Normalized defining polynomial
\( x^{28} - x^{27} + 18 x^{26} - 23 x^{25} + 29 x^{24} - 149 x^{23} - 795 x^{22} - 8 x^{21} - 2927 x^{20} + 667 x^{19} + 13846 x^{18} - 647 x^{17} + 109320 x^{16} + 98963 x^{15} + 313397 x^{14} + 494999 x^{13} + 789626 x^{12} + 291678 x^{11} + 1921156 x^{10} - 425111 x^{9} + 1044768 x^{8} + 1156338 x^{7} - 1030264 x^{6} + 642721 x^{5} + 1780368 x^{4} - 4481806 x^{3} + 897127 x^{2} + 1339909 x + 1148111 \)
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: | \(50201655190081835380839261671426578388690948486328125=5^{21}\cdot 29^{26}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $76.24$ | 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: | $5, 29$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(145=5\cdot 29\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{145}(1,·)$, $\chi_{145}(67,·)$, $\chi_{145}(136,·)$, $\chi_{145}(138,·)$, $\chi_{145}(139,·)$, $\chi_{145}(13,·)$, $\chi_{145}(141,·)$, $\chi_{145}(16,·)$, $\chi_{145}(81,·)$, $\chi_{145}(22,·)$, $\chi_{145}(24,·)$, $\chi_{145}(28,·)$, $\chi_{145}(93,·)$, $\chi_{145}(94,·)$, $\chi_{145}(33,·)$, $\chi_{145}(36,·)$, $\chi_{145}(38,·)$, $\chi_{145}(92,·)$, $\chi_{145}(42,·)$, $\chi_{145}(111,·)$, $\chi_{145}(49,·)$, $\chi_{145}(54,·)$, $\chi_{145}(57,·)$, $\chi_{145}(122,·)$, $\chi_{145}(59,·)$, $\chi_{145}(74,·)$, $\chi_{145}(62,·)$, $\chi_{145}(63,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $\frac{1}{1035241} a^{26} + \frac{167675}{1035241} a^{25} - \frac{8607}{1035241} a^{24} - \frac{49857}{1035241} a^{23} - \frac{327836}{1035241} a^{22} - \frac{226077}{1035241} a^{21} + \frac{22454}{1035241} a^{20} + \frac{89303}{1035241} a^{19} - \frac{285793}{1035241} a^{18} + \frac{14038}{1035241} a^{17} - \frac{69319}{1035241} a^{16} - \frac{323357}{1035241} a^{15} + \frac{64696}{1035241} a^{14} + \frac{455141}{1035241} a^{13} - \frac{285930}{1035241} a^{12} - \frac{332747}{1035241} a^{11} + \frac{504087}{1035241} a^{10} + \frac{50072}{1035241} a^{9} - \frac{71359}{1035241} a^{8} - \frac{479530}{1035241} a^{7} + \frac{431797}{1035241} a^{6} + \frac{22510}{1035241} a^{5} + \frac{37288}{1035241} a^{4} + \frac{214708}{1035241} a^{3} - \frac{58931}{1035241} a^{2} - \frac{343309}{1035241} a - \frac{485626}{1035241}$, $\frac{1}{26759980133816899117378771099538605124180625366083896681471977851538099946154920588201098054680201141} a^{27} - \frac{7183082423506801192360571300011230215496520557322109238349404180507443807137733635095093808017}{26759980133816899117378771099538605124180625366083896681471977851538099946154920588201098054680201141} a^{26} - \frac{11993669179159971918972113357978856094265060092140766606855163702334450937509211484167923737731901044}{26759980133816899117378771099538605124180625366083896681471977851538099946154920588201098054680201141} a^{25} - \frac{5460392075307196829231979077829791246275666421823614402958075393577853551246295705556102487324804488}{26759980133816899117378771099538605124180625366083896681471977851538099946154920588201098054680201141} a^{24} - \frac{8722618029874107909081279666902404916194292149856023237927368575885414866565184328030077617196446083}{26759980133816899117378771099538605124180625366083896681471977851538099946154920588201098054680201141} a^{23} - \frac{8421234207623751310722962248580486913455091271047310532053397126723788254627502370634793629107191748}{26759980133816899117378771099538605124180625366083896681471977851538099946154920588201098054680201141} a^{22} - \frac{5706534048240616079111743369115953195761038250376365477833928337336240201336073549059512059058706527}{26759980133816899117378771099538605124180625366083896681471977851538099946154920588201098054680201141} a^{21} + \frac{10559648182446717526873338831958650876217511278723642288819551302787751264851860228469405080856090426}{26759980133816899117378771099538605124180625366083896681471977851538099946154920588201098054680201141} a^{20} + \frac{11009336981052611878824129374179318775042409669448964167342270710345644986242945113937586441451244526}{26759980133816899117378771099538605124180625366083896681471977851538099946154920588201098054680201141} a^{19} + \frac{6610422648210135212389990020455676287630389983608830157108681472500165882365426633913584618369983956}{26759980133816899117378771099538605124180625366083896681471977851538099946154920588201098054680201141} a^{18} + \frac{11441928912760607531544554585247364232879109244098878255823153731173324041647687651626827324307552699}{26759980133816899117378771099538605124180625366083896681471977851538099946154920588201098054680201141} a^{17} + \frac{11579497638137832369568464553493678585350074307396491662750814569505215835386120000340307297412196163}{26759980133816899117378771099538605124180625366083896681471977851538099946154920588201098054680201141} a^{16} - \frac{5212609895162158159887428735130420786713059245193350514644446667622111075051585423495373621147066541}{26759980133816899117378771099538605124180625366083896681471977851538099946154920588201098054680201141} a^{15} + \frac{11131748963454073925052028484768140460520914598094419886392342905351366104339681754304948532908565921}{26759980133816899117378771099538605124180625366083896681471977851538099946154920588201098054680201141} a^{14} - \frac{6176742382289328091096127452406034068891708425525979288998772088206997132873046935713725145802467798}{26759980133816899117378771099538605124180625366083896681471977851538099946154920588201098054680201141} a^{13} + \frac{9191384041441932990252631554407085910789888773437915278313568738807999299656180599772755064115134519}{26759980133816899117378771099538605124180625366083896681471977851538099946154920588201098054680201141} a^{12} - \frac{8429429874622803590508842264699270250865765875274355592158566169453272044771912282274664167612955448}{26759980133816899117378771099538605124180625366083896681471977851538099946154920588201098054680201141} a^{11} + \frac{11392632026773918265575259752300813218871519029655510566454561348633894389379033511589948886779492435}{26759980133816899117378771099538605124180625366083896681471977851538099946154920588201098054680201141} a^{10} - \frac{9959420421246221840830024798811893894253670859870525208141265851077191430740550622114195957780036142}{26759980133816899117378771099538605124180625366083896681471977851538099946154920588201098054680201141} a^{9} + \frac{5961430075059271336477726769460749129384095888273090222370336480984142955728583587466045678064714348}{26759980133816899117378771099538605124180625366083896681471977851538099946154920588201098054680201141} a^{8} + \frac{8623667221518036440722872180225918109739104717320122978850703691631641801639556143949360835865961470}{26759980133816899117378771099538605124180625366083896681471977851538099946154920588201098054680201141} a^{7} + \frac{12809869350339408399054791867251235500414477517648425114223087109025406387610066519623269537637942348}{26759980133816899117378771099538605124180625366083896681471977851538099946154920588201098054680201141} a^{6} + \frac{1004179595203818082076909881777365876927914688962996527132016269503620510003337452341150864325921612}{26759980133816899117378771099538605124180625366083896681471977851538099946154920588201098054680201141} a^{5} - \frac{12444400948111278783793768873087603918716122950531726222380351344644058746782774788932303511727873898}{26759980133816899117378771099538605124180625366083896681471977851538099946154920588201098054680201141} a^{4} + \frac{4499169806919036814027015881292501395874500750640688257608930187302384442271049508091795248474552903}{26759980133816899117378771099538605124180625366083896681471977851538099946154920588201098054680201141} a^{3} + \frac{8157755622744723379069905428055562476782165987744856625740634314966008741283075204169291910802196277}{26759980133816899117378771099538605124180625366083896681471977851538099946154920588201098054680201141} a^{2} - \frac{12764127047147764472757686421033343738555348328159523658605125836903337214764633955050263092405963037}{26759980133816899117378771099538605124180625366083896681471977851538099946154920588201098054680201141} a + \frac{10758829350594273078578510365290895644884270704165292193561530021266315253771502527225593963415282531}{26759980133816899117378771099538605124180625366083896681471977851538099946154920588201098054680201141}$
Class group and class number
$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{562}$, which has order $8992$ (assuming GRH)
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: | 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: | \( 34681517373.86067 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A cyclic group of order 28 |
| The 28 conjugacy class representatives for $C_{28}$ |
| Character table for $C_{28}$ is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), 4.0.105125.2, 7.7.594823321.1, 14.14.27641779937927268828125.1 |
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 | $28$ | $28$ | R | $28$ | ${\href{/LocalNumberField/11.14.0.1}{14} }^{2}$ | $28$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{7}$ | ${\href{/LocalNumberField/19.7.0.1}{7} }^{4}$ | $28$ | R | ${\href{/LocalNumberField/31.14.0.1}{14} }^{2}$ | $28$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{14}$ | $28$ | $28$ | $28$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{14}$ |
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 |
|---|---|---|---|---|---|---|---|
| 5 | Data not computed | ||||||
| $29$ | 29.14.13.11 | $x^{14} + 3712$ | $14$ | $1$ | $13$ | $C_{14}$ | $[\ ]_{14}$ |
| 29.14.13.11 | $x^{14} + 3712$ | $14$ | $1$ | $13$ | $C_{14}$ | $[\ ]_{14}$ | |