Normalized defining polynomial
\( x^{28} - x^{27} - 127 x^{26} + 122 x^{25} + 6844 x^{24} - 6239 x^{23} - 208000 x^{22} + 174427 x^{21} + 3978918 x^{20} - 2921373 x^{19} - 50486029 x^{18} + 30063653 x^{17} + 435121500 x^{16} - 184317982 x^{15} - 2557698883 x^{14} + 574396734 x^{13} + 10101792966 x^{12} - 102364842 x^{11} - 25756963009 x^{10} - 5196774016 x^{9} + 39074109508 x^{8} + 15087293028 x^{7} - 29974362484 x^{6} - 15060917194 x^{5} + 8518851558 x^{4} + 3872502094 x^{3} - 1071658158 x^{2} - 256197636 x + 42607381 \)
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: | \(240112960522850526089657382557321510209178757190704345703125=3^{14}\cdot 5^{21}\cdot 29^{26}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $132.05$ | 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$$\chi_{435}(256,·)$, $\chi_{435}(1,·)$, $\chi_{435}(323,·)$, $\chi_{435}(199,·)$, $\chi_{435}(136,·)$, $\chi_{435}(364,·)$, $\chi_{435}(139,·)$, $\chi_{435}(332,·)$, $\chi_{435}(16,·)$, $\chi_{435}(212,·)$, $\chi_{435}(347,·)$, $\chi_{435}(92,·)$, $\chi_{435}(349,·)$, $\chi_{435}(286,·)$, $\chi_{435}(353,·)$, $\chi_{435}(226,·)$, $\chi_{435}(38,·)$, $\chi_{435}(167,·)$, $\chi_{435}(169,·)$, $\chi_{435}(428,·)$, $\chi_{435}(173,·)$, $\chi_{435}(49,·)$, $\chi_{435}(158,·)$, $\chi_{435}(122,·)$, $\chi_{435}(383,·)$, $\chi_{435}(94,·)$, $\chi_{435}(62,·)$, $\chi_{435}(181,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{41} a^{21} + \frac{7}{41} a^{20} + \frac{8}{41} a^{19} - \frac{7}{41} a^{18} - \frac{9}{41} a^{17} + \frac{19}{41} a^{16} + \frac{12}{41} a^{15} - \frac{1}{41} a^{14} + \frac{5}{41} a^{13} - \frac{7}{41} a^{12} - \frac{18}{41} a^{11} - \frac{6}{41} a^{10} - \frac{14}{41} a^{9} + \frac{3}{41} a^{8} + \frac{5}{41} a^{7} - \frac{7}{41} a^{6} + \frac{7}{41} a^{5} + \frac{2}{41} a^{4} + \frac{5}{41} a^{3} - \frac{20}{41} a^{2} + \frac{2}{41} a + \frac{1}{41}$, $\frac{1}{41} a^{22} + \frac{19}{41} a^{19} - \frac{1}{41} a^{18} + \frac{2}{41} a^{16} - \frac{3}{41} a^{15} + \frac{12}{41} a^{14} - \frac{1}{41} a^{13} - \frac{10}{41} a^{12} - \frac{3}{41} a^{11} - \frac{13}{41} a^{10} + \frac{19}{41} a^{9} - \frac{16}{41} a^{8} - \frac{1}{41} a^{7} + \frac{15}{41} a^{6} - \frac{6}{41} a^{5} - \frac{9}{41} a^{4} - \frac{14}{41} a^{3} + \frac{19}{41} a^{2} - \frac{13}{41} a - \frac{7}{41}$, $\frac{1}{41} a^{23} + \frac{19}{41} a^{20} - \frac{1}{41} a^{19} + \frac{2}{41} a^{17} - \frac{3}{41} a^{16} + \frac{12}{41} a^{15} - \frac{1}{41} a^{14} - \frac{10}{41} a^{13} - \frac{3}{41} a^{12} - \frac{13}{41} a^{11} + \frac{19}{41} a^{10} - \frac{16}{41} a^{9} - \frac{1}{41} a^{8} + \frac{15}{41} a^{7} - \frac{6}{41} a^{6} - \frac{9}{41} a^{5} - \frac{14}{41} a^{4} + \frac{19}{41} a^{3} - \frac{13}{41} a^{2} - \frac{7}{41} a$, $\frac{1}{2419} a^{24} + \frac{22}{2419} a^{23} + \frac{9}{2419} a^{22} - \frac{25}{2419} a^{21} - \frac{465}{2419} a^{20} - \frac{1146}{2419} a^{19} + \frac{14}{2419} a^{18} - \frac{465}{2419} a^{17} - \frac{93}{2419} a^{16} + \frac{77}{2419} a^{15} - \frac{454}{2419} a^{14} - \frac{821}{2419} a^{13} - \frac{271}{2419} a^{12} - \frac{76}{2419} a^{11} + \frac{713}{2419} a^{10} + \frac{1172}{2419} a^{9} - \frac{693}{2419} a^{8} - \frac{315}{2419} a^{7} + \frac{507}{2419} a^{6} - \frac{25}{59} a^{5} - \frac{7}{2419} a^{4} - \frac{884}{2419} a^{3} + \frac{1045}{2419} a^{2} - \frac{810}{2419} a + \frac{10}{41}$, $\frac{1}{2419} a^{25} - \frac{3}{2419} a^{23} + \frac{13}{2419} a^{22} + \frac{26}{2419} a^{21} + \frac{706}{2419} a^{20} - \frac{262}{2419} a^{19} - \frac{596}{2419} a^{18} - \frac{483}{2419} a^{17} + \frac{58}{2419} a^{16} - \frac{319}{2419} a^{15} - \frac{509}{2419} a^{14} + \frac{445}{2419} a^{13} + \frac{104}{2419} a^{12} - \frac{978}{2419} a^{11} - \frac{17}{41} a^{10} + \frac{309}{2419} a^{9} + \frac{830}{2419} a^{8} - \frac{528}{2419} a^{7} + \frac{1037}{2419} a^{6} - \frac{467}{2419} a^{5} + \frac{96}{2419} a^{4} - \frac{747}{2419} a^{3} - \frac{2}{59} a^{2} - \frac{175}{2419} a - \frac{3}{41}$, $\frac{1}{15703903681} a^{26} + \frac{2651111}{15703903681} a^{25} + \frac{457026}{15703903681} a^{24} + \frac{171524}{18943189} a^{23} + \frac{790531}{15703903681} a^{22} + \frac{50108571}{15703903681} a^{21} - \frac{5401435830}{15703903681} a^{20} - \frac{3605298997}{15703903681} a^{19} + \frac{272722654}{15703903681} a^{18} - \frac{6742305250}{15703903681} a^{17} + \frac{3227726090}{15703903681} a^{16} - \frac{1611897941}{15703903681} a^{15} + \frac{7288409062}{15703903681} a^{14} - \frac{3430748712}{15703903681} a^{13} - \frac{2838020576}{15703903681} a^{12} - \frac{2022514820}{15703903681} a^{11} + \frac{1157076465}{15703903681} a^{10} + \frac{2111064440}{15703903681} a^{9} + \frac{4569780525}{15703903681} a^{8} + \frac{5791433436}{15703903681} a^{7} + \frac{1717524566}{15703903681} a^{6} - \frac{4053481221}{15703903681} a^{5} + \frac{5413894380}{15703903681} a^{4} + \frac{1168573857}{15703903681} a^{3} - \frac{6751009701}{15703903681} a^{2} + \frac{38200436}{82219391} a + \frac{48853000}{266167859}$, $\frac{1}{98313787778868099493604179617045346175337841607334370837102226891691427730837498293844361206973791} a^{27} + \frac{2620539990171784144152464983684113771849469784277055540977188454762172491674564331865588}{98313787778868099493604179617045346175337841607334370837102226891691427730837498293844361206973791} a^{26} - \frac{8409184625036586783385764114725334435786948378118994003729779804876980823344931969727110622529}{98313787778868099493604179617045346175337841607334370837102226891691427730837498293844361206973791} a^{25} + \frac{1877521981574727173085996980892142219774368731738427173404752880766461570733868390969847501073}{98313787778868099493604179617045346175337841607334370837102226891691427730837498293844361206973791} a^{24} + \frac{1014128681952704887512574290046705284451834259747350064549086676645901108002088375278501621490335}{98313787778868099493604179617045346175337841607334370837102226891691427730837498293844361206973791} a^{23} - \frac{467148040815555015800974020685214101275499749209232604424483876579833493067184224858192152396980}{98313787778868099493604179617045346175337841607334370837102226891691427730837498293844361206973791} a^{22} - \frac{137565703147814895921693819610323859794164309307708063380836132474061119631598452680515832140615}{98313787778868099493604179617045346175337841607334370837102226891691427730837498293844361206973791} a^{21} - \frac{106755827166215397874709126324729240717665127573353660123820726363683627556831293674043605394999}{2397897262899221938868394624805984053057020527008155386270786021748571408069207275459618566023751} a^{20} - \frac{8234611624760802888070929320952187488300567175312999123710182690727267882748273207638305604084651}{98313787778868099493604179617045346175337841607334370837102226891691427730837498293844361206973791} a^{19} - \frac{22829278775380511264098906173506470488240086295988623231117121092219193672543613641321285569822607}{98313787778868099493604179617045346175337841607334370837102226891691427730837498293844361206973791} a^{18} + \frac{5079599890619222469642792621470529054047112398915339070615295207589342706736207870784092785772611}{98313787778868099493604179617045346175337841607334370837102226891691427730837498293844361206973791} a^{17} + \frac{14571182304648869373310791777702762630032225548411485509398378606610409714479212686260007156948469}{98313787778868099493604179617045346175337841607334370837102226891691427730837498293844361206973791} a^{16} - \frac{5128567407334484578860386560735763263955701088507893324114935399356329386029002761697883654688116}{98313787778868099493604179617045346175337841607334370837102226891691427730837498293844361206973791} a^{15} + \frac{37227974868455807000803513360326046265718274283656346898309047398543753432007151903508126310939393}{98313787778868099493604179617045346175337841607334370837102226891691427730837498293844361206973791} a^{14} - \frac{44160854786860159390192993340152646208538554192818677291852765477430390791180770652406223958581882}{98313787778868099493604179617045346175337841607334370837102226891691427730837498293844361206973791} a^{13} - \frac{8840663867564176857869759404884358138503968283351221580246850594884476430935789699390728699851242}{98313787778868099493604179617045346175337841607334370837102226891691427730837498293844361206973791} a^{12} - \frac{9141431645668030419143542509559074777778173208075990842921712187641820406518774437624311019805303}{98313787778868099493604179617045346175337841607334370837102226891691427730837498293844361206973791} a^{11} + \frac{24982033469327041287534276206106088193658353809483564241276641789761413629725726007786525060273380}{98313787778868099493604179617045346175337841607334370837102226891691427730837498293844361206973791} a^{10} - \frac{38067114432139031913518265944048581804942076368773445100967421874689666340590607890482747252288652}{98313787778868099493604179617045346175337841607334370837102226891691427730837498293844361206973791} a^{9} - \frac{10272321973376003159290024049946934410111174055879803506913129156946784266523017429250224224194664}{98313787778868099493604179617045346175337841607334370837102226891691427730837498293844361206973791} a^{8} - \frac{7431367596630246281037583169388719246665543664543852139542982550509699373351121974166326663662389}{98313787778868099493604179617045346175337841607334370837102226891691427730837498293844361206973791} a^{7} + \frac{17272942458062747865578291175061216983568337826064363563803195832973581755003581274540048382133768}{98313787778868099493604179617045346175337841607334370837102226891691427730837498293844361206973791} a^{6} - \frac{12420282157049717400415332643864928877997756413662793530527151267301454254002403800985120792382444}{98313787778868099493604179617045346175337841607334370837102226891691427730837498293844361206973791} a^{5} + \frac{41811711201323255920997398472541188460482938469119453434897407841583927039807738057010560287826368}{98313787778868099493604179617045346175337841607334370837102226891691427730837498293844361206973791} a^{4} - \frac{46817883295875029624633746570867308624828823830882071960067445641738525684503240526629473160123961}{98313787778868099493604179617045346175337841607334370837102226891691427730837498293844361206973791} a^{3} + \frac{45407201083256785458665233847171630446557019101621090167645848464757089101738892826025145802160431}{98313787778868099493604179617045346175337841607334370837102226891691427730837498293844361206973791} a^{2} + \frac{48077834589053747735940823824440678724159152542068597720154817005259992799312695818420777077335027}{98313787778868099493604179617045346175337841607334370837102226891691427730837498293844361206973791} a + \frac{669524036238303521901664907804649049055948423586788799665827078955978925027844064108956484724285}{1666335386082510160908545417238056714836234603514141878594952998164261486963347428709226461135149}$
Class group and class number
$C_{2}\times C_{2}$, which has order $4$ (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: | \( 322489910135565840000 \) (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
| \(\Q(\sqrt{5}) \), 4.4.946125.1, 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 | $28$ | R | R | $28$ | ${\href{/LocalNumberField/11.7.0.1}{7} }^{4}$ | $28$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{7}$ | ${\href{/LocalNumberField/19.7.0.1}{7} }^{4}$ | $28$ | R | ${\href{/LocalNumberField/31.14.0.1}{14} }^{2}$ | $28$ | ${\href{/LocalNumberField/41.1.0.1}{1} }^{28}$ | $28$ | $28$ | $28$ | ${\href{/LocalNumberField/59.1.0.1}{1} }^{28}$ |
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 | ||||||
| $29$ | 29.14.13.1 | $x^{14} - 29$ | $14$ | $1$ | $13$ | $C_{14}$ | $[\ ]_{14}$ |
| 29.14.13.1 | $x^{14} - 29$ | $14$ | $1$ | $13$ | $C_{14}$ | $[\ ]_{14}$ | |