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Properties

Label 32.0.18707220957...0000.2
Degree $32$
Signature $[0, 16]$
Discriminant $2^{191}\cdot 5^{24}$
Root discriminant $209.41$
Ramified primes $2, 5$
Class number Not computed
Class group Not computed
Galois group $C_{32}$ (as 32T33)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![28500781250, 0, 7812500000000, 0, 33203125000000, 0, 55781250000000, 0, 49207031250000, 0, 26243750000000, 0, 9185312500000, 0, 2220625000000, 0, 383057812500, 0, 48070000000, 0, 4427500000, 0, 299000000, 0, 14625000, 0, 504000, 0, 11600, 0, 160, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 + 160*x^30 + 11600*x^28 + 504000*x^26 + 14625000*x^24 + 299000000*x^22 + 4427500000*x^20 + 48070000000*x^18 + 383057812500*x^16 + 2220625000000*x^14 + 9185312500000*x^12 + 26243750000000*x^10 + 49207031250000*x^8 + 55781250000000*x^6 + 33203125000000*x^4 + 7812500000000*x^2 + 28500781250)
 
gp: K = bnfinit(x^32 + 160*x^30 + 11600*x^28 + 504000*x^26 + 14625000*x^24 + 299000000*x^22 + 4427500000*x^20 + 48070000000*x^18 + 383057812500*x^16 + 2220625000000*x^14 + 9185312500000*x^12 + 26243750000000*x^10 + 49207031250000*x^8 + 55781250000000*x^6 + 33203125000000*x^4 + 7812500000000*x^2 + 28500781250, 1)
 

Normalized defining polynomial

\( x^{32} + 160 x^{30} + 11600 x^{28} + 504000 x^{26} + 14625000 x^{24} + 299000000 x^{22} + 4427500000 x^{20} + 48070000000 x^{18} + 383057812500 x^{16} + 2220625000000 x^{14} + 9185312500000 x^{12} + 26243750000000 x^{10} + 49207031250000 x^{8} + 55781250000000 x^{6} + 33203125000000 x^{4} + 7812500000000 x^{2} + 28500781250 \)

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:  \(187072209578355573530071658587684226515959365500928000000000000000000000000=2^{191}\cdot 5^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $209.41$
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:  $2, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(640=2^{7}\cdot 5\)
Dirichlet character group:    $\lbrace$$\chi_{640}(1,·)$, $\chi_{640}(517,·)$, $\chi_{640}(9,·)$, $\chi_{640}(13,·)$, $\chi_{640}(401,·)$, $\chi_{640}(277,·)$, $\chi_{640}(409,·)$, $\chi_{640}(413,·)$, $\chi_{640}(161,·)$, $\chi_{640}(37,·)$, $\chi_{640}(169,·)$, $\chi_{640}(173,·)$, $\chi_{640}(561,·)$, $\chi_{640}(437,·)$, $\chi_{640}(569,·)$, $\chi_{640}(573,·)$, $\chi_{640}(321,·)$, $\chi_{640}(197,·)$, $\chi_{640}(329,·)$, $\chi_{640}(333,·)$, $\chi_{640}(81,·)$, $\chi_{640}(597,·)$, $\chi_{640}(89,·)$, $\chi_{640}(93,·)$, $\chi_{640}(481,·)$, $\chi_{640}(357,·)$, $\chi_{640}(489,·)$, $\chi_{640}(493,·)$, $\chi_{640}(241,·)$, $\chi_{640}(117,·)$, $\chi_{640}(249,·)$, $\chi_{640}(253,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{5} a^{4}$, $\frac{1}{5} a^{5}$, $\frac{1}{5} a^{6}$, $\frac{1}{5} a^{7}$, $\frac{1}{25} a^{8}$, $\frac{1}{25} a^{9}$, $\frac{1}{25} a^{10}$, $\frac{1}{25} a^{11}$, $\frac{1}{125} a^{12}$, $\frac{1}{125} a^{13}$, $\frac{1}{125} a^{14}$, $\frac{1}{125} a^{15}$, $\frac{1}{539375} a^{16} + \frac{16}{107875} a^{14} - \frac{343}{107875} a^{12} + \frac{34}{21575} a^{10} + \frac{103}{21575} a^{8} + \frac{403}{4315} a^{6} - \frac{287}{4315} a^{4} + \frac{233}{863} a^{2} + \frac{387}{863}$, $\frac{1}{103020625} a^{17} - \frac{46586}{20604125} a^{15} - \frac{33137}{20604125} a^{13} - \frac{35349}{4120825} a^{11} + \frac{63102}{4120825} a^{9} + \frac{78073}{824165} a^{7} - \frac{63286}{824165} a^{5} + \frac{66684}{164833} a^{3} + \frac{79783}{164833} a$, $\frac{1}{103020625} a^{18} + \frac{18}{20604125} a^{16} + \frac{68857}{20604125} a^{14} + \frac{6233}{20604125} a^{12} + \frac{73798}{4120825} a^{10} - \frac{3859}{4120825} a^{8} + \frac{53797}{824165} a^{6} + \frac{49212}{824165} a^{4} - \frac{21447}{164833} a^{2} + \frac{79}{863}$, $\frac{1}{103020625} a^{19} - \frac{24061}{20604125} a^{15} + \frac{21569}{20604125} a^{13} - \frac{41452}{4120825} a^{11} - \frac{78717}{4120825} a^{9} - \frac{49787}{824165} a^{7} - \frac{24203}{824165} a^{5} + \frac{75814}{164833} a^{3} - \frac{77562}{164833} a$, $\frac{1}{515103125} a^{20} + \frac{1}{20604125} a^{16} - \frac{39196}{20604125} a^{14} - \frac{10888}{4120825} a^{12} - \frac{54631}{4120825} a^{10} - \frac{43484}{4120825} a^{8} + \frac{1577}{824165} a^{6} - \frac{14595}{164833} a^{4} - \frac{45423}{164833} a^{2} - \frac{429}{863}$, $\frac{1}{515103125} a^{21} + \frac{28901}{20604125} a^{15} - \frac{53588}{20604125} a^{13} - \frac{42719}{4120825} a^{11} - \frac{29328}{4120825} a^{9} - \frac{59122}{824165} a^{7} + \frac{78622}{824165} a^{5} - \frac{49177}{164833} a^{3} + \frac{13645}{164833} a$, $\frac{1}{515103125} a^{22} - \frac{82}{103020625} a^{16} - \frac{59318}{20604125} a^{14} - \frac{70154}{20604125} a^{12} - \frac{296}{4120825} a^{10} - \frac{4687}{824165} a^{8} - \frac{778}{164833} a^{6} + \frac{42334}{824165} a^{4} - \frac{49194}{164833} a^{2} - \frac{402}{863}$, $\frac{1}{515103125} a^{23} + \frac{76622}{20604125} a^{15} + \frac{14773}{20604125} a^{13} + \frac{13616}{824165} a^{11} + \frac{41106}{4120825} a^{9} - \frac{30391}{824165} a^{7} - \frac{7459}{164833} a^{5} - \frac{20595}{164833} a^{3} + \frac{36937}{164833} a$, $\frac{1}{2575515625} a^{24} + \frac{31}{103020625} a^{16} + \frac{63196}{20604125} a^{14} - \frac{34487}{20604125} a^{12} - \frac{24478}{4120825} a^{10} - \frac{1456}{824165} a^{8} - \frac{49861}{824165} a^{6} + \frac{38233}{824165} a^{4} - \frac{3385}{164833} a^{2} + \frac{153}{863}$, $\frac{1}{2575515625} a^{25} + \frac{4773}{4120825} a^{15} + \frac{3762}{20604125} a^{13} + \frac{82343}{4120825} a^{11} + \frac{14554}{4120825} a^{9} + \frac{2371}{824165} a^{7} + \frac{22103}{824165} a^{5} + \frac{72240}{164833} a^{3} + \frac{28445}{164833} a$, $\frac{1}{2575515625} a^{26} - \frac{2}{4120825} a^{16} + \frac{71758}{20604125} a^{14} - \frac{15743}{20604125} a^{12} + \frac{76629}{4120825} a^{10} + \frac{15741}{824165} a^{8} + \frac{45214}{824165} a^{6} + \frac{1379}{164833} a^{4} + \frac{70847}{164833} a^{2} - \frac{235}{863}$, $\frac{1}{2575515625} a^{27} + \frac{10024}{4120825} a^{15} - \frac{24263}{20604125} a^{13} - \frac{42491}{4120825} a^{11} - \frac{12571}{824165} a^{9} - \frac{7128}{824165} a^{7} - \frac{25578}{824165} a^{5} - \frac{56446}{164833} a^{3} - \frac{11727}{164833} a$, $\frac{1}{12877578125} a^{28} + \frac{78}{103020625} a^{16} + \frac{51607}{20604125} a^{14} - \frac{20717}{20604125} a^{12} - \frac{65669}{4120825} a^{10} - \frac{51631}{4120825} a^{8} - \frac{29449}{824165} a^{6} - \frac{34863}{824165} a^{4} + \frac{8045}{164833} a^{2} - \frac{423}{863}$, $\frac{1}{12877578125} a^{29} + \frac{58989}{20604125} a^{15} - \frac{73359}{20604125} a^{13} + \frac{2169}{164833} a^{11} - \frac{28597}{4120825} a^{9} - \frac{20322}{824165} a^{7} - \frac{8709}{164833} a^{5} + \frac{81349}{164833} a^{3} - \frac{40213}{164833} a$, $\frac{1}{12877578125} a^{30} + \frac{41}{103020625} a^{16} - \frac{11666}{20604125} a^{14} + \frac{50902}{20604125} a^{12} - \frac{104}{824165} a^{10} + \frac{17383}{4120825} a^{8} - \frac{45264}{824165} a^{6} - \frac{1927}{164833} a^{4} - \frac{17484}{164833} a^{2} - \frac{332}{863}$, $\frac{1}{12877578125} a^{31} - \frac{79636}{20604125} a^{15} - \frac{73978}{20604125} a^{13} - \frac{34708}{4120825} a^{11} + \frac{67529}{4120825} a^{9} + \frac{50403}{824165} a^{7} - \frac{52237}{824165} a^{5} + \frac{50633}{164833} a^{3} - \frac{37855}{164833} a$

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

Class group and class number

Not computed

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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{32}$ (as 32T33):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 32
The 32 conjugacy class representatives for $C_{32}$
Character table for $C_{32}$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), \(\Q(\zeta_{32})^+\), 16.16.236118324143482260684800000000.1

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 $32$ R $16^{2}$ $32$ $32$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{4}$ $32$ $16^{2}$ $32$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{8}$ $32$ $16^{2}$ $32$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{4}$ $32$ $32$

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
5Data not computed