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Properties

Label 16.8.23564170227...0000.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{44}\cdot 5^{8}\cdot 7^{8}\cdot 29^{6}$
Root discriminant $140.69$
Ramified primes $2, 5, 7, 29$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $D_4.C_2^3.C_2$ (as 16T264)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![81342361, 0, -47271276, 0, -85010586, 0, -9553232, 0, 661003, 0, 53664, 0, -1154, 0, -52, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 52*x^14 - 1154*x^12 + 53664*x^10 + 661003*x^8 - 9553232*x^6 - 85010586*x^4 - 47271276*x^2 + 81342361)
 
gp: K = bnfinit(x^16 - 52*x^14 - 1154*x^12 + 53664*x^10 + 661003*x^8 - 9553232*x^6 - 85010586*x^4 - 47271276*x^2 + 81342361, 1)
 

Normalized defining polynomial

\( x^{16} - 52 x^{14} - 1154 x^{12} + 53664 x^{10} + 661003 x^{8} - 9553232 x^{6} - 85010586 x^{4} - 47271276 x^{2} + 81342361 \)

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:  \(23564170227158193941354905600000000=2^{44}\cdot 5^{8}\cdot 7^{8}\cdot 29^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $140.69$
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:  $2, 5, 7, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{24} a^{8} - \frac{1}{12} a^{6} + \frac{3}{8} a^{4} + \frac{1}{4} a^{2} - \frac{7}{24}$, $\frac{1}{24} a^{9} - \frac{1}{12} a^{7} + \frac{3}{8} a^{5} + \frac{1}{4} a^{3} - \frac{7}{24} a$, $\frac{1}{24} a^{10} + \frac{5}{24} a^{6} + \frac{5}{24} a^{2} + \frac{5}{12}$, $\frac{1}{24} a^{11} + \frac{5}{24} a^{7} + \frac{5}{24} a^{3} + \frac{5}{12} a$, $\frac{1}{24} a^{12} + \frac{5}{12} a^{6} + \frac{1}{3} a^{4} + \frac{1}{6} a^{2} + \frac{11}{24}$, $\frac{1}{24} a^{13} + \frac{5}{12} a^{7} + \frac{1}{3} a^{5} + \frac{1}{6} a^{3} + \frac{11}{24} a$, $\frac{1}{2148523719374675351414757264} a^{14} - \frac{1}{48} a^{13} + \frac{2142342325242162059816353}{537130929843668837853689316} a^{12} - \frac{1}{48} a^{11} + \frac{30223843542720327798455381}{2148523719374675351414757264} a^{10} + \frac{41891216492553259868774389}{2148523719374675351414757264} a^{8} + \frac{3}{16} a^{7} + \frac{727385505591714801989227075}{2148523719374675351414757264} a^{6} + \frac{1}{3} a^{5} + \frac{61143557695685869016705837}{716174573124891783804919088} a^{4} - \frac{3}{16} a^{3} + \frac{60739469196720643529866103}{1074261859687337675707378632} a^{2} - \frac{7}{16} a - \frac{16489978167989567419808059}{74087024806023287979819216}$, $\frac{1}{668190876725524034289989509104} a^{15} + \frac{2380897642777172682093059891}{668190876725524034289989509104} a^{13} - \frac{1}{48} a^{12} - \frac{4483359615669366353125420331}{334095438362762017144994754552} a^{11} - \frac{1}{48} a^{10} - \frac{3775793377288367936011438711}{222730292241841344763329836368} a^{9} - \frac{105786807257559196679840737537}{334095438362762017144994754552} a^{7} + \frac{3}{16} a^{6} + \frac{220854721017194338491940811501}{668190876725524034289989509104} a^{5} + \frac{1}{3} a^{4} - \frac{94995456554756248749531084169}{668190876725524034289989509104} a^{3} - \frac{3}{16} a^{2} - \frac{255973478279135152892424533}{11520532357336621280861888088} a - \frac{7}{16}$

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

Class group and class number

$C_{4}$, 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:  \( 52372122680.3 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_4.C_2^3.C_2$ (as 16T264):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 32 conjugacy class representatives for $D_4.C_2^3.C_2$
Character table for $D_4.C_2^3.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{2}) \), 4.4.2273600.4, 4.4.568400.1, \(\Q(\sqrt{2}, \sqrt{5})\), 8.8.82708111360000.2

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R R ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$7$7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$29$29.4.2.2$x^{4} - 29 x^{2} + 2523$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.8.4.1$x^{8} + 31958 x^{4} - 24389 x^{2} + 255328441$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$