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Properties

Label 32.0.22351135421...0000.3
Degree $32$
Signature $[0, 16]$
Discriminant $2^{128}\cdot 3^{16}\cdot 5^{16}$
Root discriminant $61.97$
Ramified primes $2, 3, 5$
Class number $12240$ (GRH)
Class group $[3, 4080]$ (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![65536, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -223, 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 - 223*x^16 + 65536)
 
gp: K = bnfinit(x^32 - 223*x^16 + 65536, 1)
 

Normalized defining polynomial

\( x^{32} - 223 x^{16} + 65536 \)

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:  \(2235113542185251937084439154754616201052160000000000000000=2^{128}\cdot 3^{16}\cdot 5^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $61.97$
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:  $2, 3, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(480=2^{5}\cdot 3\cdot 5\)
Dirichlet character group:    $\lbrace$$\chi_{480}(1,·)$, $\chi_{480}(131,·)$, $\chi_{480}(391,·)$, $\chi_{480}(11,·)$, $\chi_{480}(271,·)$, $\chi_{480}(19,·)$, $\chi_{480}(151,·)$, $\chi_{480}(31,·)$, $\chi_{480}(349,·)$, $\chi_{480}(449,·)$, $\chi_{480}(139,·)$, $\chi_{480}(329,·)$, $\chi_{480}(461,·)$, $\chi_{480}(209,·)$, $\chi_{480}(341,·)$, $\chi_{480}(89,·)$, $\chi_{480}(101,·)$, $\chi_{480}(221,·)$, $\chi_{480}(479,·)$, $\chi_{480}(251,·)$, $\chi_{480}(229,·)$, $\chi_{480}(359,·)$, $\chi_{480}(259,·)$, $\chi_{480}(361,·)$, $\chi_{480}(109,·)$, $\chi_{480}(239,·)$, $\chi_{480}(241,·)$, $\chi_{480}(371,·)$, $\chi_{480}(119,·)$, $\chi_{480}(121,·)$, $\chi_{480}(379,·)$, $\chi_{480}(469,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{119} a^{16} - \frac{52}{119}$, $\frac{1}{238} a^{17} + \frac{67}{238} a$, $\frac{1}{476} a^{18} - \frac{171}{476} a^{2}$, $\frac{1}{952} a^{19} + \frac{305}{952} a^{3}$, $\frac{1}{1904} a^{20} + \frac{305}{1904} a^{4}$, $\frac{1}{3808} a^{21} - \frac{1599}{3808} a^{5}$, $\frac{1}{7616} a^{22} + \frac{2209}{7616} a^{6}$, $\frac{1}{15232} a^{23} + \frac{2209}{15232} a^{7}$, $\frac{1}{30464} a^{24} - \frac{13023}{30464} a^{8}$, $\frac{1}{60928} a^{25} - \frac{13023}{60928} a^{9}$, $\frac{1}{121856} a^{26} + \frac{47905}{121856} a^{10}$, $\frac{1}{243712} a^{27} - \frac{73951}{243712} a^{11}$, $\frac{1}{487424} a^{28} - \frac{73951}{487424} a^{12}$, $\frac{1}{974848} a^{29} - \frac{73951}{974848} a^{13}$, $\frac{1}{1949696} a^{30} + \frac{900897}{1949696} a^{14}$, $\frac{1}{3899392} a^{31} - \frac{1048799}{3899392} a^{15}$

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

Class group and class number

$C_{3}\times C_{4080}$, which has order $12240$ (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:  \( -\frac{3}{7616} a^{22} + \frac{989}{7616} a^{6} \) (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:  \( 34585710193758.383 \) (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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-30}) \), \(\Q(\sqrt{30}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{15}) \), \(\Q(i, \sqrt{30})\), \(\Q(\zeta_{8})\), \(\Q(i, \sqrt{15})\), \(\Q(\sqrt{2}, \sqrt{-15})\), \(\Q(\sqrt{-2}, \sqrt{15})\), \(\Q(\sqrt{2}, \sqrt{15})\), \(\Q(\sqrt{-2}, \sqrt{-15})\), \(\Q(\zeta_{16})^+\), 4.0.2048.2, 4.0.460800.2, 4.4.460800.1, 8.0.3317760000.4, \(\Q(\zeta_{16})\), 8.0.849346560000.6, 8.0.212336640000.4, 8.0.212336640000.1, 8.8.849346560000.2, 8.0.849346560000.2, 8.0.1342177280000.1, 8.8.1342177280000.1, 8.8.173946175488.1, 8.0.173946175488.1, 16.0.721389578983833600000000.7, 16.0.7205759403792793600000000.2, 16.0.121029087867608368152576.2, 16.0.11819246862071129702400000000.6, 16.0.11819246862071129702400000000.5, 16.0.47276987448284518809600000000.3, 16.16.47276987448284518809600000000.2

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 R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/17.1.0.1}{1} }^{32}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{4}$

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
3Data not computed
5Data not computed