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Properties

Label 32.0.22351135421...0000.2
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 $29988$ (GRH)
Class group $[42, 714]$ (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![1, 0, 192, 0, 6448, 0, 76608, 0, 444972, 0, 1480992, 0, 3096208, 0, 4291632, 0, 4065711, 0, 2670144, 0, 1217920, 0, 382560, 0, 81260, 0, 11328, 0, 984, 0, 48, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 + 48*x^30 + 984*x^28 + 11328*x^26 + 81260*x^24 + 382560*x^22 + 1217920*x^20 + 2670144*x^18 + 4065711*x^16 + 4291632*x^14 + 3096208*x^12 + 1480992*x^10 + 444972*x^8 + 76608*x^6 + 6448*x^4 + 192*x^2 + 1)
 
gp: K = bnfinit(x^32 + 48*x^30 + 984*x^28 + 11328*x^26 + 81260*x^24 + 382560*x^22 + 1217920*x^20 + 2670144*x^18 + 4065711*x^16 + 4291632*x^14 + 3096208*x^12 + 1480992*x^10 + 444972*x^8 + 76608*x^6 + 6448*x^4 + 192*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{32} + 48 x^{30} + 984 x^{28} + 11328 x^{26} + 81260 x^{24} + 382560 x^{22} + 1217920 x^{20} + 2670144 x^{18} + 4065711 x^{16} + 4291632 x^{14} + 3096208 x^{12} + 1480992 x^{10} + 444972 x^{8} + 76608 x^{6} + 6448 x^{4} + 192 x^{2} + 1 \)

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:  \(2235113542185251937084439154754616201052160000000000000000=2^{128}\cdot 3^{16}\cdot 5^{16}\)
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$$\chi_{480}(1,·)$, $\chi_{480}(259,·)$, $\chi_{480}(389,·)$, $\chi_{480}(139,·)$, $\chi_{480}(269,·)$, $\chi_{480}(19,·)$, $\chi_{480}(149,·)$, $\chi_{480}(409,·)$, $\chi_{480}(29,·)$, $\chi_{480}(289,·)$, $\chi_{480}(169,·)$, $\chi_{480}(431,·)$, $\chi_{480}(49,·)$, $\chi_{480}(311,·)$, $\chi_{480}(191,·)$, $\chi_{480}(451,·)$, $\chi_{480}(71,·)$, $\chi_{480}(331,·)$, $\chi_{480}(461,·)$, $\chi_{480}(211,·)$, $\chi_{480}(341,·)$, $\chi_{480}(91,·)$, $\chi_{480}(221,·)$, $\chi_{480}(479,·)$, $\chi_{480}(101,·)$, $\chi_{480}(359,·)$, $\chi_{480}(361,·)$, $\chi_{480}(239,·)$, $\chi_{480}(241,·)$, $\chi_{480}(119,·)$, $\chi_{480}(121,·)$, $\chi_{480}(379,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $\frac{1}{1673} a^{28} + \frac{794}{1673} a^{26} - \frac{249}{1673} a^{24} - \frac{592}{1673} a^{22} + \frac{523}{1673} a^{20} + \frac{327}{1673} a^{18} - \frac{482}{1673} a^{16} + \frac{115}{1673} a^{14} + \frac{551}{1673} a^{12} - \frac{780}{1673} a^{10} - \frac{347}{1673} a^{8} - \frac{682}{1673} a^{6} - \frac{650}{1673} a^{4} + \frac{22}{1673} a^{2} - \frac{715}{1673}$, $\frac{1}{1673} a^{29} + \frac{794}{1673} a^{27} - \frac{249}{1673} a^{25} - \frac{592}{1673} a^{23} + \frac{523}{1673} a^{21} + \frac{327}{1673} a^{19} - \frac{482}{1673} a^{17} + \frac{115}{1673} a^{15} + \frac{551}{1673} a^{13} - \frac{780}{1673} a^{11} - \frac{347}{1673} a^{9} - \frac{682}{1673} a^{7} - \frac{650}{1673} a^{5} + \frac{22}{1673} a^{3} - \frac{715}{1673} a$, $\frac{1}{2603865422009975213657} a^{30} + \frac{77134621399158226}{371980774572853601951} a^{28} - \frac{286896519610723328879}{2603865422009975213657} a^{26} + \frac{855272830412179663842}{2603865422009975213657} a^{24} + \frac{598787974659798822}{2603865422009975213657} a^{22} + \frac{945412268664705890276}{2603865422009975213657} a^{20} + \frac{113730929576276043290}{371980774572853601951} a^{18} - \frac{101009165082388218764}{371980774572853601951} a^{16} - \frac{516325861568756631613}{2603865422009975213657} a^{14} + \frac{807180828711678248852}{2603865422009975213657} a^{12} - \frac{693576601612000019125}{2603865422009975213657} a^{10} + \frac{586119388012063645007}{2603865422009975213657} a^{8} + \frac{666612365395146636049}{2603865422009975213657} a^{6} + \frac{1106487728528403480432}{2603865422009975213657} a^{4} - \frac{278365585780343254981}{2603865422009975213657} a^{2} - \frac{233025987588203983513}{2603865422009975213657}$, $\frac{1}{2603865422009975213657} a^{31} + \frac{77134621399158226}{371980774572853601951} a^{29} - \frac{286896519610723328879}{2603865422009975213657} a^{27} + \frac{855272830412179663842}{2603865422009975213657} a^{25} + \frac{598787974659798822}{2603865422009975213657} a^{23} + \frac{945412268664705890276}{2603865422009975213657} a^{21} + \frac{113730929576276043290}{371980774572853601951} a^{19} - \frac{101009165082388218764}{371980774572853601951} a^{17} - \frac{516325861568756631613}{2603865422009975213657} a^{15} + \frac{807180828711678248852}{2603865422009975213657} a^{13} - \frac{693576601612000019125}{2603865422009975213657} a^{11} + \frac{586119388012063645007}{2603865422009975213657} a^{9} + \frac{666612365395146636049}{2603865422009975213657} a^{7} + \frac{1106487728528403480432}{2603865422009975213657} a^{5} - \frac{278365585780343254981}{2603865422009975213657} a^{3} - \frac{233025987588203983513}{2603865422009975213657} a$

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

Class group and class number

$C_{42}\times C_{714}$, which has order $29988$ (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:  \( -1 \) (order $2$)
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:  \( 82239790500.5115 \) (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

\(\Q(\sqrt{30}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{5}, \sqrt{6})\), \(\Q(\sqrt{2}, \sqrt{15})\), \(\Q(\sqrt{3}, \sqrt{10})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\sqrt{6}, \sqrt{10})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{3}, \sqrt{5})\), \(\Q(\zeta_{16})^+\), 4.4.460800.1, 4.4.18432.1, 4.4.51200.1, 8.8.3317760000.1, 8.8.849346560000.2, 8.8.849346560000.1, \(\Q(\zeta_{48})^+\), 8.8.849346560000.3, 8.8.2621440000.1, 8.8.212336640000.1, 8.0.108716359680000.13, 8.0.2147483648.1, 8.0.1342177280000.1, 8.0.173946175488.1, 16.16.721389578983833600000000.1, 16.0.47276987448284518809600000000.6, 16.0.47276987448284518809600000000.3, 16.0.47276987448284518809600000000.1, 16.0.121029087867608368152576.1, 16.0.11819246862071129702400000000.1, 16.0.1801439850948198400000000.1

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 R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{4}$

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