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Properties

Label 32.32.7257879674...0000.1
Degree $32$
Signature $[32, 0]$
Discriminant $2^{32}\cdot 5^{24}\cdot 17^{28}$
Root discriminant $79.78$
Ramified primes $2, 5, 17$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_4\times C_8$ (as 32T43)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -74, 1651, -7564, -89234, 357464, 1309965, -4774168, -7358192, 26633854, 19655062, -75984420, -25600339, 121984038, 13245392, -115984606, 2809454, 68207746, -7114737, -25646062, 3963550, 6296568, -1179608, -1015754, 212831, 106300, -23918, -6922, 1634, 254, -62, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 4*x^31 - 62*x^30 + 254*x^29 + 1634*x^28 - 6922*x^27 - 23918*x^26 + 106300*x^25 + 212831*x^24 - 1015754*x^23 - 1179608*x^22 + 6296568*x^21 + 3963550*x^20 - 25646062*x^19 - 7114737*x^18 + 68207746*x^17 + 2809454*x^16 - 115984606*x^15 + 13245392*x^14 + 121984038*x^13 - 25600339*x^12 - 75984420*x^11 + 19655062*x^10 + 26633854*x^9 - 7358192*x^8 - 4774168*x^7 + 1309965*x^6 + 357464*x^5 - 89234*x^4 - 7564*x^3 + 1651*x^2 - 74*x + 1)
 
gp: K = bnfinit(x^32 - 4*x^31 - 62*x^30 + 254*x^29 + 1634*x^28 - 6922*x^27 - 23918*x^26 + 106300*x^25 + 212831*x^24 - 1015754*x^23 - 1179608*x^22 + 6296568*x^21 + 3963550*x^20 - 25646062*x^19 - 7114737*x^18 + 68207746*x^17 + 2809454*x^16 - 115984606*x^15 + 13245392*x^14 + 121984038*x^13 - 25600339*x^12 - 75984420*x^11 + 19655062*x^10 + 26633854*x^9 - 7358192*x^8 - 4774168*x^7 + 1309965*x^6 + 357464*x^5 - 89234*x^4 - 7564*x^3 + 1651*x^2 - 74*x + 1, 1)
 

Normalized defining polynomial

\( x^{32} - 4 x^{31} - 62 x^{30} + 254 x^{29} + 1634 x^{28} - 6922 x^{27} - 23918 x^{26} + 106300 x^{25} + 212831 x^{24} - 1015754 x^{23} - 1179608 x^{22} + 6296568 x^{21} + 3963550 x^{20} - 25646062 x^{19} - 7114737 x^{18} + 68207746 x^{17} + 2809454 x^{16} - 115984606 x^{15} + 13245392 x^{14} + 121984038 x^{13} - 25600339 x^{12} - 75984420 x^{11} + 19655062 x^{10} + 26633854 x^{9} - 7358192 x^{8} - 4774168 x^{7} + 1309965 x^{6} + 357464 x^{5} - 89234 x^{4} - 7564 x^{3} + 1651 x^{2} - 74 x + 1 \)

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:  \(7257879674078131427258907485712138496000000000000000000000000=2^{32}\cdot 5^{24}\cdot 17^{28}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $79.78$
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:  \(340=2^{2}\cdot 5\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{340}(1,·)$, $\chi_{340}(263,·)$, $\chi_{340}(9,·)$, $\chi_{340}(149,·)$, $\chi_{340}(281,·)$, $\chi_{340}(287,·)$, $\chi_{340}(161,·)$, $\chi_{340}(169,·)$, $\chi_{340}(43,·)$, $\chi_{340}(47,·)$, $\chi_{340}(49,·)$, $\chi_{340}(307,·)$, $\chi_{340}(183,·)$, $\chi_{340}(189,·)$, $\chi_{340}(321,·)$, $\chi_{340}(67,·)$, $\chi_{340}(69,·)$, $\chi_{340}(327,·)$, $\chi_{340}(203,·)$, $\chi_{340}(81,·)$, $\chi_{340}(83,·)$, $\chi_{340}(87,·)$, $\chi_{340}(89,·)$, $\chi_{340}(101,·)$, $\chi_{340}(223,·)$, $\chi_{340}(229,·)$, $\chi_{340}(103,·)$, $\chi_{340}(127,·)$, $\chi_{340}(247,·)$, $\chi_{340}(121,·)$, $\chi_{340}(123,·)$, $\chi_{340}(21,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{2} a^{24} - \frac{1}{2} a^{18} - \frac{1}{2} a^{12} - \frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{25} - \frac{1}{2} a^{19} - \frac{1}{2} a^{13} - \frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{2} a^{26} - \frac{1}{2} a^{20} - \frac{1}{2} a^{14} - \frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{27} - \frac{1}{2} a^{21} - \frac{1}{2} a^{15} - \frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{202} a^{28} - \frac{3}{101} a^{27} + \frac{15}{202} a^{26} - \frac{11}{202} a^{25} + \frac{5}{202} a^{24} + \frac{3}{101} a^{23} + \frac{53}{202} a^{22} + \frac{34}{101} a^{21} + \frac{25}{202} a^{20} + \frac{89}{202} a^{19} - \frac{59}{202} a^{18} - \frac{40}{101} a^{17} + \frac{47}{202} a^{16} + \frac{7}{101} a^{15} - \frac{1}{202} a^{14} + \frac{75}{202} a^{13} + \frac{51}{202} a^{12} + \frac{10}{101} a^{11} - \frac{19}{202} a^{10} - \frac{7}{101} a^{9} + \frac{65}{202} a^{8} - \frac{29}{202} a^{7} + \frac{55}{202} a^{6} + \frac{31}{101} a^{5} - \frac{21}{202} a^{4} - \frac{15}{101} a^{3} - \frac{39}{202} a^{2} - \frac{63}{202} a - \frac{77}{202}$, $\frac{1}{205838} a^{29} + \frac{421}{205838} a^{28} - \frac{42947}{205838} a^{27} + \frac{14979}{205838} a^{26} - \frac{14287}{205838} a^{25} + \frac{9605}{102919} a^{24} - \frac{33139}{205838} a^{23} - \frac{65575}{205838} a^{22} - \frac{45477}{205838} a^{21} + \frac{91867}{205838} a^{20} + \frac{91777}{205838} a^{19} + \frac{29228}{102919} a^{18} + \frac{97187}{205838} a^{17} - \frac{71625}{205838} a^{16} + \frac{1331}{205838} a^{15} - \frac{5301}{205838} a^{14} - \frac{95487}{205838} a^{13} - \frac{6928}{102919} a^{12} - \frac{98337}{205838} a^{11} - \frac{97411}{205838} a^{10} - \frac{90753}{205838} a^{9} + \frac{78529}{205838} a^{8} + \frac{54029}{205838} a^{7} - \frac{29889}{102919} a^{6} + \frac{74933}{205838} a^{5} + \frac{15041}{205838} a^{4} - \frac{55269}{205838} a^{3} - \frac{40653}{205838} a^{2} - \frac{29301}{205838} a + \frac{25728}{102919}$, $\frac{1}{310668777303829828778} a^{30} + \frac{679703118014959}{310668777303829828778} a^{29} - \frac{196622568145767617}{310668777303829828778} a^{28} + \frac{45048296239249572905}{310668777303829828778} a^{27} + \frac{30183662045673894705}{155334388651914914389} a^{26} + \frac{63396656396573593831}{310668777303829828778} a^{25} + \frac{2965514878903471059}{155334388651914914389} a^{24} - \frac{66131219125536843639}{310668777303829828778} a^{23} - \frac{37197957650027869319}{310668777303829828778} a^{22} - \frac{132063354978523086181}{310668777303829828778} a^{21} + \frac{74431518695848056431}{155334388651914914389} a^{20} - \frac{31599454168400177617}{310668777303829828778} a^{19} + \frac{37560851995417978441}{155334388651914914389} a^{18} + \frac{134354775737371690615}{310668777303829828778} a^{17} + \frac{108871775999156103527}{310668777303829828778} a^{16} + \frac{10353319933116863179}{310668777303829828778} a^{15} + \frac{18756933086647093692}{155334388651914914389} a^{14} + \frac{133445191917510014371}{310668777303829828778} a^{13} + \frac{34688092784419365066}{155334388651914914389} a^{12} + \frac{106861234158362173251}{310668777303829828778} a^{11} - \frac{123536349556955651133}{310668777303829828778} a^{10} - \frac{87215353880999641523}{310668777303829828778} a^{9} + \frac{10558792282589474533}{155334388651914914389} a^{8} + \frac{1402787590789637703}{3075928488156730978} a^{7} - \frac{43118845807915545948}{155334388651914914389} a^{6} + \frac{22755841207179205665}{310668777303829828778} a^{5} + \frac{141165439366312050215}{310668777303829828778} a^{4} + \frac{112937259447690584169}{310668777303829828778} a^{3} + \frac{68222237235882434286}{155334388651914914389} a^{2} - \frac{130778371926967587851}{310668777303829828778} a - \frac{144697036817108318465}{310668777303829828778}$, $\frac{1}{250856513670515466085268286084204893395823945029658878} a^{31} - \frac{250906044743113940213666522978641}{250856513670515466085268286084204893395823945029658878} a^{30} + \frac{246293490177916140999273237264684971703568847221}{250856513670515466085268286084204893395823945029658878} a^{29} - \frac{256664648984394463564487456038654867941120902620141}{250856513670515466085268286084204893395823945029658878} a^{28} - \frac{7239681299323572518122735609470403324207617344873456}{125428256835257733042634143042102446697911972514829439} a^{27} - \frac{20095212858392787378646435224893668434332088581199225}{250856513670515466085268286084204893395823945029658878} a^{26} + \frac{41790317703995774628626767689566536577707087497120343}{250856513670515466085268286084204893395823945029658878} a^{25} - \frac{26373121151237062254008047199402385883191872310830551}{250856513670515466085268286084204893395823945029658878} a^{24} + \frac{82038068800281358468350054698712802264600845349606335}{250856513670515466085268286084204893395823945029658878} a^{23} + \frac{77492529180185908717194914560864189033908445290147091}{250856513670515466085268286084204893395823945029658878} a^{22} - \frac{13575082277701403311038691785552856200144461956196196}{125428256835257733042634143042102446697911972514829439} a^{21} + \frac{80540943677071933562601984746222946252507048668042537}{250856513670515466085268286084204893395823945029658878} a^{20} - \frac{115272366978013922215629988556692286783229632902772461}{250856513670515466085268286084204893395823945029658878} a^{19} + \frac{59920830144089549241710121936612722397472070155444451}{250856513670515466085268286084204893395823945029658878} a^{18} - \frac{48251672821185649155846490781813034416313103191993437}{250856513670515466085268286084204893395823945029658878} a^{17} + \frac{79444130201022663578738810396331240997758579138260813}{250856513670515466085268286084204893395823945029658878} a^{16} + \frac{18506418893255127143103607135257697555369476037551531}{125428256835257733042634143042102446697911972514829439} a^{15} - \frac{47463542040333715181257782268347724405810350837879693}{250856513670515466085268286084204893395823945029658878} a^{14} - \frac{107270627668363203507576886013158501179393239626324769}{250856513670515466085268286084204893395823945029658878} a^{13} - \frac{65723977344925508354580641755313882722132391483445341}{250856513670515466085268286084204893395823945029658878} a^{12} - \frac{36307300006626545043429215651477834017314584975397689}{250856513670515466085268286084204893395823945029658878} a^{11} - \frac{68829890243641222375000508979582260148388953333345529}{250856513670515466085268286084204893395823945029658878} a^{10} - \frac{57496262450655289812256805270254093963509638793658515}{125428256835257733042634143042102446697911972514829439} a^{9} - \frac{273864423661997201306531314611700132080977914952027}{2483727858123915505794735505784206865305187574551078} a^{8} - \frac{101508892750964075454865700221691643762393331739186451}{250856513670515466085268286084204893395823945029658878} a^{7} + \frac{7746596889763618165475057766358209359235077285395651}{250856513670515466085268286084204893395823945029658878} a^{6} - \frac{83613515313837624938155538283177029321664660637835815}{250856513670515466085268286084204893395823945029658878} a^{5} - \frac{28514069595400332066972538321154604545570709175636963}{250856513670515466085268286084204893395823945029658878} a^{4} + \frac{58766218702664690226494309396773142920085259379576850}{125428256835257733042634143042102446697911972514829439} a^{3} + \frac{43606415249564119523858042805371497833621840951802341}{250856513670515466085268286084204893395823945029658878} a^{2} - \frac{26827289327465193629760791533193360146927743549485235}{125428256835257733042634143042102446697911972514829439} a + \frac{10652643648267551313250890282283590868497570004803945}{125428256835257733042634143042102446697911972514829439}$

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:  \( 228509556238802780000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\times C_8$ (as 32T43):

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 $C_4\times C_8$
Character table for $C_4\times C_8$ is not computed

Intermediate fields

\(\Q(\sqrt{85}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{5}) \), 4.4.9826000.2, \(\Q(\sqrt{5}, \sqrt{17})\), 4.4.9826000.1, 4.4.122825.1, 4.4.4913.1, 4.4.578000.1, \(\Q(\zeta_{20})^+\), 8.8.96550276000000.1, 8.8.15085980625.1, 8.8.334084000000.1, 8.8.1641354692000000.2, 8.8.1641354692000000.1, \(\Q(\zeta_{17})^+\), 8.8.256461670625.1, 16.16.9321955795676176000000000000.1, 16.16.2694045224950414864000000000000.1, 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 ${\href{/LocalNumberField/7.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{8}$

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}$
5Data not computed
17Data not computed