Normalized defining polynomial
\( x^{28} - 7 x^{27} - 91 x^{26} + 623 x^{25} + 3640 x^{24} - 23205 x^{23} - 85141 x^{22} + 472657 x^{21} + 1298689 x^{20} - 5765676 x^{19} - 13490477 x^{18} + 43240799 x^{17} + 95742934 x^{16} - 195423844 x^{15} - 451889985 x^{14} + 487358557 x^{13} + 1342231212 x^{12} - 484759443 x^{11} - 2289033110 x^{10} - 289531417 x^{9} + 1935821398 x^{8} + 834820998 x^{7} - 668407222 x^{6} - 463370698 x^{5} + 34038536 x^{4} + 68490877 x^{3} + 8311506 x^{2} - 1013901 x - 63209 \)
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: | \(83709274472667248662717543234670163936409630856037139892578125=3^{14}\cdot 5^{21}\cdot 7^{48}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $162.75$ | 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: | $3, 5, 7$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(735=3\cdot 5\cdot 7^{2}\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{735}(512,·)$, $\chi_{735}(1,·)$, $\chi_{735}(323,·)$, $\chi_{735}(197,·)$, $\chi_{735}(64,·)$, $\chi_{735}(8,·)$, $\chi_{735}(631,·)$, $\chi_{735}(589,·)$, $\chi_{735}(526,·)$, $\chi_{735}(274,·)$, $\chi_{735}(211,·)$, $\chi_{735}(533,·)$, $\chi_{735}(407,·)$, $\chi_{735}(218,·)$, $\chi_{735}(92,·)$, $\chi_{735}(484,·)$, $\chi_{735}(421,·)$, $\chi_{735}(169,·)$, $\chi_{735}(106,·)$, $\chi_{735}(428,·)$, $\chi_{735}(722,·)$, $\chi_{735}(302,·)$, $\chi_{735}(113,·)$, $\chi_{735}(694,·)$, $\chi_{735}(617,·)$, $\chi_{735}(379,·)$, $\chi_{735}(316,·)$, $\chi_{735}(638,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{31} a^{20} + \frac{10}{31} a^{19} + \frac{7}{31} a^{18} + \frac{6}{31} a^{17} - \frac{11}{31} a^{16} - \frac{3}{31} a^{15} + \frac{5}{31} a^{14} + \frac{12}{31} a^{13} - \frac{3}{31} a^{12} - \frac{4}{31} a^{11} - \frac{1}{31} a^{10} + \frac{15}{31} a^{9} - \frac{6}{31} a^{8} + \frac{9}{31} a^{7} - \frac{11}{31} a^{6} + \frac{6}{31} a^{4} - \frac{4}{31} a^{3} - \frac{1}{31} a^{2} + \frac{7}{31} a$, $\frac{1}{31} a^{21} - \frac{2}{31} a^{18} - \frac{9}{31} a^{17} + \frac{14}{31} a^{16} + \frac{4}{31} a^{15} - \frac{7}{31} a^{14} + \frac{1}{31} a^{13} - \frac{5}{31} a^{12} + \frac{8}{31} a^{11} - \frac{6}{31} a^{10} - \frac{1}{31} a^{9} + \frac{7}{31} a^{8} - \frac{8}{31} a^{7} - \frac{14}{31} a^{6} + \frac{6}{31} a^{5} - \frac{2}{31} a^{4} + \frac{8}{31} a^{3} - \frac{14}{31} a^{2} - \frac{8}{31} a$, $\frac{1}{31} a^{22} - \frac{2}{31} a^{19} - \frac{9}{31} a^{18} + \frac{14}{31} a^{17} + \frac{4}{31} a^{16} - \frac{7}{31} a^{15} + \frac{1}{31} a^{14} - \frac{5}{31} a^{13} + \frac{8}{31} a^{12} - \frac{6}{31} a^{11} - \frac{1}{31} a^{10} + \frac{7}{31} a^{9} - \frac{8}{31} a^{8} - \frac{14}{31} a^{7} + \frac{6}{31} a^{6} - \frac{2}{31} a^{5} + \frac{8}{31} a^{4} - \frac{14}{31} a^{3} - \frac{8}{31} a^{2}$, $\frac{1}{31} a^{23} + \frac{11}{31} a^{19} - \frac{3}{31} a^{18} - \frac{15}{31} a^{17} + \frac{2}{31} a^{16} - \frac{5}{31} a^{15} + \frac{5}{31} a^{14} + \frac{1}{31} a^{13} - \frac{12}{31} a^{12} - \frac{9}{31} a^{11} + \frac{5}{31} a^{10} - \frac{9}{31} a^{9} + \frac{5}{31} a^{8} - \frac{7}{31} a^{7} + \frac{7}{31} a^{6} + \frac{8}{31} a^{5} - \frac{2}{31} a^{4} + \frac{15}{31} a^{3} - \frac{2}{31} a^{2} + \frac{14}{31} a$, $\frac{1}{589} a^{24} + \frac{9}{589} a^{22} - \frac{8}{589} a^{21} + \frac{2}{589} a^{20} + \frac{106}{589} a^{19} + \frac{12}{589} a^{18} - \frac{226}{589} a^{17} + \frac{235}{589} a^{16} - \frac{32}{589} a^{15} + \frac{269}{589} a^{14} + \frac{261}{589} a^{13} + \frac{6}{589} a^{12} + \frac{47}{589} a^{11} + \frac{8}{589} a^{10} - \frac{8}{31} a^{9} - \frac{1}{31} a^{8} - \frac{136}{589} a^{7} - \frac{192}{589} a^{6} + \frac{149}{589} a^{5} - \frac{44}{589} a^{4} + \frac{13}{31} a^{3} + \frac{63}{589} a^{2} + \frac{125}{589} a + \frac{3}{19}$, $\frac{1}{589} a^{25} + \frac{9}{589} a^{23} - \frac{8}{589} a^{22} + \frac{2}{589} a^{21} - \frac{8}{589} a^{20} + \frac{50}{589} a^{19} + \frac{154}{589} a^{18} + \frac{140}{589} a^{17} + \frac{44}{589} a^{16} + \frac{22}{589} a^{15} + \frac{280}{589} a^{14} - \frac{184}{589} a^{13} - \frac{200}{589} a^{12} - \frac{125}{589} a^{11} - \frac{2}{31} a^{10} + \frac{2}{31} a^{9} - \frac{41}{589} a^{8} - \frac{40}{589} a^{7} + \frac{225}{589} a^{6} - \frac{44}{589} a^{5} + \frac{8}{31} a^{4} - \frac{70}{589} a^{3} + \frac{239}{589} a^{2} - \frac{116}{589} a$, $\frac{1}{18259} a^{26} + \frac{12}{18259} a^{25} + \frac{8}{18259} a^{24} + \frac{157}{18259} a^{23} + \frac{11}{18259} a^{22} - \frac{128}{18259} a^{21} + \frac{9}{18259} a^{20} - \frac{4862}{18259} a^{19} + \frac{140}{961} a^{18} - \frac{311}{18259} a^{17} - \frac{7760}{18259} a^{16} - \frac{1875}{18259} a^{15} + \frac{462}{961} a^{14} - \frac{5006}{18259} a^{13} - \frac{7604}{18259} a^{12} - \frac{3637}{18259} a^{11} - \frac{7646}{18259} a^{10} + \frac{1270}{18259} a^{9} + \frac{300}{961} a^{8} - \frac{8042}{18259} a^{7} + \frac{3665}{18259} a^{6} - \frac{115}{589} a^{5} - \frac{2648}{18259} a^{4} + \frac{8747}{18259} a^{3} - \frac{2745}{18259} a^{2} - \frac{1460}{18259} a + \frac{206}{589}$, $\frac{1}{2337249317098578898246711927390610640680669129259765962230022157802545653492011637346783717412609} a^{27} - \frac{22829388851181287085052852818360664701226964584160906965572010093742921100470596234017014101}{2337249317098578898246711927390610640680669129259765962230022157802545653492011637346783717412609} a^{26} - \frac{1459486140931653365589426248964039046515058925958955730163545540567739579131345557335199438308}{2337249317098578898246711927390610640680669129259765962230022157802545653492011637346783717412609} a^{25} - \frac{729970588400292698017606859597324621409867815344914503748366666263329274705441223180275510611}{2337249317098578898246711927390610640680669129259765962230022157802545653492011637346783717412609} a^{24} - \frac{23734583189266836267347151676614146079534393042505204834322412808049284862567915673418029035924}{2337249317098578898246711927390610640680669129259765962230022157802545653492011637346783717412609} a^{23} - \frac{28607736019956101696515412988005509057046561502897582826848171937404419651609519091398884063}{3968165224276025294137032134788812632734582562410468526706319452975459513568780369009819554181} a^{22} - \frac{7390468212179035497947463823091787120166648037503527040410027097060899816381893790737945344063}{2337249317098578898246711927390610640680669129259765962230022157802545653492011637346783717412609} a^{21} - \frac{10516898721894892070945775297192088765228756682860758639249744738622731239504977141006400355825}{2337249317098578898246711927390610640680669129259765962230022157802545653492011637346783717412609} a^{20} + \frac{1144941559613182116637864485754486761376998426488327703107291118522685635332087675245528028402062}{2337249317098578898246711927390610640680669129259765962230022157802545653492011637346783717412609} a^{19} + \frac{958341819351609026920768541042087905829578687386635290596177557856825539294061208287461296391212}{2337249317098578898246711927390610640680669129259765962230022157802545653492011637346783717412609} a^{18} + \frac{995114612492999631358498695430631731241174409130560035750133366535019055711741432011862590401546}{2337249317098578898246711927390610640680669129259765962230022157802545653492011637346783717412609} a^{17} - \frac{512340061416296564061230479196638448923061489052584601655300323410187703257398361050567142034763}{2337249317098578898246711927390610640680669129259765962230022157802545653492011637346783717412609} a^{16} + \frac{824180573772640151535581235844487042604927937449381647183958437403691344290945480965131336455919}{2337249317098578898246711927390610640680669129259765962230022157802545653492011637346783717412609} a^{15} - \frac{697149075204368878988015076961519311359516685680000684095366588062094993868820219008950394650271}{2337249317098578898246711927390610640680669129259765962230022157802545653492011637346783717412609} a^{14} - \frac{635430233369960249678538181923721823271055571101018247232598020557584059281015563111182314784989}{2337249317098578898246711927390610640680669129259765962230022157802545653492011637346783717412609} a^{13} + \frac{44177913433781936730588327768233896125616557111305545121246321760261930453022091458905931607926}{123013121952556784118247996178453191614772059434724524327895903042239244920632191439304406179611} a^{12} - \frac{218334435995390154313890743587539909693765026582702521016113133851395663316952062180990460244678}{2337249317098578898246711927390610640680669129259765962230022157802545653492011637346783717412609} a^{11} + \frac{677772635378740406876030273564436396415042956170625558123525079977802091465995604094486865498094}{2337249317098578898246711927390610640680669129259765962230022157802545653492011637346783717412609} a^{10} + \frac{605755346991538264908456745326835067889927639947864308853043175722846994828957142479258528206982}{2337249317098578898246711927390610640680669129259765962230022157802545653492011637346783717412609} a^{9} + \frac{977749474048275841141236269919637479707327343869527328056943823692282646154730960356380576712949}{2337249317098578898246711927390610640680669129259765962230022157802545653492011637346783717412609} a^{8} - \frac{124066347739332071975157141638054828019782246017674120684641658803803688367398324542497187301544}{2337249317098578898246711927390610640680669129259765962230022157802545653492011637346783717412609} a^{7} + \frac{687371471084830585217318338310456083409703050173860578313709745299758556152007522456053136869687}{2337249317098578898246711927390610640680669129259765962230022157802545653492011637346783717412609} a^{6} - \frac{1165355048861332495253328485118179415551781678050745941432116374833389707085183209169184546095508}{2337249317098578898246711927390610640680669129259765962230022157802545653492011637346783717412609} a^{5} - \frac{283266466641182837620694150948458204803015007045249839300716717519833616242556062675741608859950}{2337249317098578898246711927390610640680669129259765962230022157802545653492011637346783717412609} a^{4} + \frac{1058351947294992458532043643977118291064805425978008628080061750949072834289157714234994891262868}{2337249317098578898246711927390610640680669129259765962230022157802545653492011637346783717412609} a^{3} - \frac{930901179932590634776049498576910959522103257305701560523870481572727482556787328402907405847466}{2337249317098578898246711927390610640680669129259765962230022157802545653492011637346783717412609} a^{2} - \frac{335264490189429477328691676454685602457488499190507617178101645195606175307688111587469528574537}{2337249317098578898246711927390610640680669129259765962230022157802545653492011637346783717412609} a - \frac{10855843502258976917595256972335126681619795794476602028350591794264906905646890165077858116165}{75395139261244480588603610560987440021957068685798902007420069606533730757806827011186571529439}$
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: | \( 16515401085022949000000 \) (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
| \(\Q(\sqrt{5}) \), \(\Q(\zeta_{15})^+\), 7.7.13841287201.1, 14.14.14967283701606751125078125.1 |
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 | $28$ | R | R | R | ${\href{/LocalNumberField/11.14.0.1}{14} }^{2}$ | $28$ | $28$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{14}$ | $28$ | ${\href{/LocalNumberField/29.7.0.1}{7} }^{4}$ | ${\href{/LocalNumberField/31.1.0.1}{1} }^{28}$ | $28$ | ${\href{/LocalNumberField/41.14.0.1}{14} }^{2}$ | $28$ | $28$ | $28$ | ${\href{/LocalNumberField/59.7.0.1}{7} }^{4}$ |
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 | Data not computed | ||||||
| 7 | Data not computed | ||||||