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Properties

Label 32.32.1870722095...0000.1
Degree $32$
Signature $[32, 0]$
Discriminant $2^{191}\cdot 5^{24}$
Root discriminant $209.41$
Ramified primes $2, 5$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
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![581850781250, 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 + 581850781250)
 
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 + 581850781250, 1)
 

Normalized defining polynomial

\( x^{32} - 160 x^{30} + 11600 x^{28} - 504000 x^{26} + 14625000 x^{24} - 299000000 x^{22} + 4427500000 x^{20} - 48070000000 x^{18} + 383057812500 x^{16} - 2220625000000 x^{14} + 9185312500000 x^{12} - 26243750000000 x^{10} + 49207031250000 x^{8} - 55781250000000 x^{6} + 33203125000000 x^{4} - 7812500000000 x^{2} + 581850781250 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $32$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[32, 0]$
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$$\chi_{640}(1,·)$, $\chi_{640}(387,·)$, $\chi_{640}(9,·)$, $\chi_{640}(523,·)$, $\chi_{640}(401,·)$, $\chi_{640}(147,·)$, $\chi_{640}(409,·)$, $\chi_{640}(283,·)$, $\chi_{640}(161,·)$, $\chi_{640}(547,·)$, $\chi_{640}(169,·)$, $\chi_{640}(43,·)$, $\chi_{640}(561,·)$, $\chi_{640}(307,·)$, $\chi_{640}(569,·)$, $\chi_{640}(443,·)$, $\chi_{640}(321,·)$, $\chi_{640}(67,·)$, $\chi_{640}(329,·)$, $\chi_{640}(203,·)$, $\chi_{640}(81,·)$, $\chi_{640}(467,·)$, $\chi_{640}(89,·)$, $\chi_{640}(603,·)$, $\chi_{640}(481,·)$, $\chi_{640}(227,·)$, $\chi_{640}(489,·)$, $\chi_{640}(363,·)$, $\chi_{640}(241,·)$, $\chi_{640}(627,·)$, $\chi_{640}(249,·)$, $\chi_{640}(123,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{5} a^{4}$, $\frac{1}{5} a^{5}$, $\frac{1}{5} a^{6}$, $\frac{1}{5} a^{7}$, $\frac{1}{25} a^{8}$, $\frac{1}{25} a^{9}$, $\frac{1}{25} a^{10}$, $\frac{1}{25} a^{11}$, $\frac{1}{125} a^{12}$, $\frac{1}{125} a^{13}$, $\frac{1}{125} a^{14}$, $\frac{1}{125} a^{15}$, $\frac{1}{119375} a^{16} - \frac{16}{23875} a^{14} - \frac{53}{23875} a^{12} - \frac{41}{4775} a^{10} + \frac{74}{4775} a^{8} + \frac{8}{955} a^{6} - \frac{4}{191} a^{4} + \frac{22}{191} a^{2} - \frac{87}{191}$, $\frac{1}{103020625} a^{17} - \frac{9324}{4120825} a^{15} + \frac{34327}{20604125} a^{13} - \frac{39769}{4120825} a^{11} - \frac{16352}{4120825} a^{9} + \frac{21973}{824165} a^{7} + \frac{76953}{824165} a^{5} + \frac{15684}{164833} a^{3} - \frac{58533}{164833} a$, $\frac{1}{103020625} a^{18} - \frac{18}{20604125} a^{16} - \frac{67507}{20604125} a^{14} - \frac{21067}{20604125} a^{12} - \frac{9448}{4120825} a^{10} + \frac{9028}{824165} a^{8} - \frac{36963}{824165} a^{6} + \frac{33544}{824165} a^{4} - \frac{42136}{164833} a^{2} + \frac{3}{191}$, $\frac{1}{103020625} a^{19} + \frac{22351}{20604125} a^{15} - \frac{63464}{20604125} a^{13} + \frac{37668}{4120825} a^{11} + \frac{56957}{4120825} a^{9} - \frac{37389}{824165} a^{7} + \frac{36328}{824165} a^{5} + \frac{50760}{164833} a^{3} + \frac{9275}{164833} a$, $\frac{1}{515103125} a^{20} - \frac{87}{103020625} a^{16} - \frac{49284}{20604125} a^{14} + \frac{73051}{20604125} a^{12} - \frac{24682}{4120825} a^{10} - \frac{49471}{4120825} a^{8} - \frac{40372}{824165} a^{6} + \frac{5021}{824165} a^{4} + \frac{2718}{164833} a^{2} - \frac{30}{191}$, $\frac{1}{515103125} a^{21} + \frac{15601}{20604125} a^{15} - \frac{72327}{20604125} a^{13} - \frac{23092}{4120825} a^{11} + \frac{11402}{4120825} a^{9} + \frac{58116}{824165} a^{7} - \frac{58221}{824165} a^{5} + \frac{48562}{164833} a^{3} - \frac{8438}{164833} a$, $\frac{1}{515103125} a^{22} + \frac{67}{20604125} a^{16} + \frac{16562}{20604125} a^{14} + \frac{45058}{20604125} a^{12} + \frac{12809}{824165} a^{10} - \frac{17511}{4120825} a^{8} - \frac{20249}{824165} a^{6} - \frac{16953}{824165} a^{4} - \frac{68848}{164833} a^{2} - \frac{1}{191}$, $\frac{1}{515103125} a^{23} - \frac{24873}{20604125} a^{15} - \frac{16202}{4120825} a^{13} + \frac{35187}{4120825} a^{11} + \frac{4184}{824165} a^{9} + \frac{36281}{824165} a^{7} - \frac{16452}{164833} a^{5} - \frac{48332}{164833} a^{3} - \frac{7435}{164833} a$, $\frac{1}{2575515625} a^{24} + \frac{154}{103020625} a^{16} + \frac{15573}{4120825} a^{14} + \frac{5484}{4120825} a^{12} - \frac{1317}{164833} a^{10} + \frac{75116}{4120825} a^{8} + \frac{18931}{824165} a^{6} - \frac{54373}{824165} a^{4} + \frac{54608}{164833} a^{2} - \frac{40}{191}$, $\frac{1}{2575515625} a^{25} + \frac{4693}{20604125} a^{15} + \frac{15718}{20604125} a^{13} - \frac{1464}{824165} a^{11} - \frac{44004}{4120825} a^{9} - \frac{68251}{824165} a^{7} - \frac{37159}{824165} a^{5} - \frac{53066}{164833} a^{3} + \frac{78580}{164833} a$, $\frac{1}{2575515625} a^{26} + \frac{164}{103020625} a^{16} + \frac{58868}{20604125} a^{14} + \frac{44522}{20604125} a^{12} - \frac{77661}{4120825} a^{10} + \frac{3092}{164833} a^{8} - \frac{58734}{824165} a^{6} + \frac{35857}{824165} a^{4} + \frac{60457}{164833} a^{2} + \frac{57}{191}$, $\frac{1}{2575515625} a^{27} - \frac{42603}{20604125} a^{15} + \frac{19216}{20604125} a^{13} + \frac{15968}{4120825} a^{11} - \frac{43133}{4120825} a^{9} - \frac{7196}{164833} a^{7} - \frac{57127}{824165} a^{5} - \frac{39224}{164833} a^{3} - \frac{76544}{164833} a$, $\frac{1}{12877578125} a^{28} - \frac{316}{103020625} a^{16} - \frac{3174}{824165} a^{14} - \frac{82414}{20604125} a^{12} - \frac{61097}{4120825} a^{10} - \frac{38569}{4120825} a^{8} - \frac{35762}{824165} a^{6} - \frac{60799}{824165} a^{4} + \frac{24907}{164833} a^{2} - \frac{61}{191}$, $\frac{1}{12877578125} a^{29} + \frac{948}{824165} a^{15} + \frac{50773}{20604125} a^{13} + \frac{12808}{824165} a^{11} + \frac{13771}{824165} a^{9} - \frac{16}{863} a^{7} + \frac{25898}{824165} a^{5} + \frac{36061}{164833} a^{3} + \frac{77058}{164833} a$, $\frac{1}{12877578125} a^{30} + \frac{269}{103020625} a^{16} - \frac{35527}{20604125} a^{14} - \frac{6877}{20604125} a^{12} - \frac{28664}{4120825} a^{10} + \frac{75488}{4120825} a^{8} + \frac{69048}{824165} a^{6} + \frac{14486}{164833} a^{4} - \frac{51529}{164833} a^{2} + \frac{77}{191}$, $\frac{1}{12877578125} a^{31} - \frac{4411}{4120825} a^{15} - \frac{10192}{20604125} a^{13} - \frac{44948}{4120825} a^{11} + \frac{4737}{824165} a^{9} - \frac{72534}{824165} a^{7} - \frac{23802}{824165} a^{5} + \frac{15133}{164833} a^{3} - \frac{12140}{164833} a$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $31$
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:  \( 1035918496783043300000000000 \) (assuming GRH)
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