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Properties

Label 32.0.22351135421...0000.5
Degree $32$
Signature $[0, 16]$
Discriminant $2^{128}\cdot 3^{16}\cdot 5^{16}$
Root discriminant $61.97$
Ramified primes $2, 3, 5$
Class number $720$ (GRH)
Class group $[3, 240]$ (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![4866436, 0, 140992, 0, -736112, 0, 1437408, 0, -1255068, 0, 239392, 0, 711048, 0, -1101648, 0, 978421, 0, -615296, 0, 283360, 0, -95680, 0, 23400, 0, -4032, 0, 464, 0, -32, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 32*x^30 + 464*x^28 - 4032*x^26 + 23400*x^24 - 95680*x^22 + 283360*x^20 - 615296*x^18 + 978421*x^16 - 1101648*x^14 + 711048*x^12 + 239392*x^10 - 1255068*x^8 + 1437408*x^6 - 736112*x^4 + 140992*x^2 + 4866436)
 
gp: K = bnfinit(x^32 - 32*x^30 + 464*x^28 - 4032*x^26 + 23400*x^24 - 95680*x^22 + 283360*x^20 - 615296*x^18 + 978421*x^16 - 1101648*x^14 + 711048*x^12 + 239392*x^10 - 1255068*x^8 + 1437408*x^6 - 736112*x^4 + 140992*x^2 + 4866436, 1)
 

Normalized defining polynomial

\( x^{32} - 32 x^{30} + 464 x^{28} - 4032 x^{26} + 23400 x^{24} - 95680 x^{22} + 283360 x^{20} - 615296 x^{18} + 978421 x^{16} - 1101648 x^{14} + 711048 x^{12} + 239392 x^{10} - 1255068 x^{8} + 1437408 x^{6} - 736112 x^{4} + 140992 x^{2} + 4866436 \)

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:  \(2235113542185251937084439154754616201052160000000000000000=2^{128}\cdot 3^{16}\cdot 5^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $61.97$
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$$\chi_{480}(1,·)$, $\chi_{480}(419,·)$, $\chi_{480}(299,·)$, $\chi_{480}(431,·)$, $\chi_{480}(179,·)$, $\chi_{480}(311,·)$, $\chi_{480}(59,·)$, $\chi_{480}(319,·)$, $\chi_{480}(449,·)$, $\chi_{480}(451,·)$, $\chi_{480}(71,·)$, $\chi_{480}(329,·)$, $\chi_{480}(439,·)$, $\chi_{480}(461,·)$, $\chi_{480}(79,·)$, $\chi_{480}(209,·)$, $\chi_{480}(211,·)$, $\chi_{480}(341,·)$, $\chi_{480}(89,·)$, $\chi_{480}(331,·)$, $\chi_{480}(91,·)$, $\chi_{480}(349,·)$, $\chi_{480}(199,·)$, $\chi_{480}(101,·)$, $\chi_{480}(229,·)$, $\chi_{480}(361,·)$, $\chi_{480}(109,·)$, $\chi_{480}(241,·)$, $\chi_{480}(221,·)$, $\chi_{480}(121,·)$, $\chi_{480}(191,·)$, $\chi_{480}(469,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{987} a^{16} - \frac{16}{987} a^{14} + \frac{104}{987} a^{12} - \frac{352}{987} a^{10} - \frac{109}{329} a^{8} + \frac{15}{47} a^{6} + \frac{16}{47} a^{4} - \frac{64}{987} a^{2} + \frac{379}{987}$, $\frac{1}{987} a^{17} - \frac{16}{987} a^{15} + \frac{104}{987} a^{13} - \frac{352}{987} a^{11} - \frac{109}{329} a^{9} + \frac{15}{47} a^{7} + \frac{16}{47} a^{5} - \frac{64}{987} a^{3} + \frac{379}{987} a$, $\frac{1}{987} a^{18} - \frac{152}{987} a^{14} + \frac{325}{987} a^{12} - \frac{37}{987} a^{10} + \frac{6}{329} a^{8} + \frac{21}{47} a^{6} + \frac{377}{987} a^{4} + \frac{114}{329} a^{2} + \frac{142}{987}$, $\frac{1}{987} a^{19} - \frac{152}{987} a^{15} + \frac{325}{987} a^{13} - \frac{37}{987} a^{11} + \frac{6}{329} a^{9} + \frac{21}{47} a^{7} + \frac{377}{987} a^{5} + \frac{114}{329} a^{3} + \frac{142}{987} a$, $\frac{1}{987} a^{20} - \frac{19}{141} a^{14} - \frac{1}{47} a^{12} - \frac{4}{21} a^{10} + \frac{29}{329} a^{8} - \frac{106}{987} a^{6} + \frac{30}{329} a^{4} + \frac{284}{987} a^{2} + \frac{362}{987}$, $\frac{1}{987} a^{21} - \frac{19}{141} a^{15} - \frac{1}{47} a^{13} - \frac{4}{21} a^{11} + \frac{29}{329} a^{9} - \frac{106}{987} a^{7} + \frac{30}{329} a^{5} + \frac{284}{987} a^{3} + \frac{362}{987} a$, $\frac{1}{987} a^{22} - \frac{25}{141} a^{14} - \frac{58}{329} a^{12} - \frac{340}{987} a^{10} - \frac{169}{987} a^{8} - \frac{152}{329} a^{6} - \frac{430}{987} a^{4} - \frac{254}{987} a^{2} + \frac{10}{141}$, $\frac{1}{987} a^{23} - \frac{25}{141} a^{15} - \frac{58}{329} a^{13} - \frac{340}{987} a^{11} - \frac{169}{987} a^{9} - \frac{152}{329} a^{7} - \frac{430}{987} a^{5} - \frac{254}{987} a^{3} + \frac{10}{141} a$, $\frac{1}{2179296} a^{24} - \frac{1}{90804} a^{22} + \frac{1}{8648} a^{20} + \frac{43}{136206} a^{18} - \frac{45}{121072} a^{16} - \frac{2545}{22701} a^{14} + \frac{3341}{136206} a^{12} + \frac{150}{7567} a^{10} - \frac{12521}{34592} a^{8} - \frac{373}{828} a^{6} - \frac{54137}{181608} a^{4} - \frac{5015}{15134} a^{2} + \frac{246193}{1089648}$, $\frac{1}{4807526976} a^{25} - \frac{31}{66771208} a^{23} + \frac{4253}{400627248} a^{21} - \frac{31835}{300470436} a^{19} - \frac{46913}{114464928} a^{17} + \frac{703739}{50078406} a^{15} - \frac{30148141}{300470436} a^{13} + \frac{8737189}{25039203} a^{11} - \frac{447373373}{1602508992} a^{9} + \frac{144010003}{600940872} a^{7} - \frac{13195417}{400627248} a^{5} - \frac{38076779}{100156812} a^{3} + \frac{112054897}{2403763488} a$, $\frac{1}{4807526976} a^{26} - \frac{13}{2403763488} a^{24} - \frac{53}{133542416} a^{22} + \frac{253}{26127864} a^{20} - \frac{226309}{2403763488} a^{18} - \frac{28807}{57232464} a^{16} + \frac{4358111}{300470436} a^{14} - \frac{15280109}{150235218} a^{12} - \frac{142324095}{534169664} a^{10} - \frac{1163478125}{2403763488} a^{8} + \frac{58285145}{1201881744} a^{6} - \frac{88984757}{200313624} a^{4} - \frac{19278961}{51143904} a^{2} - \frac{163391}{1089648}$, $\frac{1}{4807526976} a^{27} - \frac{1665}{133542416} a^{23} + \frac{85843}{300470436} a^{21} + \frac{21823}{114464928} a^{19} - \frac{1447}{100156812} a^{17} - \frac{78257101}{300470436} a^{15} + \frac{3642819}{8346401} a^{13} - \frac{122565759}{534169664} a^{11} - \frac{99793567}{300470436} a^{9} + \frac{7452913}{57232464} a^{7} - \frac{1955485}{4769372} a^{5} + \frac{155820145}{2403763488} a^{3} - \frac{28284203}{100156812} a$, $\frac{1}{4807526976} a^{28} - \frac{3}{38154976} a^{24} - \frac{125}{10731087} a^{22} + \frac{218941}{801254496} a^{20} + \frac{41}{101476} a^{18} + \frac{39395}{1201881744} a^{16} - \frac{7073606}{25039203} a^{14} - \frac{6160349}{1602508992} a^{12} - \frac{44308255}{300470436} a^{10} + \frac{390963053}{801254496} a^{8} + \frac{3233679}{8346401} a^{6} - \frac{1008352235}{2403763488} a^{4} - \frac{2543987}{14308116} a^{2} + \frac{136169}{363216}$, $\frac{1}{4807526976} a^{29} - \frac{8033}{42924348} a^{23} + \frac{26711}{114464928} a^{21} - \frac{191}{1451548} a^{19} + \frac{10831}{85848696} a^{17} + \frac{2056238}{25039203} a^{15} + \frac{26923237}{228929856} a^{13} + \frac{146193797}{300470436} a^{11} - \frac{74085895}{200313624} a^{9} - \frac{213631}{622092} a^{7} - \frac{607222919}{2403763488} a^{5} - \frac{44162719}{100156812} a^{3} - \frac{21925781}{200313624} a$, $\frac{1}{4807526976} a^{30} + \frac{11}{150235218} a^{24} - \frac{165983}{801254496} a^{22} + \frac{6311}{14308116} a^{20} + \frac{155233}{600940872} a^{18} + \frac{2749}{7154058} a^{16} + \frac{163438017}{534169664} a^{14} + \frac{66049817}{300470436} a^{12} + \frac{17018691}{66771208} a^{10} + \frac{501335}{8346401} a^{8} + \frac{766991545}{2403763488} a^{6} - \frac{2244307}{100156812} a^{4} - \frac{11089909}{200313624} a^{2} - \frac{2739}{15134}$, $\frac{1}{4807526976} a^{31} - \frac{35039}{801254496} a^{23} - \frac{8553}{33385604} a^{21} + \frac{39401}{600940872} a^{19} - \frac{11689}{50078406} a^{17} - \frac{15300671}{534169664} a^{15} + \frac{22351027}{100156812} a^{13} + \frac{48567385}{200313624} a^{11} + \frac{14756971}{50078406} a^{9} + \frac{18050599}{38154976} a^{7} + \frac{3140749}{100156812} a^{5} + \frac{16054963}{200313624} a^{3} + \frac{41508671}{150235218} a$

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

Class group and class number

$C_{3}\times C_{240}$, which has order $720$ (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:  \( 16643930584707.18 \) (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

\(\Q(\sqrt{-30}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{-5}, \sqrt{6})\), \(\Q(\sqrt{2}, \sqrt{-15})\), \(\Q(\sqrt{3}, \sqrt{-10})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\sqrt{6}, \sqrt{-10})\), \(\Q(\sqrt{2}, \sqrt{-5})\), \(\Q(\sqrt{3}, \sqrt{-5})\), \(\Q(\zeta_{16})^+\), 4.0.460800.2, 4.4.18432.1, 4.0.51200.2, 8.0.3317760000.6, 8.0.212336640000.4, 8.0.212336640000.3, \(\Q(\zeta_{48})^+\), 8.0.849346560000.4, 8.0.10485760000.2, 8.0.849346560000.5, 8.8.1342177280000.1, 8.0.173946175488.1, 8.8.108716359680000.1, 8.0.2147483648.1, 16.0.721389578983833600000000.2, 16.0.11819246862071129702400000000.5, 16.0.11819246862071129702400000000.7, 16.16.47276987448284518809600000000.1, 16.0.121029087867608368152576.1, 16.0.7205759403792793600000000.3, 16.0.47276987448284518809600000000.5

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{32}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{4}$

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