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Properties

Label 16.0.68372792983...489.23
Degree $16$
Signature $[0, 8]$
Discriminant $17^{14}\cdot 67^{8}$
Root discriminant $97.65$
Ramified primes $17, 67$
Class number $1280$ (GRH)
Class group $[2, 4, 4, 40]$ (GRH)
Galois group $C_2^3.C_4$ (as 16T41)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![149470784, -10625712, -24662408, -3001184, 2750234, -1660883, 173528, 122499, -1277, 2832, 4469, -153, 196, -19, 25, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 25*x^14 - 19*x^13 + 196*x^12 - 153*x^11 + 4469*x^10 + 2832*x^9 - 1277*x^8 + 122499*x^7 + 173528*x^6 - 1660883*x^5 + 2750234*x^4 - 3001184*x^3 - 24662408*x^2 - 10625712*x + 149470784)
 
gp: K = bnfinit(x^16 - 4*x^15 + 25*x^14 - 19*x^13 + 196*x^12 - 153*x^11 + 4469*x^10 + 2832*x^9 - 1277*x^8 + 122499*x^7 + 173528*x^6 - 1660883*x^5 + 2750234*x^4 - 3001184*x^3 - 24662408*x^2 - 10625712*x + 149470784, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 25 x^{14} - 19 x^{13} + 196 x^{12} - 153 x^{11} + 4469 x^{10} + 2832 x^{9} - 1277 x^{8} + 122499 x^{7} + 173528 x^{6} - 1660883 x^{5} + 2750234 x^{4} - 3001184 x^{3} - 24662408 x^{2} - 10625712 x + 149470784 \)

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:  \(68372792983010839504523425519489=17^{14}\cdot 67^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $97.65$
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:  $17, 67$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{3}{8} a^{3}$, $\frac{1}{8} a^{10} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} + \frac{1}{8} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{11} - \frac{1}{4} a^{6} + \frac{1}{8} a^{5} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{8} a^{12} - \frac{1}{4} a^{7} + \frac{1}{8} a^{6} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3}$, $\frac{1}{16} a^{13} - \frac{1}{16} a^{10} + \frac{3}{16} a^{7} - \frac{1}{4} a^{6} - \frac{3}{16} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{904384} a^{14} + \frac{23141}{904384} a^{13} - \frac{391}{17392} a^{12} + \frac{3591}{904384} a^{11} - \frac{41285}{904384} a^{10} - \frac{6791}{226096} a^{9} + \frac{4331}{904384} a^{8} + \frac{79531}{904384} a^{7} + \frac{177}{1087} a^{6} - \frac{5747}{904384} a^{5} - \frac{168195}{904384} a^{4} + \frac{12137}{28262} a^{3} - \frac{21643}{113048} a^{2} + \frac{38043}{113048} a + \frac{633}{28262}$, $\frac{1}{37245962058195973938063562324863566107785728} a^{15} - \frac{3361275754868249586205147554769042467}{18622981029097986969031781162431783053892864} a^{14} - \frac{957072791501969865274497742168353331291659}{37245962058195973938063562324863566107785728} a^{13} - \frac{346763810585625240219728063448007066425325}{37245962058195973938063562324863566107785728} a^{12} - \frac{444313678173467492804152239404364553851697}{18622981029097986969031781162431783053892864} a^{11} + \frac{877235714229267366542225543135319544731}{37245962058195973938063562324863566107785728} a^{10} - \frac{1192076497513136815663844037032936837919057}{37245962058195973938063562324863566107785728} a^{9} + \frac{38123641558883015082595763144730531639577}{18622981029097986969031781162431783053892864} a^{8} + \frac{5380962655803638686251139860277858016376463}{37245962058195973938063562324863566107785728} a^{7} + \frac{8499100136820336178275541292683221028518997}{37245962058195973938063562324863566107785728} a^{6} - \frac{4182838018175884260617454074025078445341577}{18622981029097986969031781162431783053892864} a^{5} - \frac{4474507370264839392830349553534842694989919}{37245962058195973938063562324863566107785728} a^{4} + \frac{929733712197160534956487040321791783072797}{4655745257274496742257945290607945763473216} a^{3} + \frac{640766678797278582173553026566741968358501}{2327872628637248371128972645303972881736608} a^{2} + \frac{1096909475547969955597095520145597073890283}{4655745257274496742257945290607945763473216} a + \frac{384753597476700411524291300190104072100385}{1163936314318624185564486322651986440868304}$

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

Class group and class number

$C_{2}\times C_{4}\times C_{4}\times C_{40}$, which has order $1280$ (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:  \( 86877550.428 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_4$ (as 16T41):

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

\(\Q(\sqrt{17}) \), 4.2.329171.1, 4.0.22054457.2, 4.2.19363.1, 8.0.8268784250602433.1 x2, 8.0.486399073564849.3

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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$17$17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$
$67$67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$