frac{40966309625101183542394349456901443656585963106262816903032001225091004442138580615517555627114700959909973243689180952993}{82099482226810191492738533382404672826255812410355634554904945505332895251254960377147807045293446593834085943100725535403} a^{17} - \frac{10898702173272469750457224460656727481803755590834186632196492063637626355476439150355269208845223346514386177571139189341}{82099482226810191492738533382404672826255812410355634554904945505332895251254960377147807045293446593834085943100725535403} a^{16} - \frac{28253940486023533979355136844375355083202063239027498196449255285981584788433428893507739874991353438037480841024507250406}{82099482226810191492738533382404672826255812410355634554904945505332895251254960377147807045293446593834085943100725535403} a^{15} + \frac{409723806185320789153696495561614264827704630835338619942078116285189704400616547071927841710394503200532653077400715479}{2002426395775858329091183741034260312835507619764771574509876719642265737835486838467019684031547477898392340075627452083} a^{14} + \frac{5218771222877016689595020055170695560414218431710955734912839379056782778508794536396402913479902505796637808360225357394}{82099482226810191492738533382404672826255812410355634554904945505332895251254960377147807045293446593834085943100725535403} a^{13} + \frac{30311040398003425842149015248874152099620775677301253083595515447455478503900132908719685446611629538817911689607635656824}{82099482226810191492738533382404672826255812410355634554904945505332895251254960377147807045293446593834085943100725535403} a^{12} - \frac{8710835302134134972719131667179535247683484670621427920279394145729869931821584521219289540196910069777275352524415839278}{82099482226810191492738533382404672826255812410355634554904945505332895251254960377147807045293446593834085943100725535403} a^{11} - \frac{6420025564231500306853457582393094495132169527142429517732225497452882611110504934317233936245340536858583960746242411269}{82099482226810191492738533382404672826255812410355634554904945505332895251254960377147807045293446593834085943100725535403} a^{10} - \frac{21815863721753774631195346603221403356459956511193136978203286042325632141160443935265443483888450440263094056318289824092}{82099482226810191492738533382404672826255812410355634554904945505332895251254960377147807045293446593834085943100725535403} a^{9} - \frac{36916045477398196497982544665744748051241241628822610631440132555044371424783034792042728671908530391446022624568521186056}{82099482226810191492738533382404672826255812410355634554904945505332895251254960377147807045293446593834085943100725535403} a^{8} - \frac{38115748962274585095187519558758303925778191874303571165388287946647701764708999269166198047366156926741926576599600490642}{82099482226810191492738533382404672826255812410355634554904945505332895251254960377147807045293446593834085943100725535403} a^{7} - \frac{27266133225185405200453985831133285613884307191758486596788786756358040106519061964723418036047702130053823629579515302408}{82099482226810191492738533382404672826255812410355634554904945505332895251254960377147807045293446593834085943100725535403} a^{6} - \frac{18431298188600384650318604431071021688056800110794268191188643035746224377204137481731578380093103817468215167813989550389}{82099482226810191492738533382404672826255812410355634554904945505332895251254960377147807045293446593834085943100725535403} a^{5} + \frac{12208758711038696254124293161946965618261653331344148944136941026177259973492701440231801978670446489105644775873765606082}{82099482226810191492738533382404672826255812410355634554904945505332895251254960377147807045293446593834085943100725535403} a^{4} - \frac{27928132972330860885076562473197657830465727986860061570109530820369579921160388139403042538143727047049547246337223876747}{82099482226810191492738533382404672826255812410355634554904945505332895251254960377147807045293446593834085943100725535403} a^{3} - \frac{20800196587031848284853599995919056571143546411003110392408560253581876988049435151550646898709563471405476184526353898190}{82099482226810191492738533382404672826255812410355634554904945505332895251254960377147807045293446593834085943100725535403} a^{2} - \frac{28163433423956272765635000753922902425272342746010329252829766129730630916221502355806774569969724069688140772189387102550}{82099482226810191492738533382404672826255812410355634554904945505332895251254960377147807045293446593834085943100725535403} a + \frac{40577751682295955767159819561760573424925438715999654784925092856336698866270794275483102465662509665998702246881116277487}{82099482226810191492738533382404672826255812410355634554904945505332895251254960377147807045293446593834085943100725535403}$
Class group and class number
Not computed
Unit group
| Rank: | $13$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -1 \) (order $2$) | magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
| |
| Fundamental units: | Not computed | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | Not computed | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2\times C_{14}$ (as 28T2):
| An abelian group of order 28 |
| The 28 conjugacy class representatives for $C_2\times C_{14}$ |
| Character table for $C_2\times C_{14}$ is not computed |
Intermediate fields
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 | ${\href{/LocalNumberField/2.14.0.1}{14} }^{2}$ | ${\href{/LocalNumberField/3.14.0.1}{14} }^{2}$ | ${\href{/LocalNumberField/5.7.0.1}{7} }^{4}$ | ${\href{/LocalNumberField/7.7.0.1}{7} }^{4}$ | ${\href{/LocalNumberField/11.14.0.1}{14} }^{2}$ | ${\href{/LocalNumberField/13.14.0.1}{14} }^{2}$ | ${\href{/LocalNumberField/17.2.0.1}{2} }^{14}$ | R | ${\href{/LocalNumberField/23.7.0.1}{7} }^{4}$ | R | ${\href{/LocalNumberField/31.14.0.1}{14} }^{2}$ | ${\href{/LocalNumberField/37.14.0.1}{14} }^{2}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{14}$ | ${\href{/LocalNumberField/43.14.0.1}{14} }^{2}$ | ${\href{/LocalNumberField/47.14.0.1}{14} }^{2}$ | ${\href{/LocalNumberField/53.14.0.1}{14} }^{2}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{14}$ |
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 | Data not computed | ||||||
| 29 | Data not computed | ||||||