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Properties

Label 32.0.18580171965...0000.1
Degree $32$
Signature $[0, 16]$
Discriminant $2^{32}\cdot 5^{16}\cdot 17^{28}$
Root discriminant $53.35$
Ramified primes $2, 5, 17$
Class number $12800$ (GRH)
Class group $[8, 40, 40]$ (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, 108, 0, 2766, 0, 30996, 0, 183789, 0, 646998, 0, 1454308, 0, 2182743, 0, 2244206, 0, 1599264, 0, 790441, 0, 268554, 0, 61576, 0, 9243, 0, 862, 0, 45, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 + 45*x^30 + 862*x^28 + 9243*x^26 + 61576*x^24 + 268554*x^22 + 790441*x^20 + 1599264*x^18 + 2244206*x^16 + 2182743*x^14 + 1454308*x^12 + 646998*x^10 + 183789*x^8 + 30996*x^6 + 2766*x^4 + 108*x^2 + 1)
 
gp: K = bnfinit(x^32 + 45*x^30 + 862*x^28 + 9243*x^26 + 61576*x^24 + 268554*x^22 + 790441*x^20 + 1599264*x^18 + 2244206*x^16 + 2182743*x^14 + 1454308*x^12 + 646998*x^10 + 183789*x^8 + 30996*x^6 + 2766*x^4 + 108*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{32} + 45 x^{30} + 862 x^{28} + 9243 x^{26} + 61576 x^{24} + 268554 x^{22} + 790441 x^{20} + 1599264 x^{18} + 2244206 x^{16} + 2182743 x^{14} + 1454308 x^{12} + 646998 x^{10} + 183789 x^{8} + 30996 x^{6} + 2766 x^{4} + 108 x^{2} + 1 \)

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:  \(18580171965640016453782803163423074549760000000000000000=2^{32}\cdot 5^{16}\cdot 17^{28}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $53.35$
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:  $2, 5, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(340=2^{2}\cdot 5\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{340}(1,·)$, $\chi_{340}(259,·)$, $\chi_{340}(9,·)$, $\chi_{340}(271,·)$, $\chi_{340}(19,·)$, $\chi_{340}(149,·)$, $\chi_{340}(151,·)$, $\chi_{340}(281,·)$, $\chi_{340}(111,·)$, $\chi_{340}(161,·)$, $\chi_{340}(291,·)$, $\chi_{340}(169,·)$, $\chi_{340}(171,·)$, $\chi_{340}(49,·)$, $\chi_{340}(179,·)$, $\chi_{340}(59,·)$, $\chi_{340}(189,·)$, $\chi_{340}(319,·)$, $\chi_{340}(321,·)$, $\chi_{340}(69,·)$, $\chi_{340}(331,·)$, $\chi_{340}(81,·)$, $\chi_{340}(339,·)$, $\chi_{340}(89,·)$, $\chi_{340}(219,·)$, $\chi_{340}(101,·)$, $\chi_{340}(229,·)$, $\chi_{340}(191,·)$, $\chi_{340}(239,·)$, $\chi_{340}(121,·)$, $\chi_{340}(251,·)$, $\chi_{340}(21,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{4} a^{24} - \frac{1}{4} a^{18} + \frac{1}{4} a^{12} - \frac{1}{4} a^{6} + \frac{1}{4}$, $\frac{1}{4} a^{25} - \frac{1}{4} a^{19} + \frac{1}{4} a^{13} - \frac{1}{4} a^{7} + \frac{1}{4} a$, $\frac{1}{4} a^{26} - \frac{1}{4} a^{20} + \frac{1}{4} a^{14} - \frac{1}{4} a^{8} + \frac{1}{4} a^{2}$, $\frac{1}{4} a^{27} - \frac{1}{4} a^{21} + \frac{1}{4} a^{15} - \frac{1}{4} a^{9} + \frac{1}{4} a^{3}$, $\frac{1}{52} a^{28} + \frac{1}{52} a^{26} - \frac{25}{52} a^{22} + \frac{3}{52} a^{20} + \frac{3}{13} a^{18} - \frac{3}{52} a^{16} + \frac{5}{52} a^{14} - \frac{3}{13} a^{12} - \frac{9}{52} a^{10} - \frac{21}{52} a^{8} - \frac{5}{13} a^{6} + \frac{9}{52} a^{4} + \frac{17}{52} a^{2} + \frac{3}{13}$, $\frac{1}{52} a^{29} + \frac{1}{52} a^{27} - \frac{25}{52} a^{23} + \frac{3}{52} a^{21} + \frac{3}{13} a^{19} - \frac{3}{52} a^{17} + \frac{5}{52} a^{15} - \frac{3}{13} a^{13} - \frac{9}{52} a^{11} - \frac{21}{52} a^{9} - \frac{5}{13} a^{7} + \frac{9}{52} a^{5} + \frac{17}{52} a^{3} + \frac{3}{13} a$, $\frac{1}{19272221147269928396} a^{30} - \frac{59661768747063517}{19272221147269928396} a^{28} + \frac{33914837478637707}{410047258452551668} a^{26} + \frac{1774089738157225373}{19272221147269928396} a^{24} + \frac{6473372205320994981}{19272221147269928396} a^{22} - \frac{4323469358533678121}{19272221147269928396} a^{20} + \frac{5489820163329942147}{19272221147269928396} a^{18} + \frac{7725559353790145575}{19272221147269928396} a^{16} - \frac{3550216724134984571}{19272221147269928396} a^{14} + \frac{5530223213172151541}{19272221147269928396} a^{12} + \frac{3143127382590898165}{19272221147269928396} a^{10} - \frac{40678737813443785}{1482478549789994492} a^{8} - \frac{1496621818872491317}{19272221147269928396} a^{6} - \frac{1278383477724446001}{19272221147269928396} a^{4} + \frac{3696283003862871853}{19272221147269928396} a^{2} - \frac{306331683232852383}{9636110573634964198}$, $\frac{1}{19272221147269928396} a^{31} - \frac{59661768747063517}{19272221147269928396} a^{29} + \frac{33914837478637707}{410047258452551668} a^{27} + \frac{1774089738157225373}{19272221147269928396} a^{25} + \frac{6473372205320994981}{19272221147269928396} a^{23} - \frac{4323469358533678121}{19272221147269928396} a^{21} + \frac{5489820163329942147}{19272221147269928396} a^{19} + \frac{7725559353790145575}{19272221147269928396} a^{17} - \frac{3550216724134984571}{19272221147269928396} a^{15} + \frac{5530223213172151541}{19272221147269928396} a^{13} + \frac{3143127382590898165}{19272221147269928396} a^{11} - \frac{40678737813443785}{1482478549789994492} a^{9} - \frac{1496621818872491317}{19272221147269928396} a^{7} - \frac{1278383477724446001}{19272221147269928396} a^{5} + \frac{3696283003862871853}{19272221147269928396} a^{3} - \frac{306331683232852383}{9636110573634964198} a$

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

Class group and class number

$C_{8}\times C_{40}\times C_{40}$, which has order $12800$ (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:  \( \frac{886567745}{105114502246} a^{31} + \frac{17320661715}{52557251123} a^{29} + \frac{261162985480}{52557251123} a^{27} + \frac{1715973249600}{52557251123} a^{25} + \frac{924165859335}{52557251123} a^{23} - \frac{63121376165020}{52557251123} a^{21} - \frac{474376200895120}{52557251123} a^{19} - \frac{138454102983779}{4042865471} a^{17} - \frac{4183093886048251}{52557251123} a^{15} - \frac{6267744733089928}{52557251123} a^{13} - \frac{6101272329495281}{52557251123} a^{11} - \frac{3783049704728980}{52557251123} a^{9} - \frac{1422312131319436}{52557251123} a^{7} - \frac{295014846552523}{52557251123} a^{5} - \frac{28032355700892}{52557251123} a^{3} - \frac{1557128623363}{105114502246} a \) (order $4$)
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:  \( 42597284566.379654 \) (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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-85}) \), \(\Q(\sqrt{85}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-17}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{5}) \), \(\Q(i, \sqrt{85})\), \(\Q(i, \sqrt{17})\), \(\Q(i, \sqrt{5})\), \(\Q(\sqrt{-5}, \sqrt{17})\), \(\Q(\sqrt{5}, \sqrt{-17})\), \(\Q(\sqrt{5}, \sqrt{17})\), \(\Q(\sqrt{-5}, \sqrt{-17})\), 4.4.4913.1, 4.0.78608.1, 4.0.1965200.1, 4.4.122825.1, 8.0.13363360000.1, 8.0.6179217664.1, 8.0.3862011040000.3, 8.0.3862011040000.2, 8.0.3862011040000.1, 8.8.15085980625.1, 8.0.3862011040000.4, 8.0.105046700288.1, \(\Q(\zeta_{17})^+\), 8.8.256461670625.1, 8.0.65654187680000.2, 16.0.14915129273081881600000000.1, 16.0.11034809241396899282944.1, 16.0.4310472359920663782400000000.3, 16.0.4310472359920663782400000000.2, 16.0.4310472359920663782400000000.1, 16.0.4310472359920663782400000000.4, 16.16.65772588499765987890625.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{4}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{16}$ 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}$
5Data not computed
17Data not computed