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Properties

Label 16.0.76785869193...3616.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 3^{8}\cdot 17^{8}$
Root discriminant $20.20$
Ramified primes $2, 3, 17$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2\times D_8$ (as 16T29)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![25, -102, 134, 34, -375, 522, 32, -1024, 1928, -2074, 1646, -978, 457, -162, 44, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 44*x^14 - 162*x^13 + 457*x^12 - 978*x^11 + 1646*x^10 - 2074*x^9 + 1928*x^8 - 1024*x^7 + 32*x^6 + 522*x^5 - 375*x^4 + 34*x^3 + 134*x^2 - 102*x + 25)
 
gp: K = bnfinit(x^16 - 8*x^15 + 44*x^14 - 162*x^13 + 457*x^12 - 978*x^11 + 1646*x^10 - 2074*x^9 + 1928*x^8 - 1024*x^7 + 32*x^6 + 522*x^5 - 375*x^4 + 34*x^3 + 134*x^2 - 102*x + 25, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 44 x^{14} - 162 x^{13} + 457 x^{12} - 978 x^{11} + 1646 x^{10} - 2074 x^{9} + 1928 x^{8} - 1024 x^{7} + 32 x^{6} + 522 x^{5} - 375 x^{4} + 34 x^{3} + 134 x^{2} - 102 x + 25 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(767858691933644783616=2^{24}\cdot 3^{8}\cdot 17^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.20$
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:  $2, 3, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{4} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} + \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{820} a^{14} + \frac{5}{164} a^{13} - \frac{1}{82} a^{12} + \frac{72}{205} a^{11} + \frac{24}{205} a^{10} + \frac{92}{205} a^{9} - \frac{37}{410} a^{8} + \frac{3}{205} a^{7} + \frac{13}{82} a^{6} - \frac{78}{205} a^{5} + \frac{53}{410} a^{4} - \frac{61}{410} a^{3} + \frac{1}{164} a^{2} - \frac{393}{820} a + \frac{20}{41}$, $\frac{1}{3675747033880} a^{15} - \frac{1211094391}{3675747033880} a^{14} + \frac{24883982399}{735149406776} a^{13} - \frac{232220027757}{3675747033880} a^{12} + \frac{611119098919}{1837873516940} a^{11} - \frac{3428283869}{15843737215} a^{10} - \frac{708717787241}{1837873516940} a^{9} + \frac{184744215339}{918936758470} a^{8} + \frac{83486373076}{459468379235} a^{7} - \frac{59888636414}{459468379235} a^{6} + \frac{188406415371}{459468379235} a^{5} + \frac{738044969791}{1837873516940} a^{4} + \frac{985831987827}{3675747033880} a^{3} - \frac{768303624823}{3675747033880} a^{2} + \frac{965499559}{2947672040} a + \frac{307289750581}{735149406776}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{198694469}{1545730460} a^{15} + \frac{1494745393}{1545730460} a^{14} - \frac{1594425355}{309146092} a^{13} + \frac{27960280593}{1545730460} a^{12} - \frac{7523163315}{154573046} a^{11} + \frac{37687516604}{386432615} a^{10} - \frac{23496851125}{154573046} a^{9} + \frac{13046468659}{77286523} a^{8} - \frac{50109932531}{386432615} a^{7} + \frac{11977089467}{386432615} a^{6} + \frac{2839298956}{77286523} a^{5} - \frac{39640340897}{772865230} a^{4} + \frac{20501473829}{1545730460} a^{3} + \frac{15324208357}{1545730460} a^{2} - \frac{419917283}{35947220} a + \frac{1161686991}{309146092} \) (order $4$)
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:  \( 20048.5497167 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times D_8$ (as 16T29):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_2\times D_8$
Character table for $C_2\times D_8$

Intermediate fields

\(\Q(\sqrt{51}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-51}) \), 4.0.2448.1, 4.0.272.1, \(\Q(i, \sqrt{51})\), 8.0.20123648.2, 8.0.1630015488.1, 8.0.1731891456.4

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

Sibling fields

Galois closure: data not computed
Degree 16 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 R ${\href{/LocalNumberField/5.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$2.8.12.14$x^{8} + 12 x^{4} + 144$$4$$2$$12$$D_4$$[2, 2]^{2}$
2.8.12.14$x^{8} + 12 x^{4} + 144$$4$$2$$12$$D_4$$[2, 2]^{2}$
3Data not computed
$17$17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$