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Properties

Label 32.0.48679729995...5376.6
Degree $32$
Signature $[0, 16]$
Discriminant $2^{128}\cdot 3^{16}\cdot 7^{16}$
Root discriminant $73.32$
Ramified primes $2, 3, 7$
Class number $1270784$ (GRH)
Class group $[8, 8, 19856]$ (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![65536, 0, 3145728, 0, 68485120, 0, 258048000, 0, 501224448, 0, 595027968, 0, 474418816, 0, 266433984, 0, 109128129, 0, 33321408, 0, 7679056, 0, 1338720, 0, 175124, 0, 16800, 0, 1128, 0, 48, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 + 48*x^30 + 1128*x^28 + 16800*x^26 + 175124*x^24 + 1338720*x^22 + 7679056*x^20 + 33321408*x^18 + 109128129*x^16 + 266433984*x^14 + 474418816*x^12 + 595027968*x^10 + 501224448*x^8 + 258048000*x^6 + 68485120*x^4 + 3145728*x^2 + 65536)
 
gp: K = bnfinit(x^32 + 48*x^30 + 1128*x^28 + 16800*x^26 + 175124*x^24 + 1338720*x^22 + 7679056*x^20 + 33321408*x^18 + 109128129*x^16 + 266433984*x^14 + 474418816*x^12 + 595027968*x^10 + 501224448*x^8 + 258048000*x^6 + 68485120*x^4 + 3145728*x^2 + 65536, 1)
 

Normalized defining polynomial

\( x^{32} + 48 x^{30} + 1128 x^{28} + 16800 x^{26} + 175124 x^{24} + 1338720 x^{22} + 7679056 x^{20} + 33321408 x^{18} + 109128129 x^{16} + 266433984 x^{14} + 474418816 x^{12} + 595027968 x^{10} + 501224448 x^{8} + 258048000 x^{6} + 68485120 x^{4} + 3145728 x^{2} + 65536 \)

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:  \(486797299958529611528622959063550881784243829894751113445376=2^{128}\cdot 3^{16}\cdot 7^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $73.32$
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:  $2, 3, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(672=2^{5}\cdot 3\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{672}(1,·)$, $\chi_{672}(517,·)$, $\chi_{672}(265,·)$, $\chi_{672}(13,·)$, $\chi_{672}(659,·)$, $\chi_{672}(407,·)$, $\chi_{672}(155,·)$, $\chi_{672}(671,·)$, $\chi_{672}(419,·)$, $\chi_{672}(421,·)$, $\chi_{672}(167,·)$, $\chi_{672}(169,·)$, $\chi_{672}(433,·)$, $\chi_{672}(181,·)$, $\chi_{672}(575,·)$, $\chi_{672}(323,·)$, $\chi_{672}(71,·)$, $\chi_{672}(587,·)$, $\chi_{672}(589,·)$, $\chi_{672}(335,·)$, $\chi_{672}(337,·)$, $\chi_{672}(83,·)$, $\chi_{672}(85,·)$, $\chi_{672}(601,·)$, $\chi_{672}(349,·)$, $\chi_{672}(97,·)$, $\chi_{672}(491,·)$, $\chi_{672}(239,·)$, $\chi_{672}(503,·)$, $\chi_{672}(505,·)$, $\chi_{672}(251,·)$, $\chi_{672}(253,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{2} a^{17} - \frac{1}{2} a$, $\frac{1}{4} a^{18} + \frac{1}{4} a^{2}$, $\frac{1}{8} a^{19} - \frac{1}{2} a^{11} + \frac{1}{8} a^{3}$, $\frac{1}{16} a^{20} - \frac{1}{2} a^{16} + \frac{1}{4} a^{12} + \frac{1}{16} a^{4}$, $\frac{1}{32} a^{21} - \frac{1}{4} a^{17} - \frac{3}{8} a^{13} - \frac{1}{2} a^{9} + \frac{1}{32} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{64} a^{22} - \frac{1}{8} a^{18} - \frac{1}{2} a^{16} + \frac{5}{16} a^{14} - \frac{1}{2} a^{12} + \frac{1}{4} a^{10} + \frac{1}{64} a^{6} + \frac{1}{4} a^{4} + \frac{1}{4} a^{2}$, $\frac{1}{128} a^{23} - \frac{1}{16} a^{19} - \frac{1}{4} a^{17} + \frac{5}{32} a^{15} + \frac{1}{4} a^{13} + \frac{1}{8} a^{11} - \frac{1}{2} a^{9} - \frac{63}{128} a^{7} + \frac{1}{8} a^{5} + \frac{1}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{256} a^{24} - \frac{1}{32} a^{20} - \frac{1}{8} a^{18} - \frac{27}{64} a^{16} - \frac{3}{8} a^{14} - \frac{7}{16} a^{12} - \frac{1}{4} a^{10} - \frac{63}{256} a^{8} - \frac{7}{16} a^{6} + \frac{1}{16} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{512} a^{25} - \frac{1}{64} a^{21} - \frac{1}{16} a^{19} - \frac{27}{128} a^{17} - \frac{3}{16} a^{15} - \frac{7}{32} a^{13} - \frac{1}{8} a^{11} + \frac{193}{512} a^{9} - \frac{7}{32} a^{7} - \frac{15}{32} a^{5} + \frac{3}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{1024} a^{26} - \frac{1}{128} a^{22} - \frac{1}{32} a^{20} - \frac{27}{256} a^{18} + \frac{13}{32} a^{16} + \frac{25}{64} a^{14} - \frac{1}{16} a^{12} - \frac{319}{1024} a^{10} + \frac{25}{64} a^{8} - \frac{15}{64} a^{6} - \frac{5}{16} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{2048} a^{27} - \frac{1}{256} a^{23} - \frac{1}{64} a^{21} - \frac{27}{512} a^{19} + \frac{13}{64} a^{17} - \frac{39}{128} a^{15} - \frac{1}{32} a^{13} - \frac{319}{2048} a^{11} + \frac{25}{128} a^{9} + \frac{49}{128} a^{7} + \frac{11}{32} a^{5} - \frac{1}{8} a^{3}$, $\frac{1}{13438976} a^{28} + \frac{277}{3359744} a^{26} + \frac{1067}{1679872} a^{24} + \frac{21}{209984} a^{22} + \frac{68549}{3359744} a^{20} - \frac{5645}{49408} a^{18} - \frac{299239}{839936} a^{16} - \frac{1809}{52496} a^{14} - \frac{4854847}{13438976} a^{12} - \frac{1075399}{3359744} a^{10} + \frac{11979}{49408} a^{8} + \frac{11649}{26248} a^{6} + \frac{18001}{52496} a^{4} + \frac{81}{13124} a^{2} - \frac{177}{3281}$, $\frac{1}{26877952} a^{29} + \frac{277}{6719488} a^{27} + \frac{1067}{3359744} a^{25} + \frac{21}{419968} a^{23} + \frac{68549}{6719488} a^{21} - \frac{5645}{98816} a^{19} - \frac{299239}{1679872} a^{17} + \frac{50687}{104992} a^{15} + \frac{8584129}{26877952} a^{13} - \frac{1075399}{6719488} a^{11} + \frac{11979}{98816} a^{9} + \frac{11649}{52496} a^{7} + \frac{18001}{104992} a^{5} - \frac{13043}{26248} a^{3} + \frac{1552}{3281} a$, $\frac{1}{945449007588843380398786299857197744799744} a^{30} - \frac{957844143599070664510233302115183}{29545281487151355637462071870537429524992} a^{28} - \frac{47540539928890030046791008248143271025}{118181125948605422549848287482149718099968} a^{26} + \frac{52254598080767032862739720452822053585}{29545281487151355637462071870537429524992} a^{24} - \frac{1358792424907824767805328665948844366107}{236362251897210845099696574964299436199936} a^{22} + \frac{82245936777350103497703723291543130609}{29545281487151355637462071870537429524992} a^{20} + \frac{3447675747361853038102306243254825565113}{59090562974302711274924143741074859049984} a^{18} - \frac{3297438428002769231532865050140723793589}{14772640743575677818731035935268714762496} a^{16} - \frac{104243017248478248593361944299937052795199}{945449007588843380398786299857197744799744} a^{14} - \frac{25783116385086218393706376913059341542789}{59090562974302711274924143741074859049984} a^{12} + \frac{26596529525316008178796384236483784865689}{59090562974302711274924143741074859049984} a^{10} + \frac{57814626900845824971022820108514357091}{461645023236739931835344872977147336328} a^{8} + \frac{1710011085059625888247506486165238388039}{3693160185893919454682758983817178690624} a^{6} + \frac{28488587291443474211373024932098919407}{57705627904592491479418109122143417041} a^{4} - \frac{50313835234128249886145302445646006487}{230822511618369965917672436488573668164} a^{2} - \frac{4207337442114008159165064255403503135}{57705627904592491479418109122143417041}$, $\frac{1}{1890898015177686760797572599714395489599488} a^{31} - \frac{957844143599070664510233302115183}{59090562974302711274924143741074859049984} a^{29} - \frac{47540539928890030046791008248143271025}{236362251897210845099696574964299436199936} a^{27} + \frac{52254598080767032862739720452822053585}{59090562974302711274924143741074859049984} a^{25} - \frac{1358792424907824767805328665948844366107}{472724503794421690199393149928598872399872} a^{23} + \frac{82245936777350103497703723291543130609}{59090562974302711274924143741074859049984} a^{21} + \frac{3447675747361853038102306243254825565113}{118181125948605422549848287482149718099968} a^{19} - \frac{3297438428002769231532865050140723793589}{29545281487151355637462071870537429524992} a^{17} + \frac{841205990340365131805424355557260692004545}{1890898015177686760797572599714395489599488} a^{15} - \frac{25783116385086218393706376913059341542789}{118181125948605422549848287482149718099968} a^{13} - \frac{32494033448986703096127759504591074184295}{118181125948605422549848287482149718099968} a^{11} - \frac{403830396335894106864322052868632979237}{923290046473479863670689745954294672656} a^{9} - \frac{1983149100834293566435252497651940302585}{7386320371787838909365517967634357381248} a^{7} + \frac{28488587291443474211373024932098919407}{115411255809184982958836218244286834082} a^{5} - \frac{50313835234128249886145302445646006487}{461645023236739931835344872977147336328} a^{3} + \frac{26749145231239241660126522433369956953}{57705627904592491479418109122143417041} a$

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

Class group and class number

$C_{8}\times C_{8}\times C_{19856}$, which has order $1270784$ (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:  \( -1 \) (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:  \( 94466336304.51273 \) (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

\(\Q(\sqrt{-14}) \), \(\Q(\sqrt{-21}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-42}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{6}, \sqrt{-14})\), \(\Q(\sqrt{2}, \sqrt{-7})\), \(\Q(\sqrt{3}, \sqrt{-14})\), \(\Q(\sqrt{2}, \sqrt{-21})\), \(\Q(\sqrt{3}, \sqrt{-7})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\sqrt{6}, \sqrt{-7})\), \(\Q(\zeta_{16})^+\), 4.0.100352.5, 4.0.903168.5, 4.4.18432.1, 8.0.12745506816.6, 8.0.10070523904.2, 8.0.815712436224.3, 8.0.3262849744896.5, 8.0.3262849744896.2, \(\Q(\zeta_{48})^+\), 8.0.3262849744896.3, \(\Q(\zeta_{32})^+\), 8.0.5156108238848.1, 8.0.417644767346688.52, 8.8.173946175488.1, 16.0.10646188457767892278050816.2, 16.0.26585452170716224216367104.1, 16.0.174427151692069147083584569344.1, 16.0.697708606768276588334338277376.6, 16.0.697708606768276588334338277376.3, \(\Q(\zeta_{96})^+\), 16.0.697708606768276588334338277376.4

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 ${\href{/LocalNumberField/5.8.0.1}{8} }^{4}$ R ${\href{/LocalNumberField/11.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{4}$

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

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
2Data not computed
3Data not computed
$7$7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$