Normalized defining polynomial
\( x^{21} - 189 x^{19} - 63 x^{18} + 14273 x^{17} + 7672 x^{16} - 559650 x^{15} - 347577 x^{14} + 12591922 x^{13} + 7983444 x^{12} - 170719178 x^{11} - 106682856 x^{10} + 1419969761 x^{9} + 876942010 x^{8} - 7150956164 x^{7} - 4423657378 x^{6} + 20583982601 x^{5} + 12754970620 x^{4} - 29178026367 x^{3} - 16933928459 x^{2} + 12876797381 x + 3691462039 \)
Invariants
| Degree: | $21$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[21, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(511602258272191839961280749067569616320187702761=7^{38}\cdot 13^{14}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $187.01$ | 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: | $7, 13$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(637=7^{2}\cdot 13\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{637}(256,·)$, $\chi_{637}(1,·)$, $\chi_{637}(198,·)$, $\chi_{637}(456,·)$, $\chi_{637}(74,·)$, $\chi_{637}(16,·)$, $\chi_{637}(529,·)$, $\chi_{637}(274,·)$, $\chi_{637}(471,·)$, $\chi_{637}(347,·)$, $\chi_{637}(92,·)$, $\chi_{637}(289,·)$, $\chi_{637}(547,·)$, $\chi_{637}(165,·)$, $\chi_{637}(107,·)$, $\chi_{637}(620,·)$, $\chi_{637}(365,·)$, $\chi_{637}(562,·)$, $\chi_{637}(438,·)$, $\chi_{637}(183,·)$, $\chi_{637}(380,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{3870596894272440222805765367148971979858788123540698254867400616127258108798094342793330213} a^{20} - \frac{419842590607002008985991886153916636478208057306058380147646673092618533264278721169127849}{3870596894272440222805765367148971979858788123540698254867400616127258108798094342793330213} a^{19} - \frac{939961657360607987901323782225075386386197948405734489216403120849369276367845040752779700}{3870596894272440222805765367148971979858788123540698254867400616127258108798094342793330213} a^{18} + \frac{672221043756540786785053306726401847897574069131531709743973875599014900630936876419547887}{3870596894272440222805765367148971979858788123540698254867400616127258108798094342793330213} a^{17} - \frac{203336047035134500404934703205399157534175826838588518162176685150997672890841480393565040}{3870596894272440222805765367148971979858788123540698254867400616127258108798094342793330213} a^{16} - \frac{1781657319625941837124486998662787816336568200140309439419879993780837179394305188023262934}{3870596894272440222805765367148971979858788123540698254867400616127258108798094342793330213} a^{15} + \frac{655830512696693651972029065522693589463427147338223402152994137554888897246845016915127947}{3870596894272440222805765367148971979858788123540698254867400616127258108798094342793330213} a^{14} - \frac{1228199459110052858343563928937649667287944597177855210485465143831879814984702185025208854}{3870596894272440222805765367148971979858788123540698254867400616127258108798094342793330213} a^{13} + \frac{123524620186055035235425044406017657724139287414655346516452073533550281983484170548837635}{3870596894272440222805765367148971979858788123540698254867400616127258108798094342793330213} a^{12} + \frac{1127890542301323908697290232032422312295027454449941036135498474255521762182078569721127466}{3870596894272440222805765367148971979858788123540698254867400616127258108798094342793330213} a^{11} - \frac{420870233096743930238519787585850947195589442545616780147793402508023420852989956149162018}{3870596894272440222805765367148971979858788123540698254867400616127258108798094342793330213} a^{10} - \frac{1053142623616829067559033269194078561309287601899353118033611487520999025256017861395749237}{3870596894272440222805765367148971979858788123540698254867400616127258108798094342793330213} a^{9} + \frac{1694714318715807843165543878196844962896369853503674599026251444001963187603689389714275233}{3870596894272440222805765367148971979858788123540698254867400616127258108798094342793330213} a^{8} + \frac{1266553333879446873633486706990924285004089044701138450646925195400407884715629914054457527}{3870596894272440222805765367148971979858788123540698254867400616127258108798094342793330213} a^{7} - \frac{1785180878927276327777673141708034176060939639857043766771021054272391513953340835024346905}{3870596894272440222805765367148971979858788123540698254867400616127258108798094342793330213} a^{6} + \frac{458503782343717733373994459391505096151202580712229624461034514809337833379740654367612936}{3870596894272440222805765367148971979858788123540698254867400616127258108798094342793330213} a^{5} - \frac{817340110614819907556608317393059399423328374588915974819429897922288655873598832198319611}{3870596894272440222805765367148971979858788123540698254867400616127258108798094342793330213} a^{4} - \frac{366615502577909336627084255380619077001083877120117565153587077114125729935871108785447120}{3870596894272440222805765367148971979858788123540698254867400616127258108798094342793330213} a^{3} + \frac{776133983715866369798648301458297744036137732137386392201053710954005189622983581803782486}{3870596894272440222805765367148971979858788123540698254867400616127258108798094342793330213} a^{2} - \frac{1093460246756252525849988761700687687499183231673381302359956086648753681527628824394256735}{3870596894272440222805765367148971979858788123540698254867400616127258108798094342793330213} a + \frac{3375751958752329100161327739388148990891438785485222057373172995161530305839267778143097}{6802454998721336068199939133829476238767641693393142802930405300750892985585403062905677}$
Class group and class number
$C_{3}$, which has order $3$ (assuming GRH)
Unit group
| Rank: | $20$ | 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: | \( 28334278031696510 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A cyclic group of order 21 |
| The 21 conjugacy class representatives for $C_{21}$ |
| Character table for $C_{21}$ is not computed |
Intermediate fields
| 3.3.8281.1, 7.7.13841287201.1 |
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 | ${\href{/LocalNumberField/2.7.0.1}{7} }^{3}$ | $21$ | $21$ | R | $21$ | R | ${\href{/LocalNumberField/17.7.0.1}{7} }^{3}$ | ${\href{/LocalNumberField/19.3.0.1}{3} }^{7}$ | ${\href{/LocalNumberField/23.7.0.1}{7} }^{3}$ | $21$ | ${\href{/LocalNumberField/31.3.0.1}{3} }^{7}$ | ${\href{/LocalNumberField/37.7.0.1}{7} }^{3}$ | $21$ | $21$ | $21$ | $21$ | ${\href{/LocalNumberField/59.7.0.1}{7} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
| 7 | Data not computed | ||||||
| 13 | Data not computed | ||||||