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Properties

Label 32.0.15728213305...5424.1
Degree $32$
Signature $[0, 16]$
Discriminant $2^{158}\cdot 3^{16}$
Root discriminant $53.08$
Ramified primes $2, 3$
Class number $153$ (GRH)
Class group $[3, 51]$ (GRH)
Galois group $C_2\times C_{16}$ (as 32T32)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, 0, -128, 0, 3424, 0, -18816, 0, 68568, 0, -142016, 0, 207712, 0, -213664, 0, 166158, 0, -97984, 0, 45184, 0, -16160, 0, 4524, 0, -960, 0, 152, 0, -16, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 16*x^30 + 152*x^28 - 960*x^26 + 4524*x^24 - 16160*x^22 + 45184*x^20 - 97984*x^18 + 166158*x^16 - 213664*x^14 + 207712*x^12 - 142016*x^10 + 68568*x^8 - 18816*x^6 + 3424*x^4 - 128*x^2 + 4)
 
gp: K = bnfinit(x^32 - 16*x^30 + 152*x^28 - 960*x^26 + 4524*x^24 - 16160*x^22 + 45184*x^20 - 97984*x^18 + 166158*x^16 - 213664*x^14 + 207712*x^12 - 142016*x^10 + 68568*x^8 - 18816*x^6 + 3424*x^4 - 128*x^2 + 4, 1)
 

Normalized defining polynomial

\( x^{32} - 16 x^{30} + 152 x^{28} - 960 x^{26} + 4524 x^{24} - 16160 x^{22} + 45184 x^{20} - 97984 x^{18} + 166158 x^{16} - 213664 x^{14} + 207712 x^{12} - 142016 x^{10} + 68568 x^{8} - 18816 x^{6} + 3424 x^{4} - 128 x^{2} + 4 \)

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:  \(15728213305806640649499960541467376826041611043077095424=2^{158}\cdot 3^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $53.08$
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:  $2, 3$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(192=2^{6}\cdot 3\)
Dirichlet character group:    $\lbrace$$\chi_{192}(1,·)$, $\chi_{192}(131,·)$, $\chi_{192}(137,·)$, $\chi_{192}(11,·)$, $\chi_{192}(17,·)$, $\chi_{192}(19,·)$, $\chi_{192}(25,·)$, $\chi_{192}(155,·)$, $\chi_{192}(161,·)$, $\chi_{192}(35,·)$, $\chi_{192}(163,·)$, $\chi_{192}(41,·)$, $\chi_{192}(43,·)$, $\chi_{192}(49,·)$, $\chi_{192}(179,·)$, $\chi_{192}(185,·)$, $\chi_{192}(59,·)$, $\chi_{192}(65,·)$, $\chi_{192}(67,·)$, $\chi_{192}(73,·)$, $\chi_{192}(83,·)$, $\chi_{192}(139,·)$, $\chi_{192}(89,·)$, $\chi_{192}(91,·)$, $\chi_{192}(97,·)$, $\chi_{192}(187,·)$, $\chi_{192}(145,·)$, $\chi_{192}(107,·)$, $\chi_{192}(113,·)$, $\chi_{192}(115,·)$, $\chi_{192}(169,·)$, $\chi_{192}(121,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{2} a^{16}$, $\frac{1}{2} a^{17}$, $\frac{1}{2} a^{18}$, $\frac{1}{2} a^{19}$, $\frac{1}{2} a^{20}$, $\frac{1}{2} a^{21}$, $\frac{1}{2} a^{22}$, $\frac{1}{2} a^{23}$, $\frac{1}{2} a^{24}$, $\frac{1}{2} a^{25}$, $\frac{1}{62} a^{26} + \frac{6}{31} a^{24} + \frac{2}{31} a^{22} + \frac{11}{62} a^{20} + \frac{3}{31} a^{18} - \frac{11}{62} a^{16} + \frac{10}{31} a^{14} - \frac{9}{31} a^{12} + \frac{11}{31} a^{10} - \frac{3}{31} a^{8} + \frac{5}{31} a^{6} + \frac{15}{31} a^{4} + \frac{7}{31} a^{2} - \frac{1}{31}$, $\frac{1}{62} a^{27} + \frac{6}{31} a^{25} + \frac{2}{31} a^{23} + \frac{11}{62} a^{21} + \frac{3}{31} a^{19} - \frac{11}{62} a^{17} + \frac{10}{31} a^{15} - \frac{9}{31} a^{13} + \frac{11}{31} a^{11} - \frac{3}{31} a^{9} + \frac{5}{31} a^{7} + \frac{15}{31} a^{5} + \frac{7}{31} a^{3} - \frac{1}{31} a$, $\frac{1}{62} a^{28} + \frac{15}{62} a^{24} - \frac{3}{31} a^{22} - \frac{1}{31} a^{20} + \frac{5}{31} a^{18} - \frac{3}{62} a^{16} - \frac{5}{31} a^{14} - \frac{5}{31} a^{12} - \frac{11}{31} a^{10} + \frac{10}{31} a^{8} - \frac{14}{31} a^{6} + \frac{13}{31} a^{4} + \frac{8}{31} a^{2} + \frac{12}{31}$, $\frac{1}{62} a^{29} + \frac{15}{62} a^{25} - \frac{3}{31} a^{23} - \frac{1}{31} a^{21} + \frac{5}{31} a^{19} - \frac{3}{62} a^{17} - \frac{5}{31} a^{15} - \frac{5}{31} a^{13} - \frac{11}{31} a^{11} + \frac{10}{31} a^{9} - \frac{14}{31} a^{7} + \frac{13}{31} a^{5} + \frac{8}{31} a^{3} + \frac{12}{31} a$, $\frac{1}{96360773033972116799042} a^{30} + \frac{319769124499034457877}{96360773033972116799042} a^{28} - \frac{298448975379448257103}{48180386516986058399521} a^{26} + \frac{18572557681951899967797}{96360773033972116799042} a^{24} + \frac{5193327782622272902803}{96360773033972116799042} a^{22} - \frac{8255421473144608572017}{48180386516986058399521} a^{20} - \frac{10418378289368478388234}{48180386516986058399521} a^{18} + \frac{16362931360001613733051}{96360773033972116799042} a^{16} - \frac{16810654737116004124420}{48180386516986058399521} a^{14} + \frac{10070570833043371677462}{48180386516986058399521} a^{12} - \frac{14408765012316091676909}{48180386516986058399521} a^{10} - \frac{20235420802166882540254}{48180386516986058399521} a^{8} - \frac{3466810361815031722727}{48180386516986058399521} a^{6} - \frac{10069047875753442733906}{48180386516986058399521} a^{4} - \frac{48967160695986614426}{48180386516986058399521} a^{2} + \frac{11881454458779171941916}{48180386516986058399521}$, $\frac{1}{96360773033972116799042} a^{31} + \frac{319769124499034457877}{96360773033972116799042} a^{29} - \frac{298448975379448257103}{48180386516986058399521} a^{27} + \frac{18572557681951899967797}{96360773033972116799042} a^{25} + \frac{5193327782622272902803}{96360773033972116799042} a^{23} - \frac{8255421473144608572017}{48180386516986058399521} a^{21} - \frac{10418378289368478388234}{48180386516986058399521} a^{19} + \frac{16362931360001613733051}{96360773033972116799042} a^{17} - \frac{16810654737116004124420}{48180386516986058399521} a^{15} + \frac{10070570833043371677462}{48180386516986058399521} a^{13} - \frac{14408765012316091676909}{48180386516986058399521} a^{11} - \frac{20235420802166882540254}{48180386516986058399521} a^{9} - \frac{3466810361815031722727}{48180386516986058399521} a^{7} - \frac{10069047875753442733906}{48180386516986058399521} a^{5} - \frac{48967160695986614426}{48180386516986058399521} a^{3} + \frac{11881454458779171941916}{48180386516986058399521} a$

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

Class group and class number

$C_{3}\times C_{51}$, which has order $153$ (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:  \( -\frac{517701136565406420796}{48180386516986058399521} a^{30} + \frac{8251430270940144606278}{48180386516986058399521} a^{28} - \frac{78191128667345673892640}{48180386516986058399521} a^{26} + \frac{492303911846553448315773}{48180386516986058399521} a^{24} - \frac{2312895014693928232679528}{48180386516986058399521} a^{22} + \frac{8230472910963778841809688}{48180386516986058399521} a^{20} - \frac{22916128329017809985172688}{48180386516986058399521} a^{18} + \frac{98845071857423609435480673}{96360773033972116799042} a^{16} - \frac{83265950141538905063633256}{48180386516986058399521} a^{14} + \frac{106087131983018123509642864}{48180386516986058399521} a^{12} - \frac{101971887212174054826194000}{48180386516986058399521} a^{10} + \frac{68415721887623632645371734}{48180386516986058399521} a^{8} - \frac{32353586528624173301383104}{48180386516986058399521} a^{6} + \frac{8401412487793961715162760}{48180386516986058399521} a^{4} - \frac{1572937276069694458774816}{48180386516986058399521} a^{2} + \frac{58759803755999892048248}{48180386516986058399521} \) (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:  \( 12949932335324.246 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{16}$ (as 32T32):

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 $C_2\times C_{16}$
Character table for $C_2\times C_{16}$ is not computed

Intermediate fields

\(\Q(\sqrt{-6}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\zeta_{16})^+\), 4.0.18432.2, 8.0.339738624.2, \(\Q(\zeta_{32})^+\), 8.0.173946175488.1, 16.0.30257271966902092038144.1, 16.0.604462909807314587353088.1, 16.16.3965881151245791007623610368.1

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 R $16^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{4}$ $16^{2}$ $16^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{8}$ $16^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{4}$ $16^{2}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{32}$ $16^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{4}$ $16^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ $16^{2}$ $16^{2}$

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