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Properties

Label 32.0.20282409603...0000.2
Degree $32$
Signature $[0, 16]$
Discriminant $2^{128}\cdot 5^{24}$
Root discriminant $53.50$
Ramified primes $2, 5$
Class number $738$ (GRH)
Class group $[3, 246]$ (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![390625, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 29375, 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 + 29375*x^16 + 390625)
 
gp: K = bnfinit(x^32 + 29375*x^16 + 390625, 1)
 

Normalized defining polynomial

\( x^{32} + 29375 x^{16} + 390625 \)

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:  \(20282409603651670423947251286016000000000000000000000000=2^{128}\cdot 5^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $53.50$
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:  $2, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(160=2^{5}\cdot 5\)
Dirichlet character group:    $\lbrace$$\chi_{160}(1,·)$, $\chi_{160}(3,·)$, $\chi_{160}(133,·)$, $\chi_{160}(129,·)$, $\chi_{160}(9,·)$, $\chi_{160}(13,·)$, $\chi_{160}(147,·)$, $\chi_{160}(151,·)$, $\chi_{160}(27,·)$, $\chi_{160}(157,·)$, $\chi_{160}(159,·)$, $\chi_{160}(37,·)$, $\chi_{160}(39,·)$, $\chi_{160}(41,·)$, $\chi_{160}(43,·)$, $\chi_{160}(49,·)$, $\chi_{160}(53,·)$, $\chi_{160}(31,·)$, $\chi_{160}(67,·)$, $\chi_{160}(71,·)$, $\chi_{160}(77,·)$, $\chi_{160}(79,·)$, $\chi_{160}(81,·)$, $\chi_{160}(83,·)$, $\chi_{160}(89,·)$, $\chi_{160}(93,·)$, $\chi_{160}(107,·)$, $\chi_{160}(111,·)$, $\chi_{160}(117,·)$, $\chi_{160}(119,·)$, $\chi_{160}(121,·)$, $\chi_{160}(123,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{5} a^{4}$, $\frac{1}{5} a^{5}$, $\frac{1}{5} a^{6}$, $\frac{1}{5} a^{7}$, $\frac{1}{25} a^{8}$, $\frac{1}{25} a^{9}$, $\frac{1}{25} a^{10}$, $\frac{1}{25} a^{11}$, $\frac{1}{125} a^{12}$, $\frac{1}{125} a^{13}$, $\frac{1}{125} a^{14}$, $\frac{1}{125} a^{15}$, $\frac{1}{13125} a^{16} - \frac{8}{21}$, $\frac{1}{13125} a^{17} - \frac{8}{21} a$, $\frac{1}{65625} a^{18} - \frac{29}{105} a^{2}$, $\frac{1}{65625} a^{19} - \frac{29}{105} a^{3}$, $\frac{1}{65625} a^{20} - \frac{8}{105} a^{4}$, $\frac{1}{65625} a^{21} - \frac{8}{105} a^{5}$, $\frac{1}{328125} a^{22} - \frac{29}{525} a^{6}$, $\frac{1}{328125} a^{23} - \frac{29}{525} a^{7}$, $\frac{1}{328125} a^{24} - \frac{8}{525} a^{8}$, $\frac{1}{328125} a^{25} - \frac{8}{525} a^{9}$, $\frac{1}{1640625} a^{26} - \frac{29}{2625} a^{10}$, $\frac{1}{1640625} a^{27} - \frac{29}{2625} a^{11}$, $\frac{1}{1640625} a^{28} - \frac{8}{2625} a^{12}$, $\frac{1}{1640625} a^{29} - \frac{8}{2625} a^{13}$, $\frac{1}{8203125} a^{30} - \frac{29}{13125} a^{14}$, $\frac{1}{8203125} a^{31} - \frac{29}{13125} a^{15}$

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

Class group and class number

$C_{3}\times C_{246}$, which has order $738$ (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:  \( -\frac{1}{65625} a^{18} - \frac{76}{105} a^{2} \) (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:  \( 2660439411454.7925 \) (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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{10}) \), \(\Q(i, \sqrt{5})\), \(\Q(\zeta_{8})\), \(\Q(i, \sqrt{10})\), \(\Q(\sqrt{2}, \sqrt{-5})\), \(\Q(\sqrt{-2}, \sqrt{-5})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{-2}, \sqrt{5})\), 4.4.51200.1, 4.0.51200.2, 4.0.2048.2, \(\Q(\zeta_{16})^+\), 8.0.40960000.1, 8.0.10485760000.3, \(\Q(\zeta_{16})\), 8.0.10485760000.1, 8.0.10485760000.2, 8.8.2621440000.1, 8.0.2621440000.1, 8.8.33554432000000.1, 8.0.33554432000000.1, 8.0.33554432000000.2, 8.8.33554432000000.2, 16.0.109951162777600000000.1, 16.0.4503599627370496000000000000.1, 16.0.4503599627370496000000000000.2, 16.0.4503599627370496000000000000.4, 16.0.4503599627370496000000000000.3, 16.16.1125899906842624000000000000.1, 16.0.1125899906842624000000000000.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{4}$ R ${\href{/LocalNumberField/7.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{4}$

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
2Data not computed
5Data not computed