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Properties

Label 32.0.26893894496...2656.1
Degree $32$
Signature $[0, 16]$
Discriminant $2^{96}\cdot 17^{24}$
Root discriminant $66.98$
Ramified primes $2, 17$
Class number $3328$ (GRH)
Class group $[4, 4, 4, 52]$ (GRH)
Galois group $C_2\times C_4^2$ (as 32T36)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 0, 0, 0, 0, 0, 0, 5393, 0, 0, 0, 0, 0, 0, 0, 1581856, 0, 0, 0, 0, 0, 0, 0, 5393, 0, 0, 0, 0, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 + 5393*x^24 + 1581856*x^16 + 5393*x^8 + 1)
 
gp: K = bnfinit(x^32 + 5393*x^24 + 1581856*x^16 + 5393*x^8 + 1, 1)
 

Normalized defining polynomial

\( x^{32} + 5393 x^{24} + 1581856 x^{16} + 5393 x^{8} + 1 \)

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:  \(26893894496165171048841611301109623910839789254478351302656=2^{96}\cdot 17^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $66.98$
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:  $2, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(272=2^{4}\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{272}(1,·)$, $\chi_{272}(259,·)$, $\chi_{272}(135,·)$, $\chi_{272}(137,·)$, $\chi_{272}(13,·)$, $\chi_{272}(271,·)$, $\chi_{272}(149,·)$, $\chi_{272}(89,·)$, $\chi_{272}(157,·)$, $\chi_{272}(33,·)$, $\chi_{272}(35,·)$, $\chi_{272}(169,·)$, $\chi_{272}(171,·)$, $\chi_{272}(47,·)$, $\chi_{272}(183,·)$, $\chi_{272}(191,·)$, $\chi_{272}(67,·)$, $\chi_{272}(69,·)$, $\chi_{272}(55,·)$, $\chi_{272}(205,·)$, $\chi_{272}(81,·)$, $\chi_{272}(203,·)$, $\chi_{272}(217,·)$, $\chi_{272}(225,·)$, $\chi_{272}(123,·)$, $\chi_{272}(101,·)$, $\chi_{272}(103,·)$, $\chi_{272}(237,·)$, $\chi_{272}(239,·)$, $\chi_{272}(115,·)$, $\chi_{272}(251,·)$, $\chi_{272}(21,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{1157} a^{16} - \frac{196}{1157} a^{8} + \frac{1}{1157}$, $\frac{1}{1157} a^{17} - \frac{196}{1157} a^{9} + \frac{1}{1157} a$, $\frac{1}{1157} a^{18} - \frac{196}{1157} a^{10} + \frac{1}{1157} a^{2}$, $\frac{1}{1157} a^{19} - \frac{196}{1157} a^{11} + \frac{1}{1157} a^{3}$, $\frac{1}{1157} a^{20} - \frac{196}{1157} a^{12} + \frac{1}{1157} a^{4}$, $\frac{1}{1157} a^{21} - \frac{196}{1157} a^{13} + \frac{1}{1157} a^{5}$, $\frac{1}{1157} a^{22} - \frac{196}{1157} a^{14} + \frac{1}{1157} a^{6}$, $\frac{1}{1157} a^{23} - \frac{196}{1157} a^{15} + \frac{1}{1157} a^{7}$, $\frac{1}{443880736} a^{24} + \frac{1470}{13871273} a^{16} - \frac{1707694}{13871273} a^{8} - \frac{128563727}{443880736}$, $\frac{1}{443880736} a^{25} + \frac{1470}{13871273} a^{17} - \frac{1707694}{13871273} a^{9} - \frac{128563727}{443880736} a$, $\frac{1}{443880736} a^{26} + \frac{1470}{13871273} a^{18} - \frac{1707694}{13871273} a^{10} - \frac{128563727}{443880736} a^{2}$, $\frac{1}{443880736} a^{27} + \frac{1470}{13871273} a^{19} - \frac{1707694}{13871273} a^{11} - \frac{128563727}{443880736} a^{3}$, $\frac{1}{443880736} a^{28} + \frac{1470}{13871273} a^{20} - \frac{1707694}{13871273} a^{12} - \frac{128563727}{443880736} a^{4}$, $\frac{1}{443880736} a^{29} + \frac{1470}{13871273} a^{21} - \frac{1707694}{13871273} a^{13} - \frac{128563727}{443880736} a^{5}$, $\frac{1}{443880736} a^{30} + \frac{1470}{13871273} a^{22} - \frac{1707694}{13871273} a^{14} - \frac{128563727}{443880736} a^{6}$, $\frac{1}{443880736} a^{31} + \frac{1470}{13871273} a^{23} - \frac{1707694}{13871273} a^{15} - \frac{128563727}{443880736} a^{7}$

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

Class group and class number

$C_{4}\times C_{4}\times C_{4}\times C_{52}$, which has order $3328$ (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:  \( -\frac{1541897}{221940368} a^{27} - \frac{519715562}{13871273} a^{19} - \frac{152440673890}{13871273} a^{11} - \frac{5924869417}{221940368} a^{3} \) (order $16$)
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:  \( 38649094918357.02 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_4^2$ (as 32T36):

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 $C_2\times C_4^2$
Character table for $C_2\times C_4^2$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{34}) \), \(\Q(\sqrt{-34}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-17}) \), \(\Q(\zeta_{8})\), \(\Q(\zeta_{16})^+\), 4.0.2048.2, \(\Q(i, \sqrt{34})\), \(\Q(i, \sqrt{17})\), \(\Q(\sqrt{2}, \sqrt{17})\), \(\Q(\sqrt{2}, \sqrt{-17})\), 4.4.591872.2, 4.0.591872.5, \(\Q(\sqrt{-2}, \sqrt{-17})\), \(\Q(\sqrt{-2}, \sqrt{17})\), 4.4.10061824.1, 4.0.10061824.1, 4.4.10061824.2, 4.0.10061824.2, 4.4.4913.1, 4.0.78608.1, 4.4.314432.1, 4.0.314432.2, \(\Q(\zeta_{16})\), 8.0.5473632256.1, 8.0.1401249857536.3, 8.8.350312464384.1, 8.0.1401249857536.2, 8.0.350312464384.1, 8.0.1401249857536.1, 8.0.404961208827904.3, 8.0.404961208827904.4, 8.0.6179217664.1, 8.0.1581879721984.3, 8.8.101240302206976.1, 8.0.101240302206976.1, 8.8.98867482624.1, 8.0.1581879721984.2, 8.0.404961208827904.2, 8.0.404961208827904.1, 8.0.98867482624.1, 8.0.1581879721984.1, 16.0.1963501163244660295991296.1, 16.0.163993580655357272581489033216.4, 16.0.2502343454824177132896256.1, 16.16.10249598790959829536343064576.1, 16.0.163993580655357272581489033216.3, 16.0.163993580655357272581489033216.2, 16.0.10249598790959829536343064576.2

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{8}$

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
2Data not computed
$17$17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$