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Properties

Label 32.0.99276740263...8624.5
Degree $32$
Signature $[0, 16]$
Discriminant $2^{48}\cdot 3^{16}\cdot 17^{30}$
Root discriminant $69.77$
Ramified primes $2, 3, 17$
Class number Not computed
Class group Not computed
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![18939904, 0, 113639424, 0, 482967552, 0, 880705536, 0, 1085493248, 0, 874786816, 0, 517296128, 0, 226539008, 0, 76865024, 0, 20297728, 0, 4252992, 0, 701760, 0, 91392, 0, 9112, 0, 680, 0, 34, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 + 34*x^30 + 680*x^28 + 9112*x^26 + 91392*x^24 + 701760*x^22 + 4252992*x^20 + 20297728*x^18 + 76865024*x^16 + 226539008*x^14 + 517296128*x^12 + 874786816*x^10 + 1085493248*x^8 + 880705536*x^6 + 482967552*x^4 + 113639424*x^2 + 18939904)
 
gp: K = bnfinit(x^32 + 34*x^30 + 680*x^28 + 9112*x^26 + 91392*x^24 + 701760*x^22 + 4252992*x^20 + 20297728*x^18 + 76865024*x^16 + 226539008*x^14 + 517296128*x^12 + 874786816*x^10 + 1085493248*x^8 + 880705536*x^6 + 482967552*x^4 + 113639424*x^2 + 18939904, 1)
 

Normalized defining polynomial

\( x^{32} + 34 x^{30} + 680 x^{28} + 9112 x^{26} + 91392 x^{24} + 701760 x^{22} + 4252992 x^{20} + 20297728 x^{18} + 76865024 x^{16} + 226539008 x^{14} + 517296128 x^{12} + 874786816 x^{10} + 1085493248 x^{8} + 880705536 x^{6} + 482967552 x^{4} + 113639424 x^{2} + 18939904 \)

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:  \(99276740263879938750515115508224780490603194567662317338624=2^{48}\cdot 3^{16}\cdot 17^{30}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $69.77$
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:  \(408=2^{3}\cdot 3\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{408}(1,·)$, $\chi_{408}(131,·)$, $\chi_{408}(257,·)$, $\chi_{408}(137,·)$, $\chi_{408}(11,·)$, $\chi_{408}(145,·)$, $\chi_{408}(403,·)$, $\chi_{408}(25,·)$, $\chi_{408}(281,·)$, $\chi_{408}(283,·)$, $\chi_{408}(161,·)$, $\chi_{408}(163,·)$, $\chi_{408}(49,·)$, $\chi_{408}(169,·)$, $\chi_{408}(299,·)$, $\chi_{408}(305,·)$, $\chi_{408}(185,·)$, $\chi_{408}(347,·)$, $\chi_{408}(139,·)$, $\chi_{408}(275,·)$, $\chi_{408}(211,·)$, $\chi_{408}(89,·)$, $\chi_{408}(91,·)$, $\chi_{408}(353,·)$, $\chi_{408}(227,·)$, $\chi_{408}(361,·)$, $\chi_{408}(107,·)$, $\chi_{408}(371,·)$, $\chi_{408}(235,·)$, $\chi_{408}(121,·)$, $\chi_{408}(217,·)$, $\chi_{408}(379,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{8} a^{6}$, $\frac{1}{8} a^{7}$, $\frac{1}{16} a^{8}$, $\frac{1}{16} a^{9}$, $\frac{1}{32} a^{10}$, $\frac{1}{32} a^{11}$, $\frac{1}{64} a^{12}$, $\frac{1}{64} a^{13}$, $\frac{1}{128} a^{14}$, $\frac{1}{128} a^{15}$, $\frac{1}{4352} a^{16}$, $\frac{1}{4352} a^{17}$, $\frac{1}{8704} a^{18}$, $\frac{1}{8704} a^{19}$, $\frac{1}{17408} a^{20}$, $\frac{1}{17408} a^{21}$, $\frac{1}{34816} a^{22}$, $\frac{1}{34816} a^{23}$, $\frac{1}{278528} a^{24} - \frac{1}{34816} a^{18} + \frac{1}{256} a^{12} - \frac{1}{32} a^{6} + \frac{1}{4}$, $\frac{1}{278528} a^{25} - \frac{1}{34816} a^{19} + \frac{1}{256} a^{13} - \frac{1}{32} a^{7} + \frac{1}{4} a$, $\frac{1}{557056} a^{26} - \frac{1}{69632} a^{20} + \frac{1}{512} a^{14} - \frac{1}{64} a^{8} + \frac{1}{8} a^{2}$, $\frac{1}{557056} a^{27} - \frac{1}{69632} a^{21} + \frac{1}{512} a^{15} - \frac{1}{64} a^{9} + \frac{1}{8} a^{3}$, $\frac{1}{642842624} a^{28} - \frac{69}{160710656} a^{26} + \frac{49}{160710656} a^{24} - \frac{49}{4726784} a^{22} + \frac{263}{20088832} a^{20} - \frac{241}{20088832} a^{18} + \frac{709}{10044416} a^{16} + \frac{323}{147712} a^{14} - \frac{1147}{147712} a^{12} - \frac{661}{73856} a^{10} + \frac{29}{18464} a^{8} - \frac{837}{18464} a^{6} + \frac{333}{9232} a^{4} - \frac{143}{2308} a^{2} + \frac{773}{2308}$, $\frac{1}{642842624} a^{29} - \frac{69}{160710656} a^{27} + \frac{49}{160710656} a^{25} - \frac{49}{4726784} a^{23} + \frac{263}{20088832} a^{21} - \frac{241}{20088832} a^{19} + \frac{709}{10044416} a^{17} + \frac{323}{147712} a^{15} - \frac{1147}{147712} a^{13} - \frac{661}{73856} a^{11} + \frac{29}{18464} a^{9} - \frac{837}{18464} a^{7} + \frac{333}{9232} a^{5} - \frac{143}{2308} a^{3} + \frac{773}{2308} a$, $\frac{1}{31985258206080727908352} a^{30} + \frac{7076152584669}{15992629103040363954176} a^{28} - \frac{780854399320071}{3998157275760090988544} a^{26} + \frac{5826370154790201}{3998157275760090988544} a^{24} + \frac{9260584561972939}{1999078637880045494272} a^{22} - \frac{7638797047769335}{499769659470011373568} a^{20} - \frac{14472775595430977}{499769659470011373568} a^{18} - \frac{10414764239601135}{249884829735005686784} a^{16} - \frac{3310315638259731}{3674776907867730688} a^{14} - \frac{24963391630336419}{3674776907867730688} a^{12} - \frac{616762865249953}{1837388453933865344} a^{10} - \frac{4194758450254935}{459347113483466336} a^{8} + \frac{6276853258721775}{459347113483466336} a^{6} - \frac{20643837479258047}{229673556741733168} a^{4} - \frac{4858269251464175}{57418389185433292} a^{2} + \frac{9313273874788873}{28709194592716646}$, $\frac{1}{31985258206080727908352} a^{31} + \frac{7076152584669}{15992629103040363954176} a^{29} - \frac{780854399320071}{3998157275760090988544} a^{27} + \frac{5826370154790201}{3998157275760090988544} a^{25} + \frac{9260584561972939}{1999078637880045494272} a^{23} - \frac{7638797047769335}{499769659470011373568} a^{21} - \frac{14472775595430977}{499769659470011373568} a^{19} - \frac{10414764239601135}{249884829735005686784} a^{17} - \frac{3310315638259731}{3674776907867730688} a^{15} - \frac{24963391630336419}{3674776907867730688} a^{13} - \frac{616762865249953}{1837388453933865344} a^{11} - \frac{4194758450254935}{459347113483466336} a^{9} + \frac{6276853258721775}{459347113483466336} a^{7} - \frac{20643837479258047}{229673556741733168} a^{5} - \frac{4858269251464175}{57418389185433292} a^{3} + \frac{9313273874788873}{28709194592716646} a$

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:  \( \frac{8862376563653}{470371444207069528064} a^{30} + \frac{5062829345284895}{7996314551520181977088} a^{28} + \frac{100477630908866135}{7996314551520181977088} a^{26} + \frac{667049026921297383}{3998157275760090988544} a^{24} + \frac{1657686115715772759}{999539318940022747136} a^{22} + \frac{12592757530253509517}{999539318940022747136} a^{20} + \frac{37715115643349410873}{499769659470011373568} a^{18} + \frac{44345503854995443385}{124942414867502843392} a^{16} + \frac{9712431125187831519}{7349553815735461376} a^{14} + \frac{13992055396039302215}{3674776907867730688} a^{12} + \frac{7774928198934191755}{918694226966932672} a^{10} + \frac{12626852064601754869}{918694226966932672} a^{8} + \frac{7491285242414234013}{459347113483466336} a^{6} + \frac{1387611017329210503}{114836778370866584} a^{4} + \frac{773969177576190803}{114836778370866584} a^{2} + \frac{90198638894591567}{57418389185433292} \) (order $6$)
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

$C_2\times C_{16}$ (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

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-51}) \), \(\Q(\sqrt{-3}, \sqrt{17})\), 4.4.4913.1, 4.0.44217.1, 8.0.1955143089.1, \(\Q(\zeta_{17})^+\), 8.0.33237432513.1, 16.0.1104726920056229495169.1, 16.16.48023489818559305679372288.1, 16.0.315082116699567604562361581568.1

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 $16^{2}$ $16^{2}$ $16^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{4}$ $16^{2}$ $16^{2}$ $16^{2}$ $16^{2}$ $16^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{4}$

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$2.8.12.6$x^{8} + 2 x^{6} + 8 x^{4} + 80$$2$$4$$12$$C_8$$[3]^{4}$
2.8.12.6$x^{8} + 2 x^{6} + 8 x^{4} + 80$$2$$4$$12$$C_8$$[3]^{4}$
2.8.12.6$x^{8} + 2 x^{6} + 8 x^{4} + 80$$2$$4$$12$$C_8$$[3]^{4}$
2.8.12.6$x^{8} + 2 x^{6} + 8 x^{4} + 80$$2$$4$$12$$C_8$$[3]^{4}$
3Data not computed
17Data not computed