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Properties

Label 32.0.15148428384...5184.2
Degree $32$
Signature $[0, 16]$
Discriminant $2^{32}\cdot 3^{16}\cdot 17^{30}$
Root discriminant $49.33$
Ramified primes $2, 3, 17$
Class number $3860$ (GRH)
Class group $[2, 1930]$ (GRH)
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![289, 0, 3468, 0, 29478, 0, 107508, 0, 265013, 0, 427142, 0, 505172, 0, 442459, 0, 300254, 0, 158576, 0, 66453, 0, 21930, 0, 5712, 0, 1139, 0, 170, 0, 17, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 + 17*x^30 + 170*x^28 + 1139*x^26 + 5712*x^24 + 21930*x^22 + 66453*x^20 + 158576*x^18 + 300254*x^16 + 442459*x^14 + 505172*x^12 + 427142*x^10 + 265013*x^8 + 107508*x^6 + 29478*x^4 + 3468*x^2 + 289)
 
gp: K = bnfinit(x^32 + 17*x^30 + 170*x^28 + 1139*x^26 + 5712*x^24 + 21930*x^22 + 66453*x^20 + 158576*x^18 + 300254*x^16 + 442459*x^14 + 505172*x^12 + 427142*x^10 + 265013*x^8 + 107508*x^6 + 29478*x^4 + 3468*x^2 + 289, 1)
 

Normalized defining polynomial

\( x^{32} + 17 x^{30} + 170 x^{28} + 1139 x^{26} + 5712 x^{24} + 21930 x^{22} + 66453 x^{20} + 158576 x^{18} + 300254 x^{16} + 442459 x^{14} + 505172 x^{12} + 427142 x^{10} + 265013 x^{8} + 107508 x^{6} + 29478 x^{4} + 3468 x^{2} + 289 \)

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:  \(1514842838499144573219529960757824409341479409296605184=2^{32}\cdot 3^{16}\cdot 17^{30}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $49.33$
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$$\chi_{204}(1,·)$, $\chi_{204}(131,·)$, $\chi_{204}(7,·)$, $\chi_{204}(137,·)$, $\chi_{204}(11,·)$, $\chi_{204}(13,·)$, $\chi_{204}(143,·)$, $\chi_{204}(145,·)$, $\chi_{204}(149,·)$, $\chi_{204}(23,·)$, $\chi_{204}(25,·)$, $\chi_{204}(157,·)$, $\chi_{204}(31,·)$, $\chi_{204}(161,·)$, $\chi_{204}(163,·)$, $\chi_{204}(167,·)$, $\chi_{204}(169,·)$, $\chi_{204}(71,·)$, $\chi_{204}(175,·)$, $\chi_{204}(49,·)$, $\chi_{204}(53,·)$, $\chi_{204}(185,·)$, $\chi_{204}(139,·)$, $\chi_{204}(199,·)$, $\chi_{204}(77,·)$, $\chi_{204}(79,·)$, $\chi_{204}(89,·)$, $\chi_{204}(91,·)$, $\chi_{204}(95,·)$, $\chi_{204}(101,·)$, $\chi_{204}(107,·)$, $\chi_{204}(121,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{17} a^{16}$, $\frac{1}{17} a^{17}$, $\frac{1}{17} a^{18}$, $\frac{1}{17} a^{19}$, $\frac{1}{17} a^{20}$, $\frac{1}{17} a^{21}$, $\frac{1}{17} a^{22}$, $\frac{1}{17} a^{23}$, $\frac{1}{68} a^{24} - \frac{1}{68} a^{18} + \frac{1}{4} a^{12} - \frac{1}{4} a^{6} + \frac{1}{4}$, $\frac{1}{68} a^{25} - \frac{1}{68} a^{19} + \frac{1}{4} a^{13} - \frac{1}{4} a^{7} + \frac{1}{4} a$, $\frac{1}{68} a^{26} - \frac{1}{68} a^{20} + \frac{1}{4} a^{14} - \frac{1}{4} a^{8} + \frac{1}{4} a^{2}$, $\frac{1}{68} a^{27} - \frac{1}{68} a^{21} + \frac{1}{4} a^{15} - \frac{1}{4} a^{9} + \frac{1}{4} a^{3}$, $\frac{1}{39236} a^{28} - \frac{69}{19618} a^{26} + \frac{49}{39236} a^{24} - \frac{49}{2308} a^{22} + \frac{263}{19618} a^{20} - \frac{241}{39236} a^{18} + \frac{709}{39236} a^{16} + \frac{323}{1154} a^{14} - \frac{1147}{2308} a^{12} - \frac{661}{2308} a^{10} + \frac{29}{1154} a^{8} - \frac{837}{2308} a^{6} + \frac{333}{2308} a^{4} - \frac{143}{1154} a^{2} + \frac{773}{2308}$, $\frac{1}{39236} a^{29} - \frac{69}{19618} a^{27} + \frac{49}{39236} a^{25} - \frac{49}{2308} a^{23} + \frac{263}{19618} a^{21} - \frac{241}{39236} a^{19} + \frac{709}{39236} a^{17} + \frac{323}{1154} a^{15} - \frac{1147}{2308} a^{13} - \frac{661}{2308} a^{11} + \frac{29}{1154} a^{9} - \frac{837}{2308} a^{7} + \frac{333}{2308} a^{5} - \frac{143}{1154} a^{3} + \frac{773}{2308} a$, $\frac{1}{976112616152365964} a^{30} + \frac{7076152584669}{976112616152365964} a^{28} - \frac{780854399320071}{488056308076182982} a^{26} + \frac{5826370154790201}{976112616152365964} a^{24} + \frac{9260584561972939}{976112616152365964} a^{22} - \frac{7638797047769335}{488056308076182982} a^{20} - \frac{14472775595430977}{976112616152365964} a^{18} - \frac{10414764239601135}{976112616152365964} a^{16} - \frac{3310315638259731}{28709194592716646} a^{14} - \frac{24963391630336419}{57418389185433292} a^{12} - \frac{616762865249953}{57418389185433292} a^{10} - \frac{4194758450254935}{28709194592716646} a^{8} + \frac{6276853258721775}{57418389185433292} a^{6} - \frac{20643837479258047}{57418389185433292} a^{4} - \frac{4858269251464175}{28709194592716646} a^{2} + \frac{9313273874788873}{28709194592716646}$, $\frac{1}{976112616152365964} a^{31} + \frac{7076152584669}{976112616152365964} a^{29} - \frac{780854399320071}{488056308076182982} a^{27} + \frac{5826370154790201}{976112616152365964} a^{25} + \frac{9260584561972939}{976112616152365964} a^{23} - \frac{7638797047769335}{488056308076182982} a^{21} - \frac{14472775595430977}{976112616152365964} a^{19} - \frac{10414764239601135}{976112616152365964} a^{17} - \frac{3310315638259731}{28709194592716646} a^{15} - \frac{24963391630336419}{57418389185433292} a^{13} - \frac{616762865249953}{57418389185433292} a^{11} - \frac{4194758450254935}{28709194592716646} a^{9} + \frac{6276853258721775}{57418389185433292} a^{7} - \frac{20643837479258047}{57418389185433292} a^{5} - \frac{4858269251464175}{28709194592716646} a^{3} + \frac{9313273874788873}{28709194592716646} a$

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

Class group and class number

$C_{2}\times C_{1930}$, which has order $3860$ (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:  \( \frac{8862376563653}{14354597296358323} a^{30} + \frac{5062829345284895}{488056308076182982} a^{28} + \frac{100477630908866135}{976112616152365964} a^{26} + \frac{667049026921297383}{976112616152365964} a^{24} + \frac{1657686115715772759}{488056308076182982} a^{22} + \frac{12592757530253509517}{976112616152365964} a^{20} + \frac{37715115643349410873}{976112616152365964} a^{18} + \frac{44345503854995443385}{488056308076182982} a^{16} + \frac{9712431125187831519}{57418389185433292} a^{14} + \frac{13992055396039302215}{57418389185433292} a^{12} + \frac{7774928198934191755}{28709194592716646} a^{10} + \frac{12626852064601754869}{57418389185433292} a^{8} + \frac{7491285242414234013}{57418389185433292} a^{6} + \frac{1387611017329210503}{28709194592716646} a^{4} + \frac{773969177576190803}{57418389185433292} a^{2} + \frac{90198638894591567}{57418389185433292} \) (order $6$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  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:  \( 73876734347.55115 \) (assuming GRH)
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, 16.0.1230789518357685955321724928.1, \(\Q(\zeta_{68})^+\)

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$2.8.8.4$x^{8} + 2 x^{7} + 2 x^{6} + 8 x^{3} + 48$$2$$4$$8$$C_8$$[2]^{4}$
2.8.8.4$x^{8} + 2 x^{7} + 2 x^{6} + 8 x^{3} + 48$$2$$4$$8$$C_8$$[2]^{4}$
2.8.8.4$x^{8} + 2 x^{7} + 2 x^{6} + 8 x^{3} + 48$$2$$4$$8$$C_8$$[2]^{4}$
2.8.8.4$x^{8} + 2 x^{7} + 2 x^{6} + 8 x^{3} + 48$$2$$4$$8$$C_8$$[2]^{4}$
3Data not computed
17Data not computed