Normalized defining polynomial
\( x^{28} - 4 x^{27} - 77 x^{26} + 298 x^{25} + 2444 x^{24} - 9196 x^{23} - 42045 x^{22} + 154538 x^{21} + 436038 x^{20} - 1566292 x^{19} - 2887069 x^{18} + 10049582 x^{17} + 12680015 x^{16} - 41758048 x^{15} - 38007063 x^{14} + 112864436 x^{13} + 79251721 x^{12} - 195680358 x^{11} - 115037594 x^{10} + 209457056 x^{9} + 112205318 x^{8} - 127991098 x^{7} - 66955254 x^{6} + 37949294 x^{5} + 20053613 x^{4} - 3584684 x^{3} - 1768008 x^{2} + 148696 x + 31321 \)
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: | \(16023667304285831340989917799610839168000000000000000000000=2^{28}\cdot 5^{21}\cdot 29^{24}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $119.88$ | 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: | $2, 5, 29$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(580=2^{2}\cdot 5\cdot 29\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{580}(1,·)$, $\chi_{580}(343,·)$, $\chi_{580}(107,·)$, $\chi_{580}(227,·)$, $\chi_{580}(81,·)$, $\chi_{580}(7,·)$, $\chi_{580}(523,·)$, $\chi_{580}(141,·)$, $\chi_{580}(401,·)$, $\chi_{580}(83,·)$, $\chi_{580}(407,·)$, $\chi_{580}(281,·)$, $\chi_{580}(103,·)$, $\chi_{580}(349,·)$, $\chi_{580}(223,·)$, $\chi_{580}(161,·)$, $\chi_{580}(547,·)$, $\chi_{580}(529,·)$, $\chi_{580}(169,·)$, $\chi_{580}(487,·)$, $\chi_{580}(429,·)$, $\chi_{580}(489,·)$, $\chi_{580}(49,·)$, $\chi_{580}(181,·)$, $\chi_{580}(567,·)$, $\chi_{580}(23,·)$, $\chi_{580}(123,·)$, $\chi_{580}(509,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $\frac{1}{41} a^{23} - \frac{15}{41} a^{22} - \frac{3}{41} a^{21} + \frac{11}{41} a^{20} + \frac{6}{41} a^{19} - \frac{6}{41} a^{18} + \frac{6}{41} a^{17} + \frac{11}{41} a^{16} - \frac{16}{41} a^{15} + \frac{2}{41} a^{14} - \frac{17}{41} a^{13} - \frac{14}{41} a^{12} + \frac{5}{41} a^{11} + \frac{12}{41} a^{10} + \frac{1}{41} a^{9} + \frac{16}{41} a^{8} - \frac{7}{41} a^{7} - \frac{7}{41} a^{6} - \frac{15}{41} a^{5} + \frac{6}{41} a^{4} - \frac{11}{41} a^{3} - \frac{5}{41} a^{2} + \frac{6}{41} a - \frac{8}{41}$, $\frac{1}{697} a^{24} + \frac{8}{697} a^{23} + \frac{21}{697} a^{22} + \frac{270}{697} a^{21} - \frac{151}{697} a^{20} - \frac{114}{697} a^{19} + \frac{114}{697} a^{18} + \frac{26}{697} a^{17} - \frac{337}{697} a^{16} - \frac{79}{697} a^{15} + \frac{193}{697} a^{14} - \frac{36}{697} a^{13} - \frac{71}{697} a^{12} - \frac{324}{697} a^{11} + \frac{318}{697} a^{10} - \frac{2}{697} a^{9} + \frac{197}{697} a^{8} + \frac{7}{41} a^{7} + \frac{234}{697} a^{6} + \frac{71}{697} a^{5} - \frac{283}{697} a^{4} + \frac{70}{697} a^{3} - \frac{150}{697} a^{2} + \frac{171}{697} a + \frac{103}{697}$, $\frac{1}{697} a^{25} + \frac{8}{697} a^{23} + \frac{2}{41} a^{22} + \frac{324}{697} a^{21} + \frac{261}{697} a^{20} - \frac{62}{697} a^{19} + \frac{202}{697} a^{18} - \frac{239}{697} a^{17} - \frac{307}{697} a^{16} + \frac{9}{697} a^{15} - \frac{84}{697} a^{14} + \frac{47}{697} a^{13} + \frac{227}{697} a^{12} - \frac{320}{697} a^{11} + \frac{157}{697} a^{10} + \frac{264}{697} a^{9} + \frac{56}{697} a^{8} + \frac{319}{697} a^{7} - \frac{67}{697} a^{6} - \frac{222}{697} a^{5} - \frac{148}{697} a^{4} + \frac{3}{17} a^{3} - \frac{278}{697} a^{2} - \frac{262}{697} a + \frac{162}{697}$, $\frac{1}{3327802946395527005877492877} a^{26} + \frac{21306286383589866871564}{56403439769415711964025303} a^{25} - \frac{1921032960000991237306159}{3327802946395527005877492877} a^{24} + \frac{36037282280364944247784555}{3327802946395527005877492877} a^{23} - \frac{1537397787448452268902161619}{3327802946395527005877492877} a^{22} - \frac{52271334121454323271457239}{195753114493854529757499581} a^{21} + \frac{21715278697616106161199875}{81165925521842122094572997} a^{20} - \frac{1296215910458651992571461597}{3327802946395527005877492877} a^{19} + \frac{456390537868672125043607942}{3327802946395527005877492877} a^{18} - \frac{44223685275104924191645840}{3327802946395527005877492877} a^{17} + \frac{778778749753993322310253136}{3327802946395527005877492877} a^{16} + \frac{1678730820342921370111543}{81165925521842122094572997} a^{15} - \frac{1086521853099071566996476498}{3327802946395527005877492877} a^{14} - \frac{310467949800067175604089304}{3327802946395527005877492877} a^{13} + \frac{274555801385351190533020193}{3327802946395527005877492877} a^{12} - \frac{50842785133066588128365120}{195753114493854529757499581} a^{11} - \frac{880929932054623792403813061}{3327802946395527005877492877} a^{10} + \frac{601435888108954113878615715}{3327802946395527005877492877} a^{9} - \frac{718639023289249748375965705}{3327802946395527005877492877} a^{8} + \frac{372983152826153464691034701}{3327802946395527005877492877} a^{7} + \frac{1090304869563016323740975812}{3327802946395527005877492877} a^{6} + \frac{165026731675020855953317502}{3327802946395527005877492877} a^{5} - \frac{18452700351600722184616829}{81165925521842122094572997} a^{4} + \frac{1254169859093402145148395470}{3327802946395527005877492877} a^{3} + \frac{1194186918528984196631539180}{3327802946395527005877492877} a^{2} + \frac{42864736645662587624635521}{3327802946395527005877492877} a + \frac{5528873663259365506259688}{3327802946395527005877492877}$, $\frac{1}{105868388445614249989656841323255168431850035394329133097} a^{27} - \frac{8572536642936610400801405001}{105868388445614249989656841323255168431850035394329133097} a^{26} - \frac{13334637758396465293931527677437634435443341953517818}{105868388445614249989656841323255168431850035394329133097} a^{25} - \frac{18102216521737046057683215976281801034373037789206003}{105868388445614249989656841323255168431850035394329133097} a^{24} + \frac{521460884220647821802654871663791058304999188177173511}{105868388445614249989656841323255168431850035394329133097} a^{23} - \frac{546674505714200912867814563202193391143776523507204393}{6227552261506720587626873019015009907755884434960537241} a^{22} - \frac{16548774962863026312867417622346748866057208559732054969}{105868388445614249989656841323255168431850035394329133097} a^{21} - \frac{22204808107355584186761992609482364916825528621692893151}{105868388445614249989656841323255168431850035394329133097} a^{20} + \frac{40309181601472557325651617735527825826457634798145145950}{105868388445614249989656841323255168431850035394329133097} a^{19} + \frac{50343688199720320532609283803710328149390707936251346080}{105868388445614249989656841323255168431850035394329133097} a^{18} + \frac{44850057876324977259976323959419403711090493023865773894}{105868388445614249989656841323255168431850035394329133097} a^{17} - \frac{44583580652205678802775175409384367978935046128884023488}{105868388445614249989656841323255168431850035394329133097} a^{16} + \frac{2650279892535069984384518250674189220290966151856618244}{105868388445614249989656841323255168431850035394329133097} a^{15} - \frac{1002570632736963452841394486130635537573411101574380793}{6227552261506720587626873019015009907755884434960537241} a^{14} + \frac{7101838026816329726359047881208523964474168386640556599}{105868388445614249989656841323255168431850035394329133097} a^{13} + \frac{6115505530626219271865280884875530276668493040492251582}{105868388445614249989656841323255168431850035394329133097} a^{12} + \frac{7212059820861688924056548721893101563094451534511155144}{105868388445614249989656841323255168431850035394329133097} a^{11} - \frac{13848172640281374131984821864982494065517236305121523114}{105868388445614249989656841323255168431850035394329133097} a^{10} - \frac{6875896964919378856385824068212313709703124394372208866}{105868388445614249989656841323255168431850035394329133097} a^{9} + \frac{267970745998848879442269361676440272225323190481007040}{105868388445614249989656841323255168431850035394329133097} a^{8} + \frac{683988716121086421895530944729846174501546770753697459}{6227552261506720587626873019015009907755884434960537241} a^{7} - \frac{48519497324198935156004948423299269691240497819401859596}{105868388445614249989656841323255168431850035394329133097} a^{6} - \frac{10567372058499089826386131890348188192590810562792880787}{105868388445614249989656841323255168431850035394329133097} a^{5} - \frac{24578203373453128787102253852487588467494177697227065238}{105868388445614249989656841323255168431850035394329133097} a^{4} + \frac{38714142263036844853190724532959331131163044830652061314}{105868388445614249989656841323255168431850035394329133097} a^{3} + \frac{11873556517527534646580446020070705434020103720389546830}{105868388445614249989656841323255168431850035394329133097} a^{2} - \frac{10337246843263210473704853880945134255014049576206746979}{105868388445614249989656841323255168431850035394329133097} a - \frac{17203342888115763067827984501126165528135569125698269156}{105868388445614249989656841323255168431850035394329133097}$
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: | \( 258467010943663670000 \) (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
| \(\Q(\sqrt{5}) \), \(\Q(\zeta_{20})^+\), 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 | R | $28$ | R | $28$ | ${\href{/LocalNumberField/11.14.0.1}{14} }^{2}$ | $28$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{7}$ | ${\href{/LocalNumberField/19.7.0.1}{7} }^{4}$ | $28$ | 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 |
|---|---|---|---|---|---|---|---|
| 2 | Data not computed | ||||||
| 5 | Data not computed | ||||||
| $29$ | 29.14.12.1 | $x^{14} + 2407 x^{7} + 1839267$ | $7$ | $2$ | $12$ | $C_{14}$ | $[\ ]_{7}^{2}$ |
| 29.14.12.1 | $x^{14} + 2407 x^{7} + 1839267$ | $7$ | $2$ | $12$ | $C_{14}$ | $[\ ]_{7}^{2}$ | |