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Properties

Label 16.8.48149758582...7616.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{32}\cdot 17^{4}\cdot 41^{10}$
Root discriminant $82.73$
Ramified primes $2, 17, 41$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $C_4.C_2^2:D_4$ (as 16T305)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-768484, 1631112, 980700, -1059392, 278674, -272904, 248706, -159212, 67535, -13616, -3718, 3624, -819, -4, 30, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 30*x^14 - 4*x^13 - 819*x^12 + 3624*x^11 - 3718*x^10 - 13616*x^9 + 67535*x^8 - 159212*x^7 + 248706*x^6 - 272904*x^5 + 278674*x^4 - 1059392*x^3 + 980700*x^2 + 1631112*x - 768484)
 
gp: K = bnfinit(x^16 - 8*x^15 + 30*x^14 - 4*x^13 - 819*x^12 + 3624*x^11 - 3718*x^10 - 13616*x^9 + 67535*x^8 - 159212*x^7 + 248706*x^6 - 272904*x^5 + 278674*x^4 - 1059392*x^3 + 980700*x^2 + 1631112*x - 768484, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 30 x^{14} - 4 x^{13} - 819 x^{12} + 3624 x^{11} - 3718 x^{10} - 13616 x^{9} + 67535 x^{8} - 159212 x^{7} + 248706 x^{6} - 272904 x^{5} + 278674 x^{4} - 1059392 x^{3} + 980700 x^{2} + 1631112 x - 768484 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(4814975858202960880562776047616=2^{32}\cdot 17^{4}\cdot 41^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $82.73$
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:  $2, 17, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{82} a^{12} + \frac{9}{41} a^{11} - \frac{11}{41} a^{10} - \frac{1}{41} a^{9} + \frac{3}{82} a^{8} - \frac{7}{41} a^{7} - \frac{15}{41} a^{6} - \frac{15}{41} a^{5} - \frac{15}{82} a^{4} + \frac{13}{41} a^{3} - \frac{16}{41} a^{2} + \frac{13}{41} a - \frac{10}{41}$, $\frac{1}{82} a^{13} - \frac{9}{41} a^{11} - \frac{8}{41} a^{10} + \frac{39}{82} a^{9} + \frac{7}{41} a^{8} - \frac{12}{41} a^{7} + \frac{9}{41} a^{6} + \frac{33}{82} a^{5} - \frac{16}{41} a^{4} - \frac{4}{41} a^{3} + \frac{14}{41} a^{2} + \frac{2}{41} a + \frac{16}{41}$, $\frac{1}{1554310} a^{14} - \frac{744}{777155} a^{13} + \frac{1203}{777155} a^{12} - \frac{177836}{777155} a^{11} + \frac{551817}{1554310} a^{10} + \frac{44581}{777155} a^{9} - \frac{125938}{777155} a^{8} - \frac{12592}{777155} a^{7} - \frac{488561}{1554310} a^{6} - \frac{14188}{777155} a^{5} - \frac{63360}{155431} a^{4} - \frac{12045}{155431} a^{3} - \frac{4678}{18955} a^{2} + \frac{184434}{777155} a + \frac{54657}{777155}$, $\frac{1}{566478141745794778518912169432869000050} a^{15} - \frac{7092997015056075982891871463031}{40462724410413912751350869245204928575} a^{14} + \frac{1594068357546123597741125730000859179}{566478141745794778518912169432869000050} a^{13} + \frac{1606546835457550873135459352587095461}{283239070872897389259456084716434500025} a^{12} + \frac{89362808272551236044837349029584374999}{566478141745794778518912169432869000050} a^{11} - \frac{134054707446684338436961533660609358}{1381654004258036045168078462031387805} a^{10} - \frac{129771404715375325769046619182928343633}{566478141745794778518912169432869000050} a^{9} - \frac{80852697502769516512849268317310420039}{283239070872897389259456084716434500025} a^{8} - \frac{32344956409918005870607723069945939561}{80925448820827825502701738490409857150} a^{7} + \frac{27537120269146358820993051984310733588}{56647814174579477851891216943286900005} a^{6} - \frac{199951596612314286652152569951871077739}{566478141745794778518912169432869000050} a^{5} - \frac{21061796348949526614931034781294741863}{56647814174579477851891216943286900005} a^{4} - \frac{47850617473328012837830523105052628448}{283239070872897389259456084716434500025} a^{3} - \frac{111158095661175047432111785578723169623}{283239070872897389259456084716434500025} a^{2} - \frac{90545545280905634655484939979209443122}{283239070872897389259456084716434500025} a + \frac{341468215456974973510131875683808338}{283239070872897389259456084716434500025}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  $11$
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:  \( 852070145.93 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4.C_2^2:D_4$ (as 16T305):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 29 conjugacy class representatives for $C_4.C_2^2:D_4$
Character table for $C_4.C_2^2:D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{41}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{82}) \), 4.4.13448.1 x2, 4.4.2624.1 x2, \(\Q(\sqrt{2}, \sqrt{41})\), 8.8.11574317056.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 16 siblings: data not computed
Degree 32 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$2$2.8.16.3$x^{8} + 2 x^{6} + 6 x^{4} + 4 x^{2} + 8 x + 28$$4$$2$$16$$C_4\times C_2$$[2, 3]^{2}$
2.8.16.3$x^{8} + 2 x^{6} + 6 x^{4} + 4 x^{2} + 8 x + 28$$4$$2$$16$$C_4\times C_2$$[2, 3]^{2}$
$17$17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.4.2.2$x^{4} - 17 x^{2} + 867$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$41$41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.8.6.1$x^{8} - 9881 x^{4} + 34857216$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$