Normalized defining polynomial
\( x^{28} - 12 x^{27} + 95 x^{26} - 544 x^{25} + 2722 x^{24} - 11790 x^{23} + 47230 x^{22} - 170465 x^{21} + 580519 x^{20} - 1817287 x^{19} + 5433646 x^{18} - 15049677 x^{17} + 40114635 x^{16} - 99220982 x^{15} + 237665411 x^{14} - 526242344 x^{13} + 1135619833 x^{12} - 2242768340 x^{11} + 4353420535 x^{10} - 7586336825 x^{9} + 13171343590 x^{8} - 19820814665 x^{7} + 30470050180 x^{6} - 37975578530 x^{5} + 50884013935 x^{4} - 48025395089 x^{3} + 54808752058 x^{2} - 30471626250 x + 28667253361 \)
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: | \(3073445894692486733947614496733715330677488092041015625=3^{14}\cdot 5^{14}\cdot 29^{26}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $88.31$ | 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: | $3, 5, 29$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(435=3\cdot 5\cdot 29\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{435}(256,·)$, $\chi_{435}(1,·)$, $\chi_{435}(194,·)$, $\chi_{435}(196,·)$, $\chi_{435}(136,·)$, $\chi_{435}(74,·)$, $\chi_{435}(16,·)$, $\chi_{435}(209,·)$, $\chi_{435}(149,·)$, $\chi_{435}(151,·)$, $\chi_{435}(344,·)$, $\chi_{435}(91,·)$, $\chi_{435}(284,·)$, $\chi_{435}(286,·)$, $\chi_{435}(226,·)$, $\chi_{435}(419,·)$, $\chi_{435}(361,·)$, $\chi_{435}(299,·)$, $\chi_{435}(239,·)$, $\chi_{435}(241,·)$, $\chi_{435}(434,·)$, $\chi_{435}(179,·)$, $\chi_{435}(181,·)$, $\chi_{435}(376,·)$, $\chi_{435}(121,·)$, $\chi_{435}(314,·)$, $\chi_{435}(59,·)$, $\chi_{435}(254,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $\frac{1}{55625454584085416418084429587888065103391461147056447485422372524311470094968077625202454217407095382875892331890079} a^{27} - \frac{22566689899321381802431913992896210326043353288465610296317034073719211210412979984521596032827172154698232572429711}{55625454584085416418084429587888065103391461147056447485422372524311470094968077625202454217407095382875892331890079} a^{26} - \frac{14980613364994495632388167860259588465167414977633885783489704555687581808089086636450693724407095200215785622736570}{55625454584085416418084429587888065103391461147056447485422372524311470094968077625202454217407095382875892331890079} a^{25} - \frac{21632709227688672231490259237898238095173818255904322410950046412442480363198281068850730000273775821389413876809323}{55625454584085416418084429587888065103391461147056447485422372524311470094968077625202454217407095382875892331890079} a^{24} + \frac{5626343062469816932348542246993731262207464564616078360692776784461970761052255731302089942376437407636399864004792}{55625454584085416418084429587888065103391461147056447485422372524311470094968077625202454217407095382875892331890079} a^{23} + \frac{27654521712348698831933264903601522239966998407353194483104160833392236290490345234278820754777726019677679406763998}{55625454584085416418084429587888065103391461147056447485422372524311470094968077625202454217407095382875892331890079} a^{22} - \frac{11626121717267957052935498372251035816046028661489273930100234356864104998980160163129497266895604504117587191510956}{55625454584085416418084429587888065103391461147056447485422372524311470094968077625202454217407095382875892331890079} a^{21} + \frac{13259968396522557699440580561496161795172311810357066770032798681879119397716499628684905517317760130984507365755900}{55625454584085416418084429587888065103391461147056447485422372524311470094968077625202454217407095382875892331890079} a^{20} - \frac{16338818716127544999779813697436010750582786738608840275658368251230872129968178639204080622135800472714360700032662}{55625454584085416418084429587888065103391461147056447485422372524311470094968077625202454217407095382875892331890079} a^{19} + \frac{25320407294193566223923126868047692598608851451784516977095669151870622761143821009127257627798674833670918647376059}{55625454584085416418084429587888065103391461147056447485422372524311470094968077625202454217407095382875892331890079} a^{18} - \frac{17961134295569209645921861713339264366595783385175999123736738821208652555385803343437973357282017157279847324253125}{55625454584085416418084429587888065103391461147056447485422372524311470094968077625202454217407095382875892331890079} a^{17} - \frac{27559397558139582873073156161806938850621606569996518136939497358509158185615292620082539570661899880928126006927760}{55625454584085416418084429587888065103391461147056447485422372524311470094968077625202454217407095382875892331890079} a^{16} - \frac{25473809897572994371436202817245195068285881090329945052068478055761147996361357729794002892288829137213588208674899}{55625454584085416418084429587888065103391461147056447485422372524311470094968077625202454217407095382875892331890079} a^{15} + \frac{21764003374092763543040052823354295677046598006603116696613058388090056764213983927844762033568801743568912620552678}{55625454584085416418084429587888065103391461147056447485422372524311470094968077625202454217407095382875892331890079} a^{14} - \frac{26670953187593414579691191644365525647154766358293616759808146904319578194127549076595945396375929575521441216355519}{55625454584085416418084429587888065103391461147056447485422372524311470094968077625202454217407095382875892331890079} a^{13} + \frac{5086909141725277593826176089988572871071843386555868259379922160142151694284454145104384091802197757562417511849926}{55625454584085416418084429587888065103391461147056447485422372524311470094968077625202454217407095382875892331890079} a^{12} - \frac{11800026944469423383326965995003360446231064306557992528108370195056538363413944502070224527797996476090994533001515}{55625454584085416418084429587888065103391461147056447485422372524311470094968077625202454217407095382875892331890079} a^{11} + \frac{17125616036185194415335827881205493210522804205992430886650912144960134343125187731872871736164060326631659135622688}{55625454584085416418084429587888065103391461147056447485422372524311470094968077625202454217407095382875892331890079} a^{10} - \frac{16777232636912119780600897109254168250152577781193524620861789679826276156575703438404950338806065159524549730114735}{55625454584085416418084429587888065103391461147056447485422372524311470094968077625202454217407095382875892331890079} a^{9} + \frac{26296049880359197890299413184053617331662205135211483244288575295216951423412700652288302709527588805499923754230988}{55625454584085416418084429587888065103391461147056447485422372524311470094968077625202454217407095382875892331890079} a^{8} + \frac{4385473639484410550593180844217287512692097208514623776776242077812153349635207564603594928453108317591771975844334}{55625454584085416418084429587888065103391461147056447485422372524311470094968077625202454217407095382875892331890079} a^{7} + \frac{7699396914585905313793514547791496324603469473625747775344609729093623647304918105551521650213893633224944480114072}{55625454584085416418084429587888065103391461147056447485422372524311470094968077625202454217407095382875892331890079} a^{6} - \frac{13497185403429187542742911663000621660384500312335246475401500063450959577696498717801174178883308027917463950657944}{55625454584085416418084429587888065103391461147056447485422372524311470094968077625202454217407095382875892331890079} a^{5} - \frac{9409738083020388473457301278841640751863243263209132894049942900861715381824468031127815957167431106993151314294626}{55625454584085416418084429587888065103391461147056447485422372524311470094968077625202454217407095382875892331890079} a^{4} + \frac{26811434374232429659129390948446363470031239555765844131241737632576840447858125357719161371606563295715621731209183}{55625454584085416418084429587888065103391461147056447485422372524311470094968077625202454217407095382875892331890079} a^{3} - \frac{13861563722479350194226683431022300367378991683926946149072571827426066545243110029278368589428575900375326679930714}{55625454584085416418084429587888065103391461147056447485422372524311470094968077625202454217407095382875892331890079} a^{2} - \frac{5608618807711527957527360706859398665220166116225595633286569872101159808687445256359175900878849997117842496282512}{55625454584085416418084429587888065103391461147056447485422372524311470094968077625202454217407095382875892331890079} a - \frac{12245997588871214739455581569796428040744995470500518488317959458778993349118342079247714202601009485042635414061649}{55625454584085416418084429587888065103391461147056447485422372524311470094968077625202454217407095382875892331890079}$
Class group and class number
$C_{2}\times C_{2}\times C_{8}\times C_{8}\times C_{26216}$, which has order $6711296$ (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: | \( 487075979.1876791 \) (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 | ${\href{/LocalNumberField/2.14.0.1}{14} }^{2}$ | R | R | ${\href{/LocalNumberField/7.14.0.1}{14} }^{2}$ | ${\href{/LocalNumberField/11.14.0.1}{14} }^{2}$ | ${\href{/LocalNumberField/13.14.0.1}{14} }^{2}$ | ${\href{/LocalNumberField/17.2.0.1}{2} }^{14}$ | ${\href{/LocalNumberField/19.14.0.1}{14} }^{2}$ | ${\href{/LocalNumberField/23.7.0.1}{7} }^{4}$ | R | ${\href{/LocalNumberField/31.14.0.1}{14} }^{2}$ | ${\href{/LocalNumberField/37.14.0.1}{14} }^{2}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{14}$ | ${\href{/LocalNumberField/43.14.0.1}{14} }^{2}$ | ${\href{/LocalNumberField/47.14.0.1}{14} }^{2}$ | ${\href{/LocalNumberField/53.7.0.1}{7} }^{4}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{14}$ |
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 |
|---|---|---|---|---|---|---|---|
| 3 | Data not computed | ||||||
| $5$ | 5.14.7.2 | $x^{14} - 15625 x^{2} + 156250$ | $2$ | $7$ | $7$ | $C_{14}$ | $[\ ]_{2}^{7}$ |
| 5.14.7.2 | $x^{14} - 15625 x^{2} + 156250$ | $2$ | $7$ | $7$ | $C_{14}$ | $[\ ]_{2}^{7}$ | |
| 29 | Data not computed | ||||||