Normalized defining polynomial
\( x^{32} - 2 x^{31} + x^{30} + 4 x^{29} + 108 x^{28} - 228 x^{27} + 404 x^{26} - 124 x^{25} + 5275 x^{24} - 10178 x^{23} + 29827 x^{22} - 29120 x^{21} + 181429 x^{20} - 275498 x^{19} + 1000869 x^{18} - 1175244 x^{17} + 4552448 x^{16} - 5579162 x^{15} + 19519380 x^{14} - 22301852 x^{13} + 72254162 x^{12} - 77576350 x^{11} + 231999798 x^{10} - 231692214 x^{9} + 641213764 x^{8} - 584431816 x^{7} + 1459356196 x^{6} - 1174733284 x^{5} + 2600147168 x^{4} - 1662828840 x^{3} + 3077744976 x^{2} - 1174279296 x + 1761923521 \)
Invariants
| Degree: | $32$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 16]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(99276740263879938750515115508224780490603194567662317338624=2^{48}\cdot 3^{16}\cdot 17^{30}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $69.77$ | 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: | $2, 3, 17$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(408=2^{3}\cdot 3\cdot 17\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{408}(1,·)$, $\chi_{408}(259,·)$, $\chi_{408}(11,·)$, $\chi_{408}(145,·)$, $\chi_{408}(275,·)$, $\chi_{408}(25,·)$, $\chi_{408}(41,·)$, $\chi_{408}(299,·)$, $\chi_{408}(49,·)$, $\chi_{408}(307,·)$, $\chi_{408}(371,·)$, $\chi_{408}(169,·)$, $\chi_{408}(19,·)$, $\chi_{408}(65,·)$, $\chi_{408}(67,·)$, $\chi_{408}(329,·)$, $\chi_{408}(331,·)$, $\chi_{408}(209,·)$, $\chi_{408}(355,·)$, $\chi_{408}(43,·)$, $\chi_{408}(377,·)$, $\chi_{408}(217,·)$, $\chi_{408}(347,·)$, $\chi_{408}(227,·)$, $\chi_{408}(131,·)$, $\chi_{408}(401,·)$, $\chi_{408}(233,·)$, $\chi_{408}(107,·)$, $\chi_{408}(113,·)$, $\chi_{408}(115,·)$, $\chi_{408}(361,·)$, $\chi_{408}(121,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $\frac{1}{438089} a^{30} - \frac{205790}{438089} a^{29} - \frac{9180}{438089} a^{28} + \frac{12640}{438089} a^{27} - \frac{133357}{438089} a^{26} + \frac{98668}{438089} a^{25} - \frac{145780}{438089} a^{24} + \frac{175641}{438089} a^{23} - \frac{65348}{438089} a^{22} - \frac{86725}{438089} a^{21} + \frac{205009}{438089} a^{20} - \frac{101474}{438089} a^{19} - \frac{89104}{438089} a^{18} + \frac{121855}{438089} a^{17} - \frac{70596}{438089} a^{16} - \frac{80363}{438089} a^{15} - \frac{151789}{438089} a^{14} + \frac{130754}{438089} a^{13} - \frac{107338}{438089} a^{12} + \frac{175280}{438089} a^{11} - \frac{22307}{438089} a^{10} + \frac{209734}{438089} a^{9} + \frac{53020}{438089} a^{8} - \frac{161656}{438089} a^{7} - \frac{204150}{438089} a^{6} + \frac{49939}{438089} a^{5} - \frac{93123}{438089} a^{4} + \frac{114256}{438089} a^{3} + \frac{51435}{438089} a^{2} + \frac{215637}{438089} a - \frac{132523}{438089}$, $\frac{1}{205376674574478129774996528705203327062488309729608350721274772358150462210964919227743109897479484485111099254858027951793} a^{31} + \frac{116013650997812581970109175583911920588211329706352560873510111613530750147159494776862905075851606386216956328913299}{205376674574478129774996528705203327062488309729608350721274772358150462210964919227743109897479484485111099254858027951793} a^{30} + \frac{102634310758999897232954964661632691407813526150171547547308226036634610555852909734341448350163457057487542379329208076958}{205376674574478129774996528705203327062488309729608350721274772358150462210964919227743109897479484485111099254858027951793} a^{29} - \frac{72342342867598792846235307455511080191671008625388634410170971945615920877358942175649600850178387228722948347675165870438}{205376674574478129774996528705203327062488309729608350721274772358150462210964919227743109897479484485111099254858027951793} a^{28} + \frac{20904387029612039960903746210859224570964056300199357415903497077207863394975188662170069283663126662593752792645630052205}{205376674574478129774996528705203327062488309729608350721274772358150462210964919227743109897479484485111099254858027951793} a^{27} + \frac{97729459576330864143639239074586230785947444727266194529022551309958163917829591761483642049620354538928105388550570578368}{205376674574478129774996528705203327062488309729608350721274772358150462210964919227743109897479484485111099254858027951793} a^{26} + \frac{43881832365035708969643095550221019556944663624123286089224060503421700748269728559092035221772966599313005812081683505122}{205376674574478129774996528705203327062488309729608350721274772358150462210964919227743109897479484485111099254858027951793} a^{25} + \frac{74006895423326030105954833781102865729501895728436148527402474820609194932018125890172189844029315959189052473326768447704}{205376674574478129774996528705203327062488309729608350721274772358150462210964919227743109897479484485111099254858027951793} a^{24} + \frac{79736811822566736378746867888991333857114651401930324316775634439217685282899879425096598462768983970164272917938761630351}{205376674574478129774996528705203327062488309729608350721274772358150462210964919227743109897479484485111099254858027951793} a^{23} - \frac{56556090443140978491193844176765730502976648985883384795669365574556733203545605045543590709676494729835185175303199447100}{205376674574478129774996528705203327062488309729608350721274772358150462210964919227743109897479484485111099254858027951793} a^{22} - \frac{72524440877668364462816212120945426567969584904958571996762459269978017708744619965113001988060132150481930981215997467192}{205376674574478129774996528705203327062488309729608350721274772358150462210964919227743109897479484485111099254858027951793} a^{21} - \frac{10086437969434747525000614506053958354476818437489733453498372066377297618306897191462010976280738351451122212859289885494}{205376674574478129774996528705203327062488309729608350721274772358150462210964919227743109897479484485111099254858027951793} a^{20} - \frac{75901056545254478557207798845120272924197238683246197408367343164185178409394655474857270392981026196906785343584750281379}{205376674574478129774996528705203327062488309729608350721274772358150462210964919227743109897479484485111099254858027951793} a^{19} + \frac{94854476209592766048978618045341682175624934372495580454499179074818103439553597733458513928743137461762621405558369634194}{205376674574478129774996528705203327062488309729608350721274772358150462210964919227743109897479484485111099254858027951793} a^{18} + \frac{92274834773963860964947664097388708736624311449179040504470734237787848771437569786894711319341130835583215662813030753965}{205376674574478129774996528705203327062488309729608350721274772358150462210964919227743109897479484485111099254858027951793} a^{17} - \frac{9815714551335195840284299819389281148060840388377695832405193624223292603380670098203957687365934494403496892115002918976}{205376674574478129774996528705203327062488309729608350721274772358150462210964919227743109897479484485111099254858027951793} a^{16} - \frac{59937622204194343553950673593934903304851267098216699035909952283388063358769996043800544526186255702618818036932855739592}{205376674574478129774996528705203327062488309729608350721274772358150462210964919227743109897479484485111099254858027951793} a^{15} + \frac{18537582453840524208232596704994301626643988458265438658326489021967835449387795188528283174765447078035149992436039442708}{205376674574478129774996528705203327062488309729608350721274772358150462210964919227743109897479484485111099254858027951793} a^{14} + \frac{47223481436160329012530581788117595801393188824654164234273631830383698286282772051723527699535681071401664621826326949531}{205376674574478129774996528705203327062488309729608350721274772358150462210964919227743109897479484485111099254858027951793} a^{13} - \frac{87897585889383132789942755670320906643197990117027285975001939030007698102672258579582379922589940333248042954580514347430}{205376674574478129774996528705203327062488309729608350721274772358150462210964919227743109897479484485111099254858027951793} a^{12} + \frac{17663833814541946564409639925765294217627105375286847120448043036402160574221478254875592752989762281236845821550904756429}{205376674574478129774996528705203327062488309729608350721274772358150462210964919227743109897479484485111099254858027951793} a^{11} + \frac{56759608732154317024130084489652192995146087563003723593107021084345216110893345625980461643146055568821628681577368936271}{205376674574478129774996528705203327062488309729608350721274772358150462210964919227743109897479484485111099254858027951793} a^{10} + \frac{91129179776235417659061318691215932343556659148489966390253314107695010395595069342033293631802391947242926195856051120680}{205376674574478129774996528705203327062488309729608350721274772358150462210964919227743109897479484485111099254858027951793} a^{9} - \frac{91687700654251588566761026756020617092069843756233869555869026990129412312632807292552574342306263420152811987372015199534}{205376674574478129774996528705203327062488309729608350721274772358150462210964919227743109897479484485111099254858027951793} a^{8} - \frac{13614643567697692406069554617269501615641831519891318909049922717924506651000344705911601812732456977175521205215936411588}{205376674574478129774996528705203327062488309729608350721274772358150462210964919227743109897479484485111099254858027951793} a^{7} + \frac{24126565453492029316362930710752139839489062597439889492223879341842253991362997112281368453001268168076715987969516425174}{205376674574478129774996528705203327062488309729608350721274772358150462210964919227743109897479484485111099254858027951793} a^{6} - \frac{27226418683468323414322591758894721768441328747484489492229616510323164704755146434134770600785666348447797013248982684142}{205376674574478129774996528705203327062488309729608350721274772358150462210964919227743109897479484485111099254858027951793} a^{5} - \frac{41342409856905722708663501855206761667217105258247849405099333609521486575510087847930557917216141984577621958461112252481}{205376674574478129774996528705203327062488309729608350721274772358150462210964919227743109897479484485111099254858027951793} a^{4} + \frac{13950672454567654545166983846788307255169372401879743758118346304265743874159412952278735589313289204512179618496700999367}{205376674574478129774996528705203327062488309729608350721274772358150462210964919227743109897479484485111099254858027951793} a^{3} + \frac{8487461544528583271315690795730938136452736166326147849627219053121880620382115428605427784722114309132523248792546042802}{205376674574478129774996528705203327062488309729608350721274772358150462210964919227743109897479484485111099254858027951793} a^{2} + \frac{50474938088155222776606150640624532037484217818386052446234615776099303804053196338603775002953106142536361368870923136945}{205376674574478129774996528705203327062488309729608350721274772358150462210964919227743109897479484485111099254858027951793} a + \frac{64425151860599110447540633966513052249457267698169734153927043565733145763810649727896106635958930203421885195361141628054}{205376674574478129774996528705203327062488309729608350721274772358150462210964919227743109897479484485111099254858027951793}$
Class group and class number
$C_{6}\times C_{99888}$, which has order $599328$ (assuming GRH)
Unit group
| Rank: | $15$ | 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: | \( 19955290291.92932 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2\times C_{16}$ (as 32T32):
| An abelian group of order 32 |
| The 32 conjugacy class representatives for $C_2\times C_{16}$ |
| Character table for $C_2\times C_{16}$ 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 | R | R | $16^{2}$ | $16^{2}$ | $16^{2}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ | R | ${\href{/LocalNumberField/19.8.0.1}{8} }^{4}$ | $16^{2}$ | $16^{2}$ | $16^{2}$ | $16^{2}$ | $16^{2}$ | ${\href{/LocalNumberField/43.8.0.1}{8} }^{4}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/53.8.0.1}{8} }^{4}$ | ${\href{/LocalNumberField/59.8.0.1}{8} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| 2 | Data not computed | ||||||
| 3 | Data not computed | ||||||
| 17 | Data not computed | ||||||