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Properties

Label 32.0.52416707214...1056.1
Degree $32$
Signature $[0, 16]$
Discriminant $2^{32}\cdot 3^{16}\cdot 17^{28}$
Root discriminant $41.33$
Ramified primes $2, 3, 17$
Class number $40$ (GRH)
Class group $[40]$ (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![1, 0, -36, 0, 1086, 0, -6636, 0, 26973, 0, -61666, 0, 99380, 0, -111861, 0, 94862, 0, -60672, 0, 30241, 0, -11638, 0, 3496, 0, -793, 0, 134, 0, -15, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 15*x^30 + 134*x^28 - 793*x^26 + 3496*x^24 - 11638*x^22 + 30241*x^20 - 60672*x^18 + 94862*x^16 - 111861*x^14 + 99380*x^12 - 61666*x^10 + 26973*x^8 - 6636*x^6 + 1086*x^4 - 36*x^2 + 1)
 
gp: K = bnfinit(x^32 - 15*x^30 + 134*x^28 - 793*x^26 + 3496*x^24 - 11638*x^22 + 30241*x^20 - 60672*x^18 + 94862*x^16 - 111861*x^14 + 99380*x^12 - 61666*x^10 + 26973*x^8 - 6636*x^6 + 1086*x^4 - 36*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{32} - 15 x^{30} + 134 x^{28} - 793 x^{26} + 3496 x^{24} - 11638 x^{22} + 30241 x^{20} - 60672 x^{18} + 94862 x^{16} - 111861 x^{14} + 99380 x^{12} - 61666 x^{10} + 26973 x^{8} - 6636 x^{6} + 1086 x^{4} - 36 x^{2} + 1 \)

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:  \(5241670721450327242974152113348873388724842246701056=2^{32}\cdot 3^{16}\cdot 17^{28}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.33$
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:  $2, 3, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(204=2^{2}\cdot 3\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{204}(1,·)$, $\chi_{204}(137,·)$, $\chi_{204}(13,·)$, $\chi_{204}(145,·)$, $\chi_{204}(19,·)$, $\chi_{204}(149,·)$, $\chi_{204}(151,·)$, $\chi_{204}(25,·)$, $\chi_{204}(155,·)$, $\chi_{204}(157,·)$, $\chi_{204}(161,·)$, $\chi_{204}(35,·)$, $\chi_{204}(169,·)$, $\chi_{204}(43,·)$, $\chi_{204}(47,·)$, $\chi_{204}(49,·)$, $\chi_{204}(179,·)$, $\chi_{204}(53,·)$, $\chi_{204}(55,·)$, $\chi_{204}(185,·)$, $\chi_{204}(59,·)$, $\chi_{204}(191,·)$, $\chi_{204}(67,·)$, $\chi_{204}(203,·)$, $\chi_{204}(77,·)$, $\chi_{204}(83,·)$, $\chi_{204}(89,·)$, $\chi_{204}(101,·)$, $\chi_{204}(103,·)$, $\chi_{204}(115,·)$, $\chi_{204}(121,·)$, $\chi_{204}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{4} a^{24} - \frac{1}{4} a^{18} + \frac{1}{4} a^{12} - \frac{1}{4} a^{6} + \frac{1}{4}$, $\frac{1}{4} a^{25} - \frac{1}{4} a^{19} + \frac{1}{4} a^{13} - \frac{1}{4} a^{7} + \frac{1}{4} a$, $\frac{1}{4} a^{26} - \frac{1}{4} a^{20} + \frac{1}{4} a^{14} - \frac{1}{4} a^{8} + \frac{1}{4} a^{2}$, $\frac{1}{4} a^{27} - \frac{1}{4} a^{21} + \frac{1}{4} a^{15} - \frac{1}{4} a^{9} + \frac{1}{4} a^{3}$, $\frac{1}{358852} a^{28} + \frac{3023}{27604} a^{26} + \frac{7889}{89713} a^{24} + \frac{3975}{358852} a^{22} + \frac{84017}{358852} a^{20} + \frac{18516}{89713} a^{18} - \frac{132151}{358852} a^{16} + \frac{62195}{358852} a^{14} + \frac{33355}{89713} a^{12} - \frac{114209}{358852} a^{10} - \frac{47527}{358852} a^{8} + \frac{30317}{89713} a^{6} - \frac{138651}{358852} a^{4} - \frac{130001}{358852} a^{2} - \frac{34733}{89713}$, $\frac{1}{358852} a^{29} + \frac{3023}{27604} a^{27} + \frac{7889}{89713} a^{25} + \frac{3975}{358852} a^{23} + \frac{84017}{358852} a^{21} + \frac{18516}{89713} a^{19} - \frac{132151}{358852} a^{17} + \frac{62195}{358852} a^{15} + \frac{33355}{89713} a^{13} - \frac{114209}{358852} a^{11} - \frac{47527}{358852} a^{9} + \frac{30317}{89713} a^{7} - \frac{138651}{358852} a^{5} - \frac{130001}{358852} a^{3} - \frac{34733}{89713} a$, $\frac{1}{17570529103923849716} a^{30} + \frac{7900196601225}{8785264551961924858} a^{28} - \frac{999041555612202613}{8785264551961924858} a^{26} - \frac{1338024522544717873}{17570529103923849716} a^{24} + \frac{3835392211117073523}{8785264551961924858} a^{22} + \frac{3694594152601151133}{8785264551961924858} a^{20} + \frac{4996032877804513497}{17570529103923849716} a^{18} - \frac{147626888757663837}{8785264551961924858} a^{16} - \frac{3820195192929595923}{8785264551961924858} a^{14} + \frac{4830698173074491843}{17570529103923849716} a^{12} + \frac{1838877410860332189}{8785264551961924858} a^{10} + \frac{1778620435172716663}{8785264551961924858} a^{8} + \frac{3463142646295211545}{17570529103923849716} a^{6} - \frac{3220148525250131219}{8785264551961924858} a^{4} + \frac{178917159753353623}{675789580920148066} a^{2} + \frac{889741426692540796}{4392632275980962429}$, $\frac{1}{17570529103923849716} a^{31} + \frac{7900196601225}{8785264551961924858} a^{29} - \frac{999041555612202613}{8785264551961924858} a^{27} - \frac{1338024522544717873}{17570529103923849716} a^{25} + \frac{3835392211117073523}{8785264551961924858} a^{23} + \frac{3694594152601151133}{8785264551961924858} a^{21} + \frac{4996032877804513497}{17570529103923849716} a^{19} - \frac{147626888757663837}{8785264551961924858} a^{17} - \frac{3820195192929595923}{8785264551961924858} a^{15} + \frac{4830698173074491843}{17570529103923849716} a^{13} + \frac{1838877410860332189}{8785264551961924858} a^{11} + \frac{1778620435172716663}{8785264551961924858} a^{9} + \frac{3463142646295211545}{17570529103923849716} a^{7} - \frac{3220148525250131219}{8785264551961924858} a^{5} + \frac{178917159753353623}{675789580920148066} a^{3} + \frac{889741426692540796}{4392632275980962429} a$

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

Class group and class number

$C_{40}$, which has order $40$ (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:  \( -\frac{2996814042794625635}{17570529103923849716} a^{31} + \frac{45062338851799879863}{17570529103923849716} a^{29} - \frac{403224884666926751635}{17570529103923849716} a^{27} + \frac{1195607463592421012995}{8785264551961924858} a^{25} - \frac{10563959124639323765711}{17570529103923849716} a^{23} + \frac{35259854287434061112435}{17570529103923849716} a^{21} - \frac{3534462243250914068935}{675789580920148066} a^{19} + \frac{185099129365526453587327}{17570529103923849716} a^{17} - \frac{290787438949352501206315}{17570529103923849716} a^{15} + \frac{172610670898559810840895}{8785264551961924858} a^{13} - \frac{309268137292545588766747}{17570529103923849716} a^{11} + \frac{194467955203039354540475}{17570529103923849716} a^{9} - \frac{43135277737412749781565}{8785264551961924858} a^{7} + \frac{1681472366991125311839}{1351579161840296132} a^{5} - \frac{3533117998425916751535}{17570529103923849716} a^{3} + \frac{117170196485415320475}{17570529103923849716} a \) (order $12$)
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:  \( 550321919281.4646 \) (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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-17}) \), \(\Q(\sqrt{-51}) \), \(\Q(\sqrt{51}) \), \(\Q(\zeta_{12})\), \(\Q(i, \sqrt{17})\), \(\Q(i, \sqrt{51})\), \(\Q(\sqrt{-3}, \sqrt{17})\), \(\Q(\sqrt{-3}, \sqrt{-17})\), \(\Q(\sqrt{3}, \sqrt{17})\), \(\Q(\sqrt{3}, \sqrt{-17})\), 4.4.4913.1, 4.0.78608.1, 4.0.44217.1, 4.4.707472.1, 8.0.1731891456.1, 8.0.6179217664.1, 8.0.500516630784.2, 8.0.1955143089.1, 8.0.500516630784.3, 8.8.500516630784.1, 8.0.500516630784.1, \(\Q(\zeta_{17})^+\), 8.0.105046700288.1, 8.0.33237432513.1, 8.8.8508782723328.1, 16.0.250516897691366976454656.1, 16.0.11034809241396899282944.1, 16.0.72399383432805056195395584.3, 16.0.1104726920056229495169.1, 16.0.72399383432805056195395584.2, 16.16.72399383432805056195395584.1, 16.0.72399383432805056195395584.1

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 ${\href{/LocalNumberField/5.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{16}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{8}$

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
$2$2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
3Data not computed
17Data not computed