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Properties

Label 16.0.83558672988...3376.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{62}\cdot 7^{6}\cdot 17^{2}\cdot 73^{2}$
Root discriminant $74.15$
Ramified primes $2, 7, 17, 73$
Class number $15008$ (GRH)
Class group $[2, 2, 2, 1876]$ (GRH)
Galois group 16T1086

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6575522, -5546400, 7393712, -2790640, 3884800, -721360, 1196888, -180656, 246024, -44128, 33752, -6304, 2668, -400, 96, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 96*x^14 - 400*x^13 + 2668*x^12 - 6304*x^11 + 33752*x^10 - 44128*x^9 + 246024*x^8 - 180656*x^7 + 1196888*x^6 - 721360*x^5 + 3884800*x^4 - 2790640*x^3 + 7393712*x^2 - 5546400*x + 6575522)
 
gp: K = bnfinit(x^16 - 8*x^15 + 96*x^14 - 400*x^13 + 2668*x^12 - 6304*x^11 + 33752*x^10 - 44128*x^9 + 246024*x^8 - 180656*x^7 + 1196888*x^6 - 721360*x^5 + 3884800*x^4 - 2790640*x^3 + 7393712*x^2 - 5546400*x + 6575522, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 96 x^{14} - 400 x^{13} + 2668 x^{12} - 6304 x^{11} + 33752 x^{10} - 44128 x^{9} + 246024 x^{8} - 180656 x^{7} + 1196888 x^{6} - 721360 x^{5} + 3884800 x^{4} - 2790640 x^{3} + 7393712 x^{2} - 5546400 x + 6575522 \)

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:  \(835586729888343128721772773376=2^{62}\cdot 7^{6}\cdot 17^{2}\cdot 73^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $74.15$
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:  $2, 7, 17, 73$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3689} a^{14} + \frac{1129}{3689} a^{13} + \frac{44}{119} a^{12} - \frac{180}{527} a^{11} - \frac{666}{3689} a^{10} + \frac{60}{527} a^{9} - \frac{718}{3689} a^{8} + \frac{1388}{3689} a^{7} + \frac{226}{527} a^{6} - \frac{143}{3689} a^{5} - \frac{1472}{3689} a^{4} - \frac{827}{3689} a^{3} - \frac{1040}{3689} a^{2} - \frac{1742}{3689} a - \frac{120}{3689}$, $\frac{1}{44713901801044955291996230575241922553914681} a^{15} + \frac{1886690256776177759703220883634568775847}{44713901801044955291996230575241922553914681} a^{14} - \frac{19091569775659995515211228830609470022814178}{44713901801044955291996230575241922553914681} a^{13} - \frac{21694107145911686274298515376117955346105850}{44713901801044955291996230575241922553914681} a^{12} - \frac{13893633862909439302583657097618321901892846}{44713901801044955291996230575241922553914681} a^{11} - \frac{16328014650230601789284450834423558020832046}{44713901801044955291996230575241922553914681} a^{10} + \frac{10613779740934688380440788867487716652907006}{44713901801044955291996230575241922553914681} a^{9} - \frac{8026434323275992247917258295631808464018315}{44713901801044955291996230575241922553914681} a^{8} + \frac{17583279007383275617273371647976110995522783}{44713901801044955291996230575241922553914681} a^{7} - \frac{20200244166834739109506905271392291666319249}{44713901801044955291996230575241922553914681} a^{6} + \frac{11534774512310214338517750309002067142971652}{44713901801044955291996230575241922553914681} a^{5} - \frac{51515132782971306998398003846887352422961}{44713901801044955291996230575241922553914681} a^{4} - \frac{20348982029171556304807465078048642956603563}{44713901801044955291996230575241922553914681} a^{3} - \frac{11167661110587684413620419052651181651163323}{44713901801044955291996230575241922553914681} a^{2} + \frac{2200736986544141966793059966860692648156933}{6387700257292136470285175796463131793416383} a - \frac{2165442399312804279885267536767188984901133}{44713901801044955291996230575241922553914681}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{1876}$, which has order $15008$ (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:  \( -1 \) (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:  \( 20726.065235 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1086:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 97 conjugacy class representatives for t16n1086 are not computed
Character table for t16n1086 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.7168.1, \(\Q(\zeta_{16})^+\), 4.4.14336.1, 8.8.3288334336.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$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}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
$17$$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
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}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
$73$73.4.2.2$x^{4} - 73 x^{2} + 58619$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
73.4.0.1$x^{4} - x + 13$$1$$4$$0$$C_4$$[\ ]^{4}$
73.4.0.1$x^{4} - x + 13$$1$$4$$0$$C_4$$[\ ]^{4}$
73.4.0.1$x^{4} - x + 13$$1$$4$$0$$C_4$$[\ ]^{4}$