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Properties

Label 32.0.13969459638...0000.1
Degree $32$
Signature $[0, 16]$
Discriminant $2^{124}\cdot 3^{16}\cdot 5^{16}$
Root discriminant $56.82$
Ramified primes $2, 3, 5$
Class number $340$ (GRH)
Class group $[340]$ (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![16, 0, -384, 0, 7072, 0, -41472, 0, 161928, 0, -405120, 0, 740368, 0, -931104, 0, 868896, 0, -571296, 0, 277384, 0, -91200, 0, 21794, 0, -3360, 0, 372, 0, -24, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 24*x^30 + 372*x^28 - 3360*x^26 + 21794*x^24 - 91200*x^22 + 277384*x^20 - 571296*x^18 + 868896*x^16 - 931104*x^14 + 740368*x^12 - 405120*x^10 + 161928*x^8 - 41472*x^6 + 7072*x^4 - 384*x^2 + 16)
 
gp: K = bnfinit(x^32 - 24*x^30 + 372*x^28 - 3360*x^26 + 21794*x^24 - 91200*x^22 + 277384*x^20 - 571296*x^18 + 868896*x^16 - 931104*x^14 + 740368*x^12 - 405120*x^10 + 161928*x^8 - 41472*x^6 + 7072*x^4 - 384*x^2 + 16, 1)
 

Normalized defining polynomial

\( x^{32} - 24 x^{30} + 372 x^{28} - 3360 x^{26} + 21794 x^{24} - 91200 x^{22} + 277384 x^{20} - 571296 x^{18} + 868896 x^{16} - 931104 x^{14} + 740368 x^{12} - 405120 x^{10} + 161928 x^{8} - 41472 x^{6} + 7072 x^{4} - 384 x^{2} + 16 \)

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:  \(139694596386578246067777447172163512565760000000000000000=2^{124}\cdot 3^{16}\cdot 5^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $56.82$
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:  $2, 3, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(480=2^{5}\cdot 3\cdot 5\)
Dirichlet character group:    $\lbrace$$\chi_{480}(1,·)$, $\chi_{480}(259,·)$, $\chi_{480}(139,·)$, $\chi_{480}(401,·)$, $\chi_{480}(131,·)$, $\chi_{480}(409,·)$, $\chi_{480}(281,·)$, $\chi_{480}(289,·)$, $\chi_{480}(419,·)$, $\chi_{480}(41,·)$, $\chi_{480}(299,·)$, $\chi_{480}(49,·)$, $\chi_{480}(179,·)$, $\chi_{480}(59,·)$, $\chi_{480}(19,·)$, $\chi_{480}(449,·)$, $\chi_{480}(11,·)$, $\chi_{480}(161,·)$, $\chi_{480}(329,·)$, $\chi_{480}(331,·)$, $\chi_{480}(209,·)$, $\chi_{480}(211,·)$, $\chi_{480}(89,·)$, $\chi_{480}(91,·)$, $\chi_{480}(379,·)$, $\chi_{480}(361,·)$, $\chi_{480}(451,·)$, $\chi_{480}(241,·)$, $\chi_{480}(371,·)$, $\chi_{480}(169,·)$, $\chi_{480}(121,·)$, $\chi_{480}(251,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{2} a^{11}$, $\frac{1}{2} a^{12}$, $\frac{1}{2} a^{13}$, $\frac{1}{2} a^{14}$, $\frac{1}{2} a^{15}$, $\frac{1}{4} a^{16}$, $\frac{1}{4} a^{17}$, $\frac{1}{4} a^{18}$, $\frac{1}{4} a^{19}$, $\frac{1}{4} a^{20}$, $\frac{1}{4} a^{21}$, $\frac{1}{4} a^{22}$, $\frac{1}{4} a^{23}$, $\frac{1}{20224} a^{24} - \frac{21}{316} a^{22} - \frac{17}{158} a^{20} + \frac{97}{1264} a^{18} - \frac{4}{79} a^{16} + \frac{13}{79} a^{14} + \frac{289}{2528} a^{12} + \frac{4}{79} a^{10} - \frac{37}{632} a^{6} - \frac{20}{79} a^{4} + \frac{26}{79} a^{2} + \frac{433}{2528}$, $\frac{1}{20224} a^{25} - \frac{21}{316} a^{23} - \frac{17}{158} a^{21} + \frac{97}{1264} a^{19} - \frac{4}{79} a^{17} + \frac{13}{79} a^{15} + \frac{289}{2528} a^{13} + \frac{4}{79} a^{11} - \frac{37}{632} a^{7} - \frac{20}{79} a^{5} + \frac{26}{79} a^{3} + \frac{433}{2528} a$, $\frac{1}{151093504} a^{26} - \frac{1005}{151093504} a^{24} + \frac{50973}{1180418} a^{22} + \frac{940861}{9443344} a^{20} + \frac{1149247}{9443344} a^{18} - \frac{567}{14942} a^{16} + \frac{1532977}{18886688} a^{14} + \frac{1464483}{18886688} a^{12} - \frac{57249}{1180418} a^{10} + \frac{24295}{4721672} a^{8} + \frac{386089}{4721672} a^{6} + \frac{166572}{590209} a^{4} - \frac{2250575}{18886688} a^{2} + \frac{7766179}{18886688}$, $\frac{1}{151093504} a^{27} - \frac{1005}{151093504} a^{25} + \frac{50973}{1180418} a^{23} + \frac{940861}{9443344} a^{21} + \frac{1149247}{9443344} a^{19} - \frac{567}{14942} a^{17} + \frac{1532977}{18886688} a^{15} + \frac{1464483}{18886688} a^{13} - \frac{57249}{1180418} a^{11} + \frac{24295}{4721672} a^{9} + \frac{386089}{4721672} a^{7} + \frac{166572}{590209} a^{5} - \frac{2250575}{18886688} a^{3} + \frac{7766179}{18886688} a$, $\frac{1}{102592489216} a^{28} - \frac{109}{102592489216} a^{26} - \frac{422911}{51296244608} a^{24} - \frac{147868779}{6412030576} a^{22} + \frac{560173051}{6412030576} a^{20} - \frac{9917387}{458002184} a^{18} + \frac{1363222265}{12824061152} a^{16} + \frac{2375886803}{12824061152} a^{14} + \frac{152344055}{916004368} a^{12} - \frac{381501605}{3206015288} a^{10} - \frac{47087165}{458002184} a^{8} - \frac{254386901}{1603007644} a^{6} + \frac{1303093585}{12824061152} a^{4} - \frac{5011971485}{12824061152} a^{2} + \frac{2937862177}{6412030576}$, $\frac{1}{102592489216} a^{29} - \frac{109}{102592489216} a^{27} - \frac{422911}{51296244608} a^{25} - \frac{147868779}{6412030576} a^{23} + \frac{560173051}{6412030576} a^{21} - \frac{9917387}{458002184} a^{19} + \frac{1363222265}{12824061152} a^{17} + \frac{2375886803}{12824061152} a^{15} + \frac{152344055}{916004368} a^{13} - \frac{381501605}{3206015288} a^{11} - \frac{47087165}{458002184} a^{9} - \frac{254386901}{1603007644} a^{7} + \frac{1303093585}{12824061152} a^{5} - \frac{5011971485}{12824061152} a^{3} + \frac{2937862177}{6412030576} a$, $\frac{1}{31519593054320896} a^{30} - \frac{3753}{7879898263580224} a^{28} - \frac{30976763}{15759796527160448} a^{26} + \frac{149944201035}{7879898263580224} a^{24} - \frac{33118981699817}{492493641473764} a^{22} + \frac{49048881375}{4539111902984} a^{20} - \frac{202830953231047}{3939949131790112} a^{18} + \frac{9651035886981}{984987282947528} a^{16} - \frac{53091110743645}{281424937985008} a^{14} - \frac{55948476446255}{492493641473764} a^{12} + \frac{2332272264899}{35178117248126} a^{10} + \frac{101762350802281}{492493641473764} a^{8} + \frac{874772379478497}{3939949131790112} a^{6} + \frac{6901111841975}{984987282947528} a^{4} + \frac{473374149533829}{1969974565895056} a^{2} + \frac{29262369609873}{140712468992504}$, $\frac{1}{31519593054320896} a^{31} - \frac{3753}{7879898263580224} a^{29} - \frac{30976763}{15759796527160448} a^{27} + \frac{149944201035}{7879898263580224} a^{25} - \frac{33118981699817}{492493641473764} a^{23} + \frac{49048881375}{4539111902984} a^{21} - \frac{202830953231047}{3939949131790112} a^{19} + \frac{9651035886981}{984987282947528} a^{17} - \frac{53091110743645}{281424937985008} a^{15} - \frac{55948476446255}{492493641473764} a^{13} + \frac{2332272264899}{35178117248126} a^{11} + \frac{101762350802281}{492493641473764} a^{9} + \frac{874772379478497}{3939949131790112} a^{7} + \frac{6901111841975}{984987282947528} a^{5} + \frac{473374149533829}{1969974565895056} a^{3} + \frac{29262369609873}{140712468992504} a$

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

Class group and class number

$C_{340}$, which has order $340$ (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:  \( \frac{138348853213023}{31519593054320896} a^{30} - \frac{3308339308506173}{31519593054320896} a^{28} + \frac{25590034949140803}{15759796527160448} a^{26} - \frac{460448972663930175}{31519593054320896} a^{24} + \frac{185991817977597123}{1969974565895056} a^{22} - \frac{386411732987333595}{984987282947528} a^{20} + \frac{4667797627210088289}{3939949131790112} a^{18} - \frac{9495136860775454533}{3939949131790112} a^{16} + \frac{7131797277368893923}{1969974565895056} a^{14} - \frac{14984888895327517899}{3939949131790112} a^{12} + \frac{2919866961549038217}{984987282947528} a^{10} - \frac{770948977255591605}{492493641473764} a^{8} + \frac{2396961647767810203}{3939949131790112} a^{6} - \frac{572587403775997101}{3939949131790112} a^{4} + \frac{49985305337986131}{1969974565895056} a^{2} - \frac{1472070413755327}{3939949131790112} \) (order $6$)
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:  \( 10112591187794.762 \) (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

\(\Q(\sqrt{-30}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{5}, \sqrt{-6})\), \(\Q(\sqrt{2}, \sqrt{-15})\), \(\Q(\sqrt{-3}, \sqrt{10})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\sqrt{-6}, \sqrt{10})\), \(\Q(\zeta_{16})^+\), 4.0.460800.2, 4.4.51200.1, 4.0.18432.2, 8.0.207360000.1, 8.0.212336640000.4, 8.0.212336640000.2, 8.8.2621440000.1, 8.0.212336640000.5, 8.0.339738624.2, 8.0.212336640000.7, 8.8.108716359680000.1, 8.0.2147483648.1, 8.8.173946175488.1, 8.0.1342177280000.1, 16.0.45086848686489600000000.1, 16.0.11819246862071129702400000000.7, 16.0.11819246862071129702400000000.6, 16.16.11819246862071129702400000000.1, 16.0.1801439850948198400000000.1, 16.0.11819246862071129702400000000.3, 16.0.30257271966902092038144.2

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 R ${\href{/LocalNumberField/7.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{4}$

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
5Data not computed