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Properties

Label 32.0.55951614452...0896.1
Degree $32$
Signature $[0, 16]$
Discriminant $2^{32}\cdot 11^{16}\cdot 17^{28}$
Root discriminant $79.14$
Ramified primes $2, 11, 17$
Class number $2312$ (GRH)
Class group $[17, 136]$ (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![23670130201, 0, -41004582111, 0, 36301701182, 0, -21689088233, 0, 9723925436, 0, -3471939654, 0, 1019045945, 0, -251354668, 0, 52856810, 0, -9556173, 0, 1499568, 0, -202718, 0, 24189, 0, -2352, 0, 218, 0, -12, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 12*x^30 + 218*x^28 - 2352*x^26 + 24189*x^24 - 202718*x^22 + 1499568*x^20 - 9556173*x^18 + 52856810*x^16 - 251354668*x^14 + 1019045945*x^12 - 3471939654*x^10 + 9723925436*x^8 - 21689088233*x^6 + 36301701182*x^4 - 41004582111*x^2 + 23670130201)
 
gp: K = bnfinit(x^32 - 12*x^30 + 218*x^28 - 2352*x^26 + 24189*x^24 - 202718*x^22 + 1499568*x^20 - 9556173*x^18 + 52856810*x^16 - 251354668*x^14 + 1019045945*x^12 - 3471939654*x^10 + 9723925436*x^8 - 21689088233*x^6 + 36301701182*x^4 - 41004582111*x^2 + 23670130201, 1)
 

Normalized defining polynomial

\( x^{32} - 12 x^{30} + 218 x^{28} - 2352 x^{26} + 24189 x^{24} - 202718 x^{22} + 1499568 x^{20} - 9556173 x^{18} + 52856810 x^{16} - 251354668 x^{14} + 1019045945 x^{12} - 3471939654 x^{10} + 9723925436 x^{8} - 21689088233 x^{6} + 36301701182 x^{4} - 41004582111 x^{2} + 23670130201 \)

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

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:  \(5595161445268686921572205116053290617295140691904726508240896=2^{32}\cdot 11^{16}\cdot 17^{28}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $79.14$
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:  $2, 11, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(748=2^{2}\cdot 11\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{748}(1,·)$, $\chi_{748}(263,·)$, $\chi_{748}(395,·)$, $\chi_{748}(529,·)$, $\chi_{748}(659,·)$, $\chi_{748}(21,·)$, $\chi_{748}(155,·)$, $\chi_{748}(285,·)$, $\chi_{748}(287,·)$, $\chi_{748}(417,·)$, $\chi_{748}(681,·)$, $\chi_{748}(43,·)$, $\chi_{748}(307,·)$, $\chi_{748}(441,·)$, $\chi_{748}(705,·)$, $\chi_{748}(67,·)$, $\chi_{748}(331,·)$, $\chi_{748}(461,·)$, $\chi_{748}(727,·)$, $\chi_{748}(463,·)$, $\chi_{748}(593,·)$, $\chi_{748}(87,·)$, $\chi_{748}(89,·)$, $\chi_{748}(219,·)$, $\chi_{748}(353,·)$, $\chi_{748}(485,·)$, $\chi_{748}(747,·)$, $\chi_{748}(111,·)$, $\chi_{748}(373,·)$, $\chi_{748}(375,·)$, $\chi_{748}(637,·)$, $\chi_{748}(661,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{4} a^{24} - \frac{1}{4} a^{18} + \frac{1}{4} a^{12} - \frac{1}{4} a^{6} + \frac{1}{4}$, $\frac{1}{4} a^{25} - \frac{1}{4} a^{19} + \frac{1}{4} a^{13} - \frac{1}{4} a^{7} + \frac{1}{4} a$, $\frac{1}{268} a^{26} - \frac{5}{67} a^{24} + \frac{18}{67} a^{22} - \frac{57}{268} a^{20} + \frac{31}{67} a^{18} + \frac{7}{67} a^{16} + \frac{1}{268} a^{14} + \frac{8}{67} a^{12} + \frac{24}{67} a^{10} - \frac{21}{268} a^{8} - \frac{29}{67} a^{6} - \frac{6}{67} a^{4} - \frac{103}{268} a^{2} - \frac{13}{67}$, $\frac{1}{268} a^{27} - \frac{5}{67} a^{25} + \frac{18}{67} a^{23} - \frac{57}{268} a^{21} + \frac{31}{67} a^{19} + \frac{7}{67} a^{17} + \frac{1}{268} a^{15} + \frac{8}{67} a^{13} + \frac{24}{67} a^{11} - \frac{21}{268} a^{9} - \frac{29}{67} a^{7} - \frac{6}{67} a^{5} - \frac{103}{268} a^{3} - \frac{13}{67} a$, $\frac{1}{358852} a^{28} + \frac{58}{89713} a^{26} - \frac{943}{179426} a^{24} + \frac{25323}{358852} a^{22} + \frac{23508}{89713} a^{20} + \frac{17179}{179426} a^{18} - \frac{61819}{358852} a^{16} - \frac{2891}{6901} a^{14} + \frac{36307}{179426} a^{12} + \frac{58207}{358852} a^{10} - \frac{18504}{89713} a^{8} - \frac{52483}{179426} a^{6} + \frac{31101}{358852} a^{4} - \frac{4090}{89713} a^{2} + \frac{82357}{179426}$, $\frac{1}{358852} a^{29} + \frac{58}{89713} a^{27} - \frac{943}{179426} a^{25} + \frac{25323}{358852} a^{23} + \frac{23508}{89713} a^{21} + \frac{17179}{179426} a^{19} - \frac{61819}{358852} a^{17} - \frac{2891}{6901} a^{15} + \frac{36307}{179426} a^{13} + \frac{58207}{358852} a^{11} - \frac{18504}{89713} a^{9} - \frac{52483}{179426} a^{7} + \frac{31101}{358852} a^{5} - \frac{4090}{89713} a^{3} + \frac{82357}{179426} a$, $\frac{1}{255222070724238365249779009819503932706884023923829865711252} a^{30} + \frac{113850422259842011800080058198416079348027602960916977}{255222070724238365249779009819503932706884023923829865711252} a^{28} - \frac{98071871701575819512218900656763365851407525955128988605}{255222070724238365249779009819503932706884023923829865711252} a^{26} - \frac{1926386264004976010921035648670630472902796796067735922941}{63805517681059591312444752454875983176721005980957466427813} a^{24} - \frac{97043314736454175237641278669631852897315891401531236036105}{255222070724238365249779009819503932706884023923829865711252} a^{22} + \frac{45642132343666104333433512969467558503741634338884244149721}{255222070724238365249779009819503932706884023923829865711252} a^{20} + \frac{23127081846786605751309179642876038970336721186583850933756}{63805517681059591312444752454875983176721005980957466427813} a^{18} - \frac{53987766936304035178819146740276654182007296730865247605811}{255222070724238365249779009819503932706884023923829865711252} a^{16} - \frac{21682728478345422475151532163034720832469278342134766216901}{255222070724238365249779009819503932706884023923829865711252} a^{14} + \frac{29306703757897249479967129895858970287645944857579283212582}{63805517681059591312444752454875983176721005980957466427813} a^{12} - \frac{16644430491386932632699894682848999739330312492049566964673}{255222070724238365249779009819503932706884023923829865711252} a^{10} + \frac{1710924325618731468705308756471502696381620908923657803769}{19632466978787566557675308447654148669760309532602297362404} a^{8} + \frac{1932456877221290421591726510275641829718917827036573096118}{63805517681059591312444752454875983176721005980957466427813} a^{6} - \frac{75594804444385778178893900778990385055313201542901865244799}{255222070724238365249779009819503932706884023923829865711252} a^{4} + \frac{24952142675076858414792168324759687247604168472247146135}{19632466978787566557675308447654148669760309532602297362404} a^{2} + \frac{19440621840687453706804362615933241904459935565591160189285}{255222070724238365249779009819503932706884023923829865711252}$, $\frac{1}{39266170802994796732043750439740499550886813964705148669541831452} a^{31} + \frac{383078414756350450886393449193687761932074279091275074185}{381224959252376667301395635337286403406668096744710181257687684} a^{29} - \frac{38744027720279873763153966146742429616713080591962296045021653}{39266170802994796732043750439740499550886813964705148669541831452} a^{27} - \frac{1354443169497504763446918656775834755891142782592969871850460967}{39266170802994796732043750439740499550886813964705148669541831452} a^{25} - \frac{886855675983549391589669162572635643008302373551884643550591283}{39266170802994796732043750439740499550886813964705148669541831452} a^{23} - \frac{3082400261242028537124260110340130363114951883091291475322491199}{39266170802994796732043750439740499550886813964705148669541831452} a^{21} + \frac{7665218689115868065624877316975913310300880019383108747366922515}{39266170802994796732043750439740499550886813964705148669541831452} a^{19} + \frac{18117922453859359184235472661254444554421777971992867132664213127}{39266170802994796732043750439740499550886813964705148669541831452} a^{17} - \frac{11227439988089339029610772096407797716891263712937726214701093565}{39266170802994796732043750439740499550886813964705148669541831452} a^{15} + \frac{298359236199244153710294369574918376471778945214310842947255029}{3020474677153445902464903879980038426991293381900396051503217804} a^{13} + \frac{18615687320383557582054766500400541891533205434906963014495685813}{39266170802994796732043750439740499550886813964705148669541831452} a^{11} - \frac{19569043774222387889800928071068469807332781800126698748545940823}{39266170802994796732043750439740499550886813964705148669541831452} a^{9} - \frac{3565427060260698457373243509552569008929793859154913711223343009}{39266170802994796732043750439740499550886813964705148669541831452} a^{7} - \frac{8746972642226823698069275437727706258865183855075654094289442633}{39266170802994796732043750439740499550886813964705148669541831452} a^{5} + \frac{251890923915206123846827646588671040994232415808552286519216379}{39266170802994796732043750439740499550886813964705148669541831452} a^{3} + \frac{2210707752017070036582037641341026562983772728159710118333992739}{19633085401497398366021875219870249775443406982352574334770915726} a$

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

Class group and class number

$C_{17}\times C_{136}$, which has order $2312$ (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:  $15$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{2494067624241462626795437996973}{280696378676679095202445835382877809658} a^{31} - \frac{13548503169693542247908917692830}{140348189338339547601222917691438904829} a^{29} + \frac{493580512196307381696002422951409}{280696378676679095202445835382877809658} a^{27} - \frac{5152816803696336315207816943397169}{280696378676679095202445835382877809658} a^{25} + \frac{25526942075082002360271757634359842}{140348189338339547601222917691438904829} a^{23} - \frac{32131510170576706304963208372856545}{21592029128975315015572756567913677666} a^{21} + \frac{227088696774041472329669034936636429}{21592029128975315015572756567913677666} a^{19} - \frac{9071094843068572669556189068154629396}{140348189338339547601222917691438904829} a^{17} + \frac{94894636668984379358583954645546105643}{280696378676679095202445835382877809658} a^{15} - \frac{424881762819824718145680403870762474389}{280696378676679095202445835382877809658} a^{13} + \frac{795007545993502452161273831445676722118}{140348189338339547601222917691438904829} a^{11} - \frac{4895277350289120368971955239523032047081}{280696378676679095202445835382877809658} a^{9} + \frac{11966996245980712566286105045457245099393}{280696378676679095202445835382877809658} a^{7} - \frac{843923701069819338913723442703421899274}{10796014564487657507786378283956838833} a^{5} + \frac{26835824089162626680449346873718713742489}{280696378676679095202445835382877809658} a^{3} - \frac{7800715906062786258427799555001583377856}{140348189338339547601222917691438904829} a \) (order $4$)
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:  \( 107850976760852.73 \) (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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{11}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-17}) \), \(\Q(\sqrt{187}) \), \(\Q(\sqrt{-187}) \), \(\Q(i, \sqrt{11})\), \(\Q(i, \sqrt{17})\), \(\Q(i, \sqrt{187})\), \(\Q(\sqrt{11}, \sqrt{17})\), \(\Q(\sqrt{11}, \sqrt{-17})\), \(\Q(\sqrt{-11}, \sqrt{17})\), \(\Q(\sqrt{-11}, \sqrt{-17})\), 4.4.4913.1, 4.0.78608.1, 4.4.9511568.1, 4.0.594473.1, 8.0.313044726016.11, 8.0.6179217664.1, 8.0.90469925818624.17, 8.8.90469925818624.1, 8.0.90469925818624.4, 8.0.353398147729.1, 8.0.90469925818624.7, 8.0.6007768511393.1, 8.8.1537988738916608.1, 8.0.105046700288.1, \(\Q(\zeta_{17})^+\), 16.0.8184807477627329436545253376.1, 16.0.2365409361034298207161578225664.2, 16.0.11034809241396899282944.1, 16.0.2365409361034298207161578225664.1, 16.16.2365409361034298207161578225664.1, 16.0.36093282486485263170800449.1, 16.0.2365409361034298207161578225664.3

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{4}$ R ${\href{/LocalNumberField/13.2.0.1}{2} }^{16}$ 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$2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
11Data not computed
17Data not computed