Normalized defining polynomial
\( x^{28} - 56 x^{25} - 7 x^{24} - 14 x^{23} + 1162 x^{22} + 286 x^{21} + 539 x^{20} - 11074 x^{19} - 4074 x^{18} - 6433 x^{17} + 48363 x^{16} + 27475 x^{15} + 29189 x^{14} - 79527 x^{13} - 90503 x^{12} - 49217 x^{11} + 5593 x^{10} + 93345 x^{9} + 89726 x^{8} + 166311 x^{7} + 114709 x^{6} + 86730 x^{5} + 11690 x^{4} - 17626 x^{3} + 25221 x^{2} - 9401 x + 6241 \)
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: | \(8602002546566250227508255640254075178338897178281=3^{14}\cdot 7^{50}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $55.93$ | 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: | $3, 7$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(147=3\cdot 7^{2}\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{147}(64,·)$, $\chi_{147}(1,·)$, $\chi_{147}(134,·)$, $\chi_{147}(71,·)$, $\chi_{147}(8,·)$, $\chi_{147}(139,·)$, $\chi_{147}(76,·)$, $\chi_{147}(13,·)$, $\chi_{147}(146,·)$, $\chi_{147}(83,·)$, $\chi_{147}(20,·)$, $\chi_{147}(85,·)$, $\chi_{147}(22,·)$, $\chi_{147}(92,·)$, $\chi_{147}(29,·)$, $\chi_{147}(97,·)$, $\chi_{147}(34,·)$, $\chi_{147}(104,·)$, $\chi_{147}(41,·)$, $\chi_{147}(106,·)$, $\chi_{147}(43,·)$, $\chi_{147}(113,·)$, $\chi_{147}(50,·)$, $\chi_{147}(118,·)$, $\chi_{147}(55,·)$, $\chi_{147}(125,·)$, $\chi_{147}(62,·)$, $\chi_{147}(127,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $\frac{1}{67} a^{23} + \frac{3}{67} a^{22} + \frac{5}{67} a^{21} - \frac{9}{67} a^{20} + \frac{6}{67} a^{19} + \frac{14}{67} a^{18} - \frac{1}{67} a^{17} - \frac{16}{67} a^{16} + \frac{11}{67} a^{15} - \frac{15}{67} a^{14} + \frac{11}{67} a^{13} + \frac{16}{67} a^{12} + \frac{26}{67} a^{11} - \frac{20}{67} a^{10} - \frac{26}{67} a^{9} + \frac{1}{67} a^{8} - \frac{29}{67} a^{7} - \frac{11}{67} a^{6} + \frac{18}{67} a^{5} + \frac{13}{67} a^{4} + \frac{13}{67} a^{3} - \frac{21}{67} a^{2} + \frac{32}{67} a - \frac{22}{67}$, $\frac{1}{67} a^{24} - \frac{4}{67} a^{22} - \frac{24}{67} a^{21} + \frac{33}{67} a^{20} - \frac{4}{67} a^{19} + \frac{24}{67} a^{18} - \frac{13}{67} a^{17} - \frac{8}{67} a^{16} + \frac{19}{67} a^{15} - \frac{11}{67} a^{14} - \frac{17}{67} a^{13} - \frac{22}{67} a^{12} - \frac{31}{67} a^{11} - \frac{33}{67} a^{10} + \frac{12}{67} a^{9} - \frac{32}{67} a^{8} + \frac{9}{67} a^{7} - \frac{16}{67} a^{6} + \frac{26}{67} a^{5} - \frac{26}{67} a^{4} + \frac{7}{67} a^{3} + \frac{28}{67} a^{2} + \frac{16}{67} a - \frac{1}{67}$, $\frac{1}{24991} a^{25} - \frac{180}{24991} a^{24} + \frac{48}{24991} a^{23} - \frac{8394}{24991} a^{22} - \frac{5504}{24991} a^{21} - \frac{1789}{24991} a^{20} + \frac{7354}{24991} a^{19} - \frac{7357}{24991} a^{18} - \frac{3348}{24991} a^{17} - \frac{8820}{24991} a^{16} - \frac{8219}{24991} a^{15} - \frac{626}{24991} a^{14} - \frac{8785}{24991} a^{13} + \frac{1612}{24991} a^{12} - \frac{5295}{24991} a^{11} - \frac{9895}{24991} a^{10} + \frac{8650}{24991} a^{9} - \frac{6775}{24991} a^{8} - \frac{1067}{24991} a^{7} + \frac{4277}{24991} a^{6} + \frac{6615}{24991} a^{5} + \frac{4559}{24991} a^{4} - \frac{12348}{24991} a^{3} + \frac{10634}{24991} a^{2} - \frac{3428}{24991} a - \frac{2371}{24991}$, $\frac{1}{4018632481368851789} a^{26} - \frac{9164187088120}{4018632481368851789} a^{25} + \frac{9756}{1974289} a^{24} + \frac{15742965953823085}{4018632481368851789} a^{23} + \frac{301231894285664662}{4018632481368851789} a^{22} + \frac{1854086237406800177}{4018632481368851789} a^{21} + \frac{1198119516069590475}{4018632481368851789} a^{20} + \frac{1661002814701538776}{4018632481368851789} a^{19} + \frac{1935970873799544448}{4018632481368851789} a^{18} + \frac{961526220515910162}{4018632481368851789} a^{17} - \frac{1068721411255719431}{4018632481368851789} a^{16} - \frac{149752326454546722}{4018632481368851789} a^{15} + \frac{1826110108905038293}{4018632481368851789} a^{14} + \frac{1062781666359313279}{4018632481368851789} a^{13} + \frac{3293706990172020}{10773813622972793} a^{12} + \frac{795250459292100193}{4018632481368851789} a^{11} + \frac{1469651075958454424}{4018632481368851789} a^{10} - \frac{1234340535840071214}{4018632481368851789} a^{9} + \frac{642378447644951881}{4018632481368851789} a^{8} + \frac{990350867192513289}{4018632481368851789} a^{7} + \frac{351312950846234035}{4018632481368851789} a^{6} - \frac{160676330058184782}{4018632481368851789} a^{5} + \frac{709261909156301780}{4018632481368851789} a^{4} - \frac{1518603145442654385}{4018632481368851789} a^{3} + \frac{1748179704066416138}{4018632481368851789} a^{2} + \frac{147954540001039751}{4018632481368851789} a - \frac{7777273381888533}{50868765586947491}$, $\frac{1}{782264708700309765510040504606662037229469731219867} a^{27} + \frac{46570544129674953548597494147345}{782264708700309765510040504606662037229469731219867} a^{26} - \frac{15571210237899180873231819841615163966596509594}{782264708700309765510040504606662037229469731219867} a^{25} + \frac{5431185816534497959576448086597312658376869693962}{782264708700309765510040504606662037229469731219867} a^{24} + \frac{5345063811545835627305215989836266743339307443857}{782264708700309765510040504606662037229469731219867} a^{23} - \frac{17643520014139582266001129270227195024286840603538}{782264708700309765510040504606662037229469731219867} a^{22} + \frac{334527106746745653492232316276616148518406142932478}{782264708700309765510040504606662037229469731219867} a^{21} + \frac{267747318502634658780543676799492881411436764536136}{782264708700309765510040504606662037229469731219867} a^{20} - \frac{178247683767725129887999732779477146406617534075002}{782264708700309765510040504606662037229469731219867} a^{19} - \frac{118002627939745659131896032085348004978145414135377}{782264708700309765510040504606662037229469731219867} a^{18} - \frac{387721114163107634374880317769466604574837349838871}{782264708700309765510040504606662037229469731219867} a^{17} + \frac{251594806548042183853740916102347961988790687888222}{782264708700309765510040504606662037229469731219867} a^{16} + \frac{315594152559265547115289102358598174676656995485796}{782264708700309765510040504606662037229469731219867} a^{15} + \frac{294223761357691292785032530434396013982190785371200}{782264708700309765510040504606662037229469731219867} a^{14} - \frac{21624304918643621982866880144013875852833789396280}{782264708700309765510040504606662037229469731219867} a^{13} + \frac{112307926374413900417568654118953540781517211079680}{782264708700309765510040504606662037229469731219867} a^{12} - \frac{284673818578287419318440859883003216272235259561001}{782264708700309765510040504606662037229469731219867} a^{11} + \frac{66037714383462223528495968409832649010373499962904}{782264708700309765510040504606662037229469731219867} a^{10} - \frac{25407370584508019756528125550487107001212313026458}{782264708700309765510040504606662037229469731219867} a^{9} + \frac{212997727842107637610137322940990189168835572837086}{782264708700309765510040504606662037229469731219867} a^{8} + \frac{382273517765169514408764544838909694248115665824183}{782264708700309765510040504606662037229469731219867} a^{7} + \frac{254758651519916051192699650667445315722141046465001}{782264708700309765510040504606662037229469731219867} a^{6} - \frac{226342105561449984160812958926182792712703653664380}{782264708700309765510040504606662037229469731219867} a^{5} - \frac{209007935390394731906414941355904987535722446464279}{782264708700309765510040504606662037229469731219867} a^{4} - \frac{319274686229800315414800049874098367085205307485340}{782264708700309765510040504606662037229469731219867} a^{3} - \frac{58196733586223682722059289603256238231454330776084}{782264708700309765510040504606662037229469731219867} a^{2} - \frac{143405751459412164995689368382476853955682039751209}{782264708700309765510040504606662037229469731219867} a + \frac{3435055002879791884600267703063537550803500727000}{9902084920257085639367601324134962496575566217973}$
Class group and class number
$C_{203}$, which has order $203$ (assuming GRH)
Unit group
| Rank: | $13$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( \frac{19484256624830066084479257549146132}{384313971639714861166486517483616837367} a^{27} + \frac{23328403931461934586061990919318}{5736029427458430763678903246024131901} a^{26} - \frac{5655625859506795731327806991442230}{384313971639714861166486517483616837367} a^{25} - \frac{1085972559700393341024194990764530295}{384313971639714861166486517483616837367} a^{24} - \frac{222036171646406629478318164483563865}{384313971639714861166486517483616837367} a^{23} + \frac{33296530476837137387280139528892527}{384313971639714861166486517483616837367} a^{22} + \frac{22366582829669234472744771410677140515}{384313971639714861166486517483616837367} a^{21} + \frac{7330137531720316711425613221144859322}{384313971639714861166486517483616837367} a^{20} + \frac{4273825358624491215052055672157386632}{384313971639714861166486517483616837367} a^{19} - \frac{210392141229235345885710042539200520084}{384313971639714861166486517483616837367} a^{18} - \frac{96252763176466774627613656742524547413}{384313971639714861166486517483616837367} a^{17} - \frac{65335643796080313174454350346530745104}{384313971639714861166486517483616837367} a^{16} + \frac{895370921593694027819081519052215296443}{384313971639714861166486517483616837367} a^{15} + \frac{7702947804201029881771825302087026560}{4864733818224238748942867309919200473} a^{14} + \frac{292283491654081221832669361459408075834}{384313971639714861166486517483616837367} a^{13} - \frac{1386412506441237599875895846208488191235}{384313971639714861166486517483616837367} a^{12} - \frac{1843369365384312021180612591556002187684}{384313971639714861166486517483616837367} a^{11} - \frac{419737301740435974203965496160546950677}{384313971639714861166486517483616837367} a^{10} + \frac{41108641134642716326708557166932453056}{384313971639714861166486517483616837367} a^{9} + \frac{1551177707596446060803269897516068972624}{384313971639714861166486517483616837367} a^{8} + \frac{1364601089720989340066862228966628860875}{384313971639714861166486517483616837367} a^{7} + \frac{3012337905816300510527454308246005547310}{384313971639714861166486517483616837367} a^{6} + \frac{2522796857281906483060616587693390030566}{384313971639714861166486517483616837367} a^{5} + \frac{1698998269373640968427978378626612129699}{384313971639714861166486517483616837367} a^{4} + \frac{671711984447844853653502508932888447877}{384313971639714861166486517483616837367} a^{3} - \frac{25983582601985266877220218123311081900}{384313971639714861166486517483616837367} a^{2} + \frac{735194648674975386751851126891452695942}{384313971639714861166486517483616837367} a + \frac{1231915933935924231178229402334538943}{4864733818224238748942867309919200473} \) (order $6$) | 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: | \( 156859247232.13297 \) (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 | ${\href{/LocalNumberField/5.14.0.1}{14} }^{2}$ | R | ${\href{/LocalNumberField/11.14.0.1}{14} }^{2}$ | ${\href{/LocalNumberField/13.14.0.1}{14} }^{2}$ | ${\href{/LocalNumberField/17.14.0.1}{14} }^{2}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{14}$ | ${\href{/LocalNumberField/23.14.0.1}{14} }^{2}$ | ${\href{/LocalNumberField/29.14.0.1}{14} }^{2}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{14}$ | ${\href{/LocalNumberField/37.7.0.1}{7} }^{4}$ | ${\href{/LocalNumberField/41.14.0.1}{14} }^{2}$ | ${\href{/LocalNumberField/43.7.0.1}{7} }^{4}$ | ${\href{/LocalNumberField/47.14.0.1}{14} }^{2}$ | ${\href{/LocalNumberField/53.14.0.1}{14} }^{2}$ | ${\href{/LocalNumberField/59.14.0.1}{14} }^{2}$ |
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 | ||||||
| $7$ | 7.14.25.59 | $x^{14} - 168 x^{13} + 70 x^{12} - 147 x^{11} + 147 x^{10} - 98 x^{9} + 49 x^{8} + 168 x^{7} - 49 x^{4} - 147 x^{3} - 49 x^{2} + 98 x + 126$ | $14$ | $1$ | $25$ | $C_{14}$ | $[2]_{2}$ |
| 7.14.25.59 | $x^{14} - 168 x^{13} + 70 x^{12} - 147 x^{11} + 147 x^{10} - 98 x^{9} + 49 x^{8} + 168 x^{7} - 49 x^{4} - 147 x^{3} - 49 x^{2} + 98 x + 126$ | $14$ | $1$ | $25$ | $C_{14}$ | $[2]_{2}$ | |