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Properties

Label 32.0.20868191876...7216.1
Degree $32$
Signature $[0, 16]$
Discriminant $2^{128}\cdot 23^{16}$
Root discriminant $76.73$
Ramified primes $2, 23$
Class number Not computed
Class group Not computed
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![2821109907456, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -3322751, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 3322751*x^16 + 2821109907456)
 
gp: K = bnfinit(x^32 - 3322751*x^16 + 2821109907456, 1)
 

Normalized defining polynomial

\( x^{32} - 3322751 x^{16} + 2821109907456 \)

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:  \(2086819187651930534747998459703375504007855816232843362697216=2^{128}\cdot 23^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $76.73$
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:  $2, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(736=2^{5}\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{736}(1,·)$, $\chi_{736}(643,·)$, $\chi_{736}(645,·)$, $\chi_{736}(137,·)$, $\chi_{736}(139,·)$, $\chi_{736}(275,·)$, $\chi_{736}(277,·)$, $\chi_{736}(413,·)$, $\chi_{736}(415,·)$, $\chi_{736}(551,·)$, $\chi_{736}(553,·)$, $\chi_{736}(45,·)$, $\chi_{736}(47,·)$, $\chi_{736}(689,·)$, $\chi_{736}(691,·)$, $\chi_{736}(183,·)$, $\chi_{736}(185,·)$, $\chi_{736}(321,·)$, $\chi_{736}(323,·)$, $\chi_{736}(459,·)$, $\chi_{736}(461,·)$, $\chi_{736}(597,·)$, $\chi_{736}(599,·)$, $\chi_{736}(91,·)$, $\chi_{736}(93,·)$, $\chi_{736}(735,·)$, $\chi_{736}(229,·)$, $\chi_{736}(231,·)$, $\chi_{736}(367,·)$, $\chi_{736}(369,·)$, $\chi_{736}(505,·)$, $\chi_{736}(507,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{102949} a^{16} + \frac{37283}{102949}$, $\frac{1}{617694} a^{17} + \frac{243181}{617694} a$, $\frac{1}{3706164} a^{18} + \frac{1478569}{3706164} a^{2}$, $\frac{1}{22236984} a^{19} - \frac{5933759}{22236984} a^{3}$, $\frac{1}{133421904} a^{20} - \frac{50407727}{133421904} a^{4}$, $\frac{1}{800531424} a^{21} + \frac{83014177}{800531424} a^{5}$, $\frac{1}{4803188544} a^{22} - \frac{2318580095}{4803188544} a^{6}$, $\frac{1}{28819131264} a^{23} - \frac{11924957183}{28819131264} a^{7}$, $\frac{1}{172914787584} a^{24} - \frac{40744088447}{172914787584} a^{8}$, $\frac{1}{1037488725504} a^{25} - \frac{386573663615}{1037488725504} a^{9}$, $\frac{1}{6224932353024} a^{26} + \frac{650915061889}{6224932353024} a^{10}$, $\frac{1}{37349594118144} a^{27} - \frac{5574017291135}{37349594118144} a^{11}$, $\frac{1}{224097564708864} a^{28} - \frac{80273205527423}{224097564708864} a^{12}$, $\frac{1}{1344585388253184} a^{29} + \frac{143824359181441}{1344585388253184} a^{13}$, $\frac{1}{8067512329519104} a^{30} - \frac{2545346417324927}{8067512329519104} a^{14}$, $\frac{1}{48405073977114624} a^{31} + \frac{21657190571232385}{48405073977114624} a^{15}$

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:  \( \frac{1}{617694} a^{17} - \frac{1609901}{617694} a \) (order $32$)
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

$C_2^2\times C_8$ (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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-23}) \), \(\Q(\sqrt{23}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-46}) \), \(\Q(\sqrt{46}) \), \(\Q(i, \sqrt{23})\), \(\Q(\zeta_{8})\), \(\Q(i, \sqrt{46})\), \(\Q(\sqrt{2}, \sqrt{-23})\), \(\Q(\sqrt{-2}, \sqrt{-23})\), \(\Q(\sqrt{2}, \sqrt{23})\), \(\Q(\sqrt{-2}, \sqrt{23})\), \(\Q(\zeta_{16})^+\), 4.0.2048.2, 4.0.1083392.5, 4.4.1083392.2, 8.0.18339659776.1, \(\Q(\zeta_{16})\), 8.0.4694952902656.2, 8.0.1173738225664.2, 8.0.1173738225664.1, 8.8.4694952902656.1, 8.0.4694952902656.1, 8.0.600953971539968.31, 8.8.600953971539968.4, \(\Q(\zeta_{32})^+\), 8.0.2147483648.1, 16.0.22042582758157999811854336.1, 16.0.1444582703638642675669685764096.1, \(\Q(\zeta_{32})\), 16.0.361145675909660668917421441024.1, 16.0.361145675909660668917421441024.2, 16.0.1444582703638642675669685764096.2, 16.16.1444582703638642675669685764096.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{4}$ R ${\href{/LocalNumberField/29.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{4}$

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
2Data not computed
$23$23.8.4.1$x^{8} + 11638 x^{4} - 12167 x^{2} + 33860761$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
23.8.4.1$x^{8} + 11638 x^{4} - 12167 x^{2} + 33860761$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
23.8.4.1$x^{8} + 11638 x^{4} - 12167 x^{2} + 33860761$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
23.8.4.1$x^{8} + 11638 x^{4} - 12167 x^{2} + 33860761$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$