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Properties

Label 32.32.1217670149...0000.1
Degree $32$
Signature $[32, 0]$
Discriminant $2^{48}\cdot 5^{16}\cdot 17^{28}$
Root discriminant $75.45$
Ramified primes $2, 5, 17$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^2\times C_8$ (as 32T37)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1361, 44248, -182100, -1375256, 3719918, 13786936, -28450908, -64933250, 108780553, 166790036, -236469810, -256163602, 317228010, 250455230, -276907093, -162666214, 162802674, 72088572, -65844764, -22098426, 18500129, 4693052, -3607182, -682632, 481690, 66098, -42773, -4028, 2388, 138, -75, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 2*x^31 - 75*x^30 + 138*x^29 + 2388*x^28 - 4028*x^27 - 42773*x^26 + 66098*x^25 + 481690*x^24 - 682632*x^23 - 3607182*x^22 + 4693052*x^21 + 18500129*x^20 - 22098426*x^19 - 65844764*x^18 + 72088572*x^17 + 162802674*x^16 - 162666214*x^15 - 276907093*x^14 + 250455230*x^13 + 317228010*x^12 - 256163602*x^11 - 236469810*x^10 + 166790036*x^9 + 108780553*x^8 - 64933250*x^7 - 28450908*x^6 + 13786936*x^5 + 3719918*x^4 - 1375256*x^3 - 182100*x^2 + 44248*x + 1361)
 
gp: K = bnfinit(x^32 - 2*x^31 - 75*x^30 + 138*x^29 + 2388*x^28 - 4028*x^27 - 42773*x^26 + 66098*x^25 + 481690*x^24 - 682632*x^23 - 3607182*x^22 + 4693052*x^21 + 18500129*x^20 - 22098426*x^19 - 65844764*x^18 + 72088572*x^17 + 162802674*x^16 - 162666214*x^15 - 276907093*x^14 + 250455230*x^13 + 317228010*x^12 - 256163602*x^11 - 236469810*x^10 + 166790036*x^9 + 108780553*x^8 - 64933250*x^7 - 28450908*x^6 + 13786936*x^5 + 3719918*x^4 - 1375256*x^3 - 182100*x^2 + 44248*x + 1361, 1)
 

Normalized defining polynomial

\( x^{32} - 2 x^{31} - 75 x^{30} + 138 x^{29} + 2388 x^{28} - 4028 x^{27} - 42773 x^{26} + 66098 x^{25} + 481690 x^{24} - 682632 x^{23} - 3607182 x^{22} + 4693052 x^{21} + 18500129 x^{20} - 22098426 x^{19} - 65844764 x^{18} + 72088572 x^{17} + 162802674 x^{16} - 162666214 x^{15} - 276907093 x^{14} + 250455230 x^{13} + 317228010 x^{12} - 256163602 x^{11} - 236469810 x^{10} + 166790036 x^{9} + 108780553 x^{8} - 64933250 x^{7} - 28450908 x^{6} + 13786936 x^{5} + 3719918 x^{4} - 1375256 x^{3} - 182100 x^{2} + 44248 x + 1361 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $32$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[32, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1217670149940184118315109788118094613693071360000000000000000=2^{48}\cdot 5^{16}\cdot 17^{28}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $75.45$
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:  $2, 5, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(680=2^{3}\cdot 5\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{680}(1,·)$, $\chi_{680}(389,·)$, $\chi_{680}(9,·)$, $\chi_{680}(661,·)$, $\chi_{680}(529,·)$, $\chi_{680}(149,·)$, $\chi_{680}(409,·)$, $\chi_{680}(281,·)$, $\chi_{680}(161,·)$, $\chi_{680}(421,·)$, $\chi_{680}(169,·)$, $\chi_{680}(429,·)$, $\chi_{680}(49,·)$, $\chi_{680}(569,·)$, $\chi_{680}(189,·)$, $\chi_{680}(321,·)$, $\chi_{680}(69,·)$, $\chi_{680}(461,·)$, $\chi_{680}(81,·)$, $\chi_{680}(341,·)$, $\chi_{680}(441,·)$, $\chi_{680}(89,·)$, $\chi_{680}(349,·)$, $\chi_{680}(101,·)$, $\chi_{680}(229,·)$, $\chi_{680}(361,·)$, $\chi_{680}(621,·)$, $\chi_{680}(501,·)$, $\chi_{680}(489,·)$, $\chi_{680}(121,·)$, $\chi_{680}(509,·)$, $\chi_{680}(21,·)$$\rbrace$
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{4} a^{24} - \frac{1}{2} a^{21} - \frac{1}{4} a^{18} - \frac{1}{2} a^{16} + \frac{1}{4} a^{12} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4}$, $\frac{1}{4} a^{25} - \frac{1}{2} a^{22} - \frac{1}{4} a^{19} - \frac{1}{2} a^{17} + \frac{1}{4} a^{13} - \frac{1}{4} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a$, $\frac{1}{4} a^{26} - \frac{1}{2} a^{23} - \frac{1}{4} a^{20} - \frac{1}{2} a^{18} + \frac{1}{4} a^{14} - \frac{1}{4} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{2}$, $\frac{1}{4} a^{27} - \frac{1}{4} a^{21} - \frac{1}{2} a^{19} - \frac{1}{2} a^{18} + \frac{1}{4} a^{15} - \frac{1}{2} a^{12} - \frac{1}{4} a^{9} - \frac{1}{2} a^{7} + \frac{1}{4} a^{3} - \frac{1}{2}$, $\frac{1}{52} a^{28} + \frac{1}{52} a^{27} - \frac{3}{52} a^{26} - \frac{5}{52} a^{25} - \frac{5}{52} a^{24} - \frac{7}{26} a^{23} - \frac{23}{52} a^{22} - \frac{15}{52} a^{21} + \frac{1}{52} a^{20} + \frac{1}{4} a^{19} - \frac{15}{52} a^{18} + \frac{11}{26} a^{17} - \frac{1}{52} a^{16} - \frac{15}{52} a^{15} + \frac{21}{52} a^{14} - \frac{11}{52} a^{13} - \frac{11}{52} a^{12} + \frac{4}{13} a^{11} - \frac{21}{52} a^{10} - \frac{1}{4} a^{9} - \frac{19}{52} a^{8} - \frac{1}{52} a^{7} - \frac{5}{52} a^{6} - \frac{23}{52} a^{4} + \frac{23}{52} a^{3} - \frac{3}{52} a^{2} + \frac{1}{52} a - \frac{19}{52}$, $\frac{1}{12428} a^{29} + \frac{57}{12428} a^{28} + \frac{1483}{12428} a^{27} - \frac{274}{3107} a^{26} + \frac{1197}{12428} a^{25} + \frac{915}{12428} a^{24} - \frac{4837}{12428} a^{23} - \frac{5359}{12428} a^{22} + \frac{3321}{12428} a^{21} + \frac{1002}{3107} a^{20} + \frac{5263}{12428} a^{19} - \frac{5433}{12428} a^{18} + \frac{1491}{12428} a^{17} - \frac{3633}{12428} a^{16} - \frac{317}{956} a^{15} - \frac{997}{6214} a^{14} + \frac{1167}{12428} a^{13} + \frac{5913}{12428} a^{12} + \frac{4515}{12428} a^{11} - \frac{4673}{12428} a^{10} + \frac{5467}{12428} a^{9} + \frac{657}{6214} a^{8} - \frac{659}{12428} a^{7} + \frac{3633}{12428} a^{6} - \frac{1921}{12428} a^{5} - \frac{4567}{12428} a^{4} + \frac{5913}{12428} a^{3} + \frac{2627}{6214} a^{2} - \frac{3}{52} a - \frac{2091}{12428}$, $\frac{1}{4380846051244} a^{30} + \frac{29627601}{4380846051244} a^{29} + \frac{1496467699}{1095211512811} a^{28} - \frac{258137258965}{4380846051244} a^{27} - \frac{57889579587}{1095211512811} a^{26} - \frac{223006331507}{2190423025622} a^{25} + \frac{6110490377}{1095211512811} a^{24} + \frac{2158121501231}{4380846051244} a^{23} - \frac{715526305643}{2190423025622} a^{22} - \frac{1001103224789}{4380846051244} a^{21} - \frac{607134676391}{2190423025622} a^{20} - \frac{170721484364}{1095211512811} a^{19} + \frac{751537068183}{2190423025622} a^{18} + \frac{1570900383073}{4380846051244} a^{17} + \frac{804427829267}{2190423025622} a^{16} + \frac{1682139909877}{4380846051244} a^{15} - \frac{528574533899}{2190423025622} a^{14} - \frac{354027861349}{2190423025622} a^{13} - \frac{17036887875}{46604745226} a^{12} + \frac{830677619747}{4380846051244} a^{11} + \frac{927254430605}{2190423025622} a^{10} - \frac{133952622133}{336988157788} a^{9} + \frac{403985702529}{1095211512811} a^{8} + \frac{453489342340}{1095211512811} a^{7} + \frac{260662481324}{1095211512811} a^{6} + \frac{1654420499293}{4380846051244} a^{5} - \frac{1041597021669}{2190423025622} a^{4} - \frac{1823573546141}{4380846051244} a^{3} + \frac{908936142409}{2190423025622} a^{2} + \frac{1026446950515}{2190423025622} a - \frac{107028201301}{4380846051244}$, $\frac{1}{6400294590616270371798481665786755769996284967823735598392327485814802921338191284} a^{31} + \frac{184386556535462836025116563717588439716499405926423693832918541288787}{6400294590616270371798481665786755769996284967823735598392327485814802921338191284} a^{30} - \frac{60013849763598950957719074951170095825495109778823955303079524682115962837496}{1600073647654067592949620416446688942499071241955933899598081871453700730334547821} a^{29} - \frac{333993522960664739005456317916776365978431175714368693093493589420103483220347}{246165176562164245069172371761029068076780191070143676861243364839030881589930434} a^{28} + \frac{163651991626661798289234928589374989228088053292386022791706292185248248696850335}{3200147295308135185899240832893377884998142483911867799196163742907401460669095642} a^{27} - \frac{419893951178209371065594056319078241674503138284959883194278012563424295481743291}{6400294590616270371798481665786755769996284967823735598392327485814802921338191284} a^{26} - \frac{239862564169194162777952560002052892055026424430610226296552549695368292977375437}{6400294590616270371798481665786755769996284967823735598392327485814802921338191284} a^{25} + \frac{135991799783363697571658629923645925855357576965617436101949018545548171308236637}{1600073647654067592949620416446688942499071241955933899598081871453700730334547821} a^{24} + \frac{38694182772441058223839884834169896657618085401509149385782705335458819281952521}{123082588281082122534586185880514534038390095535071838430621682419515440794965217} a^{23} + \frac{849872568875762747477560507358277993341008379411387761012916135843357879970759025}{3200147295308135185899240832893377884998142483911867799196163742907401460669095642} a^{22} - \frac{1158542855027055456088340400521234124048769325535379705136019251970998282160376193}{3200147295308135185899240832893377884998142483911867799196163742907401460669095642} a^{21} + \frac{1041166712613742666719211133514745895038001484657224689044533281675622336233095595}{6400294590616270371798481665786755769996284967823735598392327485814802921338191284} a^{20} - \frac{2553670142840804415263617747247928292505954579310587432937547147973962357396614243}{6400294590616270371798481665786755769996284967823735598392327485814802921338191284} a^{19} - \frac{6729768872453432759317208668331544719318463326539742619017986464262421573926087}{68088240325705003955302996444539954999960478381103570195663058359731945971682886} a^{18} + \frac{177527117089624607083904139264606092694221260586218190272838100973050105924981198}{1600073647654067592949620416446688942499071241955933899598081871453700730334547821} a^{17} - \frac{1097794736736074187274283345703191243381574783794134726862890594794850059140327745}{3200147295308135185899240832893377884998142483911867799196163742907401460669095642} a^{16} + \frac{407053679463685967772604739807558735975986088493504884811733745515072280821714938}{1600073647654067592949620416446688942499071241955933899598081871453700730334547821} a^{15} - \frac{53901231333976906803791596072520551444268473609234765420006929459354785290373225}{136176480651410007910605992889079909999920956762207140391326116719463891943365772} a^{14} + \frac{3150924307232553563310438921868816159685107684176083044325476520676823530603475527}{6400294590616270371798481665786755769996284967823735598392327485814802921338191284} a^{13} - \frac{11185599261575603709371706069238957851394083761205468081119480295329644428617262}{1600073647654067592949620416446688942499071241955933899598081871453700730334547821} a^{12} - \frac{1581854601086994620309912490621904211029463386255248695964074602415329868796704045}{3200147295308135185899240832893377884998142483911867799196163742907401460669095642} a^{11} - \frac{14655311053541018371855291913802678928973893629440306604734022631310371086540415}{34044120162852501977651498222269977499980239190551785097831529179865972985841443} a^{10} + \frac{54136805237644142891163085459825270480141422324812950341417736177605108620161969}{246165176562164245069172371761029068076780191070143676861243364839030881589930434} a^{9} + \frac{1783352350663856705132496624328914064058251995951196454765467509509333024982449395}{6400294590616270371798481665786755769996284967823735598392327485814802921338191284} a^{8} - \frac{3061135837753481024212060289110097277326559610409430201153902525798490617386659143}{6400294590616270371798481665786755769996284967823735598392327485814802921338191284} a^{7} + \frac{534190383925630132543496472857233913813638351640898742675732687611167346289353143}{3200147295308135185899240832893377884998142483911867799196163742907401460669095642} a^{6} - \frac{1438671590514293111103898026173770212182400578332341494818073102572432888889589637}{3200147295308135185899240832893377884998142483911867799196163742907401460669095642} a^{5} - \frac{1180970666297598368839198010021876380982329893983481749363829689536306648172047717}{3200147295308135185899240832893377884998142483911867799196163742907401460669095642} a^{4} + \frac{1201456350117376900361102349717625204574856331835343514423857113053096025261067383}{3200147295308135185899240832893377884998142483911867799196163742907401460669095642} a^{3} - \frac{616945795935978571902479045274349361918136496952863763605961830425511033323748335}{6400294590616270371798481665786755769996284967823735598392327485814802921338191284} a^{2} - \frac{247339639787694322522963647976253678506747400755348744090326665938933170804517727}{1600073647654067592949620416446688942499071241955933899598081871453700730334547821} a - \frac{2628313076298842480586816646869924327607222339286282575397662078058862451458458953}{6400294590616270371798481665786755769996284967823735598392327485814802921338191284}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $31$
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:  \( 83633567199842760000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_8$ (as 32T37):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 32
The 32 conjugacy class representatives for $C_2^2\times C_8$
Character table for $C_2^2\times C_8$ is not computed

Intermediate fields

\(\Q(\sqrt{170}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{85}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{34}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}, \sqrt{85})\), \(\Q(\sqrt{10}, \sqrt{17})\), \(\Q(\sqrt{5}, \sqrt{34})\), \(\Q(\sqrt{2}, \sqrt{17})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{5}, \sqrt{17})\), \(\Q(\sqrt{10}, \sqrt{34})\), 4.4.4913.1, 4.4.7860800.1, 4.4.314432.1, 4.4.122825.1, 8.8.213813760000.1, 8.8.61792176640000.1, 8.8.61792176640000.2, 8.8.98867482624.1, 8.8.61792176640000.3, 8.8.15085980625.1, 8.8.61792176640000.4, 8.8.1680747204608.1, 8.8.256461670625.1, \(\Q(\zeta_{17})^+\), 8.8.1050467002880000.1, 16.16.3818273093908961689600000000.1, 16.16.1103480924139689928294400000000.1, 16.16.1103480924139689928294400000000.3, 16.16.2824911165797606216433664.1, 16.16.1103480924139689928294400000000.4, 16.16.1103480924139689928294400000000.2, 16.16.65772588499765987890625.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{4}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{16}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{8}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
5Data not computed
17Data not computed