Normalized defining polynomial
\( x^{28} - 12 x^{27} - 17 x^{26} + 704 x^{25} - 1230 x^{24} - 16910 x^{23} + 53678 x^{22} + 208815 x^{21} - 952529 x^{20} - 1292583 x^{19} + 9401590 x^{18} + 1835875 x^{17} - 55400397 x^{16} + 27000714 x^{15} + 193853027 x^{14} - 190074984 x^{13} - 373374103 x^{12} + 552455692 x^{11} + 303130711 x^{10} - 780400889 x^{9} + 66439726 x^{8} + 483771191 x^{7} - 213323444 x^{6} - 85110098 x^{5} + 72075535 x^{4} - 7307537 x^{3} - 3799902 x^{2} + 793446 x - 20591 \)
Invariants
| Degree: | $28$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[28, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(17726902035997940785736680176345909279067179524672108809=17^{14}\cdot 29^{26}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $94.01$ | 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: | $17, 29$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(493=17\cdot 29\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{493}(256,·)$, $\chi_{493}(1,·)$, $\chi_{493}(67,·)$, $\chi_{493}(324,·)$, $\chi_{493}(390,·)$, $\chi_{493}(458,·)$, $\chi_{493}(460,·)$, $\chi_{493}(16,·)$, $\chi_{493}(339,·)$, $\chi_{493}(341,·)$, $\chi_{493}(86,·)$, $\chi_{493}(407,·)$, $\chi_{493}(152,·)$, $\chi_{493}(154,·)$, $\chi_{493}(477,·)$, $\chi_{493}(33,·)$, $\chi_{493}(35,·)$, $\chi_{493}(103,·)$, $\chi_{493}(169,·)$, $\chi_{493}(426,·)$, $\chi_{493}(492,·)$, $\chi_{493}(237,·)$, $\chi_{493}(239,·)$, $\chi_{493}(52,·)$, $\chi_{493}(373,·)$, $\chi_{493}(120,·)$, $\chi_{493}(441,·)$, $\chi_{493}(254,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{59} a^{24} - \frac{7}{59} a^{23} - \frac{8}{59} a^{22} - \frac{8}{59} a^{21} + \frac{21}{59} a^{20} - \frac{4}{59} a^{19} + \frac{11}{59} a^{18} + \frac{20}{59} a^{17} - \frac{6}{59} a^{16} - \frac{17}{59} a^{15} - \frac{3}{59} a^{14} - \frac{10}{59} a^{13} + \frac{3}{59} a^{12} + \frac{17}{59} a^{11} - \frac{5}{59} a^{10} + \frac{14}{59} a^{9} - \frac{4}{59} a^{8} - \frac{1}{59} a^{7} - \frac{5}{59} a^{6} + \frac{3}{59} a^{5} + \frac{6}{59} a^{4} - \frac{21}{59} a^{3} + \frac{17}{59} a^{2} - \frac{17}{59} a$, $\frac{1}{59} a^{25} + \frac{2}{59} a^{23} - \frac{5}{59} a^{22} + \frac{24}{59} a^{21} + \frac{25}{59} a^{20} - \frac{17}{59} a^{19} - \frac{21}{59} a^{18} + \frac{16}{59} a^{17} - \frac{4}{59} a^{15} + \frac{28}{59} a^{14} - \frac{8}{59} a^{13} - \frac{21}{59} a^{12} - \frac{4}{59} a^{11} - \frac{21}{59} a^{10} - \frac{24}{59} a^{9} - \frac{29}{59} a^{8} - \frac{12}{59} a^{7} + \frac{27}{59} a^{6} + \frac{27}{59} a^{5} + \frac{21}{59} a^{4} - \frac{12}{59} a^{3} - \frac{16}{59} a^{2} - \frac{1}{59} a$, $\frac{1}{20591} a^{26} - \frac{148}{20591} a^{25} - \frac{13}{20591} a^{24} - \frac{1730}{20591} a^{23} - \frac{2361}{20591} a^{22} + \frac{3024}{20591} a^{21} - \frac{3796}{20591} a^{20} - \frac{1044}{20591} a^{19} + \frac{9095}{20591} a^{18} - \frac{7624}{20591} a^{17} - \frac{799}{20591} a^{16} + \frac{4887}{20591} a^{15} + \frac{4212}{20591} a^{14} - \frac{2817}{20591} a^{13} + \frac{8605}{20591} a^{12} - \frac{1100}{20591} a^{11} - \frac{4452}{20591} a^{10} - \frac{7484}{20591} a^{9} - \frac{6103}{20591} a^{8} + \frac{4237}{20591} a^{7} - \frac{153}{349} a^{6} + \frac{7721}{20591} a^{5} + \frac{5994}{20591} a^{4} - \frac{5831}{20591} a^{3} - \frac{2431}{20591} a^{2} + \frac{108}{20591} a$, $\frac{1}{196024196309198099322859925982239120077191081177547826143781096569928082171203033771750887} a^{27} + \frac{4463162297482936739840016290832464448447196549221695010414647880365818864509962176190}{196024196309198099322859925982239120077191081177547826143781096569928082171203033771750887} a^{26} + \frac{1224559776111183663607793060683115886859991292325719866360037021913375755589995851846211}{196024196309198099322859925982239120077191081177547826143781096569928082171203033771750887} a^{25} + \frac{1034609140815795901704001691906567027347831026147055524139100060976463112779170838095693}{196024196309198099322859925982239120077191081177547826143781096569928082171203033771750887} a^{24} + \frac{85887728959807696017542585969042341764888805093082349250970972923401109062799399796044616}{196024196309198099322859925982239120077191081177547826143781096569928082171203033771750887} a^{23} - \frac{76198556278944676197549855573849286753520811310796158827853046916886782759503941606391140}{196024196309198099322859925982239120077191081177547826143781096569928082171203033771750887} a^{22} + \frac{8823227515011322058924443294650994901688869465604460955501877540315809136231883679527536}{196024196309198099322859925982239120077191081177547826143781096569928082171203033771750887} a^{21} - \frac{70823288462621535706931003783577850313631520622577463439515329654952176363747098550313655}{196024196309198099322859925982239120077191081177547826143781096569928082171203033771750887} a^{20} - \frac{10823540064214647525794210722114720571002661016310343348384510669692813531052296180244530}{196024196309198099322859925982239120077191081177547826143781096569928082171203033771750887} a^{19} + \frac{52953016859146000460971893347719452725182304811949672650120250408460582707401860117451862}{196024196309198099322859925982239120077191081177547826143781096569928082171203033771750887} a^{18} - \frac{1609083859558361160870899349985767794310319943116664088409025592251375300898840564511155}{3322444005240645751234913999698968136901543748771997053284425365592001392732254809690693} a^{17} + \frac{54659157386445886440197955213331923729055692966909480326176599779070420498590418062621768}{196024196309198099322859925982239120077191081177547826143781096569928082171203033771750887} a^{16} + \frac{64200259084097291051145959840282389592428780338906408395049885671282499562183857499793724}{196024196309198099322859925982239120077191081177547826143781096569928082171203033771750887} a^{15} - \frac{49172379449353338858664470608250632422502464838821854079237922276568767060632848898503275}{196024196309198099322859925982239120077191081177547826143781096569928082171203033771750887} a^{14} + \frac{21049163330463352233873730356398365874061677455715582445631924373994526976334345452080868}{196024196309198099322859925982239120077191081177547826143781096569928082171203033771750887} a^{13} - \frac{46321473858431828227322033282040466221313180944754333209180850205641316279365396589903066}{196024196309198099322859925982239120077191081177547826143781096569928082171203033771750887} a^{12} + \frac{37623403972494719065774884666321542392574656755384914789046343342854433883193896903725129}{196024196309198099322859925982239120077191081177547826143781096569928082171203033771750887} a^{11} + \frac{62829856907165387433609900873783410048770347707893394923913679989293992918868456679577686}{196024196309198099322859925982239120077191081177547826143781096569928082171203033771750887} a^{10} + \frac{29735144855344798355399418806436454542340095655352511169339020529025253052517237427247073}{196024196309198099322859925982239120077191081177547826143781096569928082171203033771750887} a^{9} - \frac{10555662740972614466653449548458211845045098877735277181443474417347854772813054226984388}{196024196309198099322859925982239120077191081177547826143781096569928082171203033771750887} a^{8} + \frac{1123285024664626390094792935940297416008979552711726219194799755898179931683570206187605}{4781077958760929251777071365420466343346123931159703076677587721217758101736659360286607} a^{7} + \frac{22595285134763023125570467050762760712734403129727350590357251844484976246058415362795176}{196024196309198099322859925982239120077191081177547826143781096569928082171203033771750887} a^{6} - \frac{39502295095294636295795612122361360603355689500768100247843221579004270138025476467223418}{196024196309198099322859925982239120077191081177547826143781096569928082171203033771750887} a^{5} + \frac{34890738555302793182611396421487271265214248795108863411628930099800563974905603554599905}{196024196309198099322859925982239120077191081177547826143781096569928082171203033771750887} a^{4} - \frac{15987773004528100495134382420368490618883793144215580433502810914867662000072179185693844}{196024196309198099322859925982239120077191081177547826143781096569928082171203033771750887} a^{3} - \frac{19830927707982837190336578202367673764944021647544335243086072006468668392449080063844828}{196024196309198099322859925982239120077191081177547826143781096569928082171203033771750887} a^{2} + \frac{33039734779098927064413055570669007766560461746711390440977255256388676140171660167032988}{196024196309198099322859925982239120077191081177547826143781096569928082171203033771750887} a - \frac{467338851366639557630511039674141545420033681081635782570678139180705959485871413366}{9519896863153712754254767907446900105735082374704862616860817666452726053674082549257}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $27$ | 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: | 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) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 5165284741490759000 \) (assuming GRH) | 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 | ${\href{/LocalNumberField/2.14.0.1}{14} }^{2}$ | ${\href{/LocalNumberField/3.14.0.1}{14} }^{2}$ | ${\href{/LocalNumberField/5.14.0.1}{14} }^{2}$ | ${\href{/LocalNumberField/7.14.0.1}{14} }^{2}$ | ${\href{/LocalNumberField/11.14.0.1}{14} }^{2}$ | ${\href{/LocalNumberField/13.7.0.1}{7} }^{4}$ | R | ${\href{/LocalNumberField/19.14.0.1}{14} }^{2}$ | ${\href{/LocalNumberField/23.14.0.1}{14} }^{2}$ | R | ${\href{/LocalNumberField/31.14.0.1}{14} }^{2}$ | ${\href{/LocalNumberField/37.14.0.1}{14} }^{2}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{14}$ | ${\href{/LocalNumberField/43.14.0.1}{14} }^{2}$ | ${\href{/LocalNumberField/47.14.0.1}{14} }^{2}$ | ${\href{/LocalNumberField/53.7.0.1}{7} }^{4}$ | ${\href{/LocalNumberField/59.1.0.1}{1} }^{28}$ |
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 |
|---|---|---|---|---|---|---|---|
| $17$ | 17.4.2.1 | $x^{4} + 85 x^{2} + 2601$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 17.4.2.1 | $x^{4} + 85 x^{2} + 2601$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 17.4.2.1 | $x^{4} + 85 x^{2} + 2601$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 17.4.2.1 | $x^{4} + 85 x^{2} + 2601$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 17.4.2.1 | $x^{4} + 85 x^{2} + 2601$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 17.4.2.1 | $x^{4} + 85 x^{2} + 2601$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 17.4.2.1 | $x^{4} + 85 x^{2} + 2601$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 29 | Data not computed | ||||||