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Properties

Label 16.0.37909967031...8041.4
Degree $16$
Signature $[0, 8]$
Discriminant $3^{12}\cdot 61^{10}$
Root discriminant $29.76$
Ramified primes $3, 61$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $C_2^2.D_4$ (as 16T33)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2352, -9072, 8184, 1566, 6685, 4294, -5305, 1038, -1226, -682, 1042, -193, -83, 57, -4, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 - 4*x^14 + 57*x^13 - 83*x^12 - 193*x^11 + 1042*x^10 - 682*x^9 - 1226*x^8 + 1038*x^7 - 5305*x^6 + 4294*x^5 + 6685*x^4 + 1566*x^3 + 8184*x^2 - 9072*x + 2352)
 
gp: K = bnfinit(x^16 - 5*x^15 - 4*x^14 + 57*x^13 - 83*x^12 - 193*x^11 + 1042*x^10 - 682*x^9 - 1226*x^8 + 1038*x^7 - 5305*x^6 + 4294*x^5 + 6685*x^4 + 1566*x^3 + 8184*x^2 - 9072*x + 2352, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} - 4 x^{14} + 57 x^{13} - 83 x^{12} - 193 x^{11} + 1042 x^{10} - 682 x^{9} - 1226 x^{8} + 1038 x^{7} - 5305 x^{6} + 4294 x^{5} + 6685 x^{4} + 1566 x^{3} + 8184 x^{2} - 9072 x + 2352 \)

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:  \(379099670317033992358041=3^{12}\cdot 61^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.76$
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:  $3, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4}$, $\frac{1}{6} a^{11} + \frac{1}{6} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{6} a^{6} - \frac{1}{2} a^{5} - \frac{1}{3} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{12} a^{12} - \frac{1}{12} a^{10} - \frac{1}{2} a^{9} + \frac{1}{4} a^{7} + \frac{5}{12} a^{6} - \frac{5}{12} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{372} a^{13} + \frac{1}{62} a^{12} + \frac{23}{372} a^{11} + \frac{1}{31} a^{10} + \frac{11}{31} a^{9} + \frac{21}{124} a^{8} + \frac{35}{372} a^{7} - \frac{7}{62} a^{6} - \frac{17}{372} a^{5} - \frac{41}{124} a^{4} + \frac{55}{124} a^{3} + \frac{15}{62} a^{2} + \frac{5}{31} a + \frac{5}{31}$, $\frac{1}{372} a^{14} - \frac{13}{372} a^{12} - \frac{1}{186} a^{11} + \frac{5}{31} a^{10} + \frac{139}{372} a^{9} + \frac{51}{124} a^{8} - \frac{32}{93} a^{7} - \frac{13}{372} a^{6} - \frac{7}{124} a^{5} - \frac{89}{372} a^{4} - \frac{13}{31} a^{3} - \frac{9}{31} a^{2} + \frac{6}{31} a + \frac{1}{31}$, $\frac{1}{3736636874735410662601568760} a^{15} + \frac{4466083044434163884131}{40178891126187211425823320} a^{14} + \frac{63775350119561147339315}{373663687473541066260156876} a^{13} + \frac{106823536878295018000097677}{3736636874735410662601568760} a^{12} + \frac{81189966224160975301247041}{1245545624911803554200522920} a^{11} + \frac{284929608890614633902837611}{3736636874735410662601568760} a^{10} - \frac{29675722706881597014086101}{373663687473541066260156876} a^{9} + \frac{11810074376952357186858739}{51897734371325148091688455} a^{8} + \frac{190821831098245344275574071}{467079609341926332825196095} a^{7} - \frac{705632769625960935893173619}{1868318437367705331300784380} a^{6} + \frac{523274215682353878145245827}{1245545624911803554200522920} a^{5} + \frac{109254363689558044450836899}{467079609341926332825196095} a^{4} - \frac{3372596382338575558621931}{9365004698584989129327240} a^{3} - \frac{12731410000519693586718011}{622772812455901777100261460} a^{2} - \frac{14562716634764094006728471}{155693203113975444275065365} a - \frac{1595321011672219448887258}{22241886159139349182152195}$

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:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{10529781561404857}{28588712573181234120} a^{15} + \frac{49534279694231279}{28588712573181234120} a^{14} + \frac{5200008647313697}{2858871257318123412} a^{13} - \frac{558687838143420169}{28588712573181234120} a^{12} + \frac{710995992697356389}{28588712573181234120} a^{11} + \frac{1987544435756533003}{28588712573181234120} a^{10} - \frac{492728200188417019}{1429435628659061706} a^{9} + \frac{2405770166797093247}{14294356286590617060} a^{8} + \frac{4675784197902868627}{14294356286590617060} a^{7} - \frac{1267644837950298757}{14294356286590617060} a^{6} + \frac{56224423413290179813}{28588712573181234120} a^{5} - \frac{3914328568248514988}{3573589071647654265} a^{4} - \frac{21996974887082272539}{9529570857727078040} a^{3} - \frac{4361115445948649269}{2382392714431769510} a^{2} - \frac{8797711947636960061}{2382392714431769510} a + \frac{2803846593718697217}{1191196357215884755} \) (order $6$)
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:  \( 505125.641002 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2.D_4$ (as 16T33):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $C_2^2.D_4$
Character table for $C_2^2.D_4$

Intermediate fields

\(\Q(\sqrt{-183}) \), \(\Q(\sqrt{61}) \), \(\Q(\sqrt{-3}) \), 4.0.549.1 x2, \(\Q(\sqrt{-3}, \sqrt{61})\), 4.2.11163.1 x2, 8.0.10093618089.2 x2, 8.0.1121513121.2, 8.0.615710703429.1 x2

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 8 siblings: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$3$3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
61Data not computed