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Properties

Label 16.0.43690605516...1401.1
Degree $16$
Signature $[0, 8]$
Discriminant $29^{14}\cdot 59^{8}$
Root discriminant $146.23$
Ramified primes $29, 59$
Class number $42$ (GRH)
Class group $[42]$ (GRH)
Galois group $C_2^3.C_4$ (as 16T36)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![15280961, -27672790, 45995976, -1515286, -7791049, -5431518, 10535912, -6421886, 2156282, -468046, 67960, -6092, -16, 274, -36, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 36*x^14 + 274*x^13 - 16*x^12 - 6092*x^11 + 67960*x^10 - 468046*x^9 + 2156282*x^8 - 6421886*x^7 + 10535912*x^6 - 5431518*x^5 - 7791049*x^4 - 1515286*x^3 + 45995976*x^2 - 27672790*x + 15280961)
 
gp: K = bnfinit(x^16 - 4*x^15 - 36*x^14 + 274*x^13 - 16*x^12 - 6092*x^11 + 67960*x^10 - 468046*x^9 + 2156282*x^8 - 6421886*x^7 + 10535912*x^6 - 5431518*x^5 - 7791049*x^4 - 1515286*x^3 + 45995976*x^2 - 27672790*x + 15280961, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 36 x^{14} + 274 x^{13} - 16 x^{12} - 6092 x^{11} + 67960 x^{10} - 468046 x^{9} + 2156282 x^{8} - 6421886 x^{7} + 10535912 x^{6} - 5431518 x^{5} - 7791049 x^{4} - 1515286 x^{3} + 45995976 x^{2} - 27672790 x + 15280961 \)

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:  \(43690605516556003276152578767301401=29^{14}\cdot 59^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $146.23$
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:  $29, 59$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{116} a^{8} - \frac{1}{58} a^{7} - \frac{5}{29} a^{6} + \frac{5}{58} a^{5} - \frac{7}{58} a^{4} + \frac{13}{58} a^{3} + \frac{33}{116} a^{2} + \frac{2}{29} a + \frac{7}{116}$, $\frac{1}{116} a^{9} - \frac{6}{29} a^{7} + \frac{7}{29} a^{6} + \frac{3}{58} a^{5} - \frac{1}{58} a^{4} + \frac{27}{116} a^{3} + \frac{4}{29} a^{2} + \frac{23}{116} a + \frac{7}{58}$, $\frac{1}{116} a^{10} - \frac{5}{29} a^{7} - \frac{5}{58} a^{6} + \frac{3}{58} a^{5} - \frac{19}{116} a^{4} - \frac{14}{29} a^{3} + \frac{3}{116} a^{2} + \frac{8}{29} a - \frac{3}{58}$, $\frac{1}{116} a^{11} + \frac{2}{29} a^{7} + \frac{3}{29} a^{6} + \frac{7}{116} a^{5} + \frac{3}{29} a^{4} - \frac{57}{116} a^{3} - \frac{1}{29} a^{2} + \frac{19}{58} a - \frac{17}{58}$, $\frac{1}{1624} a^{12} - \frac{1}{406} a^{11} - \frac{3}{812} a^{10} - \frac{1}{232} a^{9} - \frac{1}{1624} a^{8} - \frac{59}{406} a^{7} + \frac{351}{1624} a^{6} - \frac{179}{812} a^{5} + \frac{33}{1624} a^{4} + \frac{659}{1624} a^{3} + \frac{613}{1624} a^{2} - \frac{495}{1624} a + \frac{185}{1624}$, $\frac{1}{1624} a^{13} + \frac{3}{812} a^{11} - \frac{3}{1624} a^{10} - \frac{1}{1624} a^{9} - \frac{1}{812} a^{8} + \frac{359}{1624} a^{7} + \frac{187}{812} a^{6} - \frac{111}{1624} a^{5} - \frac{51}{232} a^{4} - \frac{279}{1624} a^{3} + \frac{487}{1624} a^{2} - \frac{535}{1624} a + \frac{27}{812}$, $\frac{1}{1830248} a^{14} - \frac{177}{915124} a^{13} - \frac{5}{261464} a^{12} + \frac{2041}{1830248} a^{11} - \frac{6841}{1830248} a^{10} + \frac{1927}{1830248} a^{9} - \frac{3}{39788} a^{8} - \frac{20299}{457562} a^{7} + \frac{193139}{915124} a^{6} + \frac{377995}{1830248} a^{5} + \frac{51873}{915124} a^{4} + \frac{119263}{915124} a^{3} + \frac{79372}{228781} a^{2} + \frac{322187}{1830248} a + \frac{440213}{1830248}$, $\frac{1}{27967071136078101888825556495305631635896} a^{15} - \frac{1285777475015020786751271103039981}{13983535568039050944412778247652815817948} a^{14} - \frac{45806369200448470365248125459105433}{303989903653022846617669092340278604738} a^{13} + \frac{6479328798245215996113935559277044013}{27967071136078101888825556495305631635896} a^{12} + \frac{79027217701378357668935046312576893885}{27967071136078101888825556495305631635896} a^{11} - \frac{1294883525387314152500342030441191708}{499411984572823248014742080273314850641} a^{10} - \frac{62453150793060357536871170699546338429}{27967071136078101888825556495305631635896} a^{9} + \frac{5971785234914794173722685071277610813}{6991767784019525472206389123826407908974} a^{8} - \frac{2887563526071543757340022028448612426297}{27967071136078101888825556495305631635896} a^{7} + \frac{315640086658283474581774257749680050841}{27967071136078101888825556495305631635896} a^{6} + \frac{401397247278547326435218346634447659685}{27967071136078101888825556495305631635896} a^{5} - \frac{102564150958715524365889073313367341161}{3995295876582585984117936642186518805128} a^{4} - \frac{1914615916194432059904844506036236730411}{3995295876582585984117936642186518805128} a^{3} - \frac{1114457044509322954341985490677969513503}{3495883892009762736103194561913203954487} a^{2} + \frac{1956450426927469912328883975581755296495}{6991767784019525472206389123826407908974} a + \frac{1957546198682166435209274204317155607813}{6991767784019525472206389123826407908974}$

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

Class group and class number

$C_{42}$, which has order $42$ (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:  \( 2123693498.09 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_4$ (as 16T36):

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 $C_2^3.C_4$
Character table for $C_2^3.C_4$

Intermediate fields

\(\Q(\sqrt{29}) \), \(\Q(\sqrt{-59}) \), \(\Q(\sqrt{-1711}) \), 4.4.84898109.1, 4.0.24389.1, \(\Q(\sqrt{29}, \sqrt{-59})\), 8.4.209022978441500549.1 x2, 8.0.60046819431629.1 x2, 8.0.7207688911775881.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ R

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
$29$29.8.7.2$x^{8} - 116$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
29.8.7.2$x^{8} - 116$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$59$59.2.1.2$x^{2} + 177$$2$$1$$1$$C_2$$[\ ]_{2}$
59.2.1.2$x^{2} + 177$$2$$1$$1$$C_2$$[\ ]_{2}$
59.2.1.2$x^{2} + 177$$2$$1$$1$$C_2$$[\ ]_{2}$
59.2.1.2$x^{2} + 177$$2$$1$$1$$C_2$$[\ ]_{2}$
59.4.2.1$x^{4} + 177 x^{2} + 13924$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
59.4.2.1$x^{4} + 177 x^{2} + 13924$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$