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Properties

Label 16.0.92420260574...2593.1
Degree $16$
Signature $[0, 8]$
Discriminant $17^{7}\cdot 83^{8}$
Root discriminant $31.47$
Ramified primes $17, 83$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_4.D_4:C_4$ (as 16T289)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3949, 2330, 5666, -5376, -461, -2639, 2612, 1049, -1876, 479, 492, -315, -32, 61, -7, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 7*x^14 + 61*x^13 - 32*x^12 - 315*x^11 + 492*x^10 + 479*x^9 - 1876*x^8 + 1049*x^7 + 2612*x^6 - 2639*x^5 - 461*x^4 - 5376*x^3 + 5666*x^2 + 2330*x + 3949)
 
gp: K = bnfinit(x^16 - 4*x^15 - 7*x^14 + 61*x^13 - 32*x^12 - 315*x^11 + 492*x^10 + 479*x^9 - 1876*x^8 + 1049*x^7 + 2612*x^6 - 2639*x^5 - 461*x^4 - 5376*x^3 + 5666*x^2 + 2330*x + 3949, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 7 x^{14} + 61 x^{13} - 32 x^{12} - 315 x^{11} + 492 x^{10} + 479 x^{9} - 1876 x^{8} + 1049 x^{7} + 2612 x^{6} - 2639 x^{5} - 461 x^{4} - 5376 x^{3} + 5666 x^{2} + 2330 x + 3949 \)

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:  \(924202605744142035432593=17^{7}\cdot 83^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.47$
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:  $17, 83$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{121} a^{14} - \frac{16}{121} a^{13} + \frac{5}{11} a^{12} + \frac{29}{121} a^{11} - \frac{28}{121} a^{10} + \frac{2}{121} a^{9} - \frac{6}{121} a^{8} + \frac{49}{121} a^{7} + \frac{10}{121} a^{6} + \frac{4}{121} a^{5} + \frac{54}{121} a^{4} - \frac{56}{121} a^{3} - \frac{3}{11} a^{2} + \frac{1}{121} a + \frac{2}{11}$, $\frac{1}{189773862245392752300773107} a^{15} + \frac{487805653430186404236389}{189773862245392752300773107} a^{14} + \frac{11529280911451671308951966}{189773862245392752300773107} a^{13} + \frac{90416235661550341091449644}{189773862245392752300773107} a^{12} - \frac{34541021128727740509698671}{189773862245392752300773107} a^{11} + \frac{44656517498460590230416802}{189773862245392752300773107} a^{10} - \frac{11136085251492859004539294}{27110551749341821757253301} a^{9} - \frac{38186039878731126186114986}{189773862245392752300773107} a^{8} - \frac{388144857966399873768437}{17252169295035704754615737} a^{7} + \frac{46398461510405788066242647}{189773862245392752300773107} a^{6} + \frac{22156011349898569612188359}{189773862245392752300773107} a^{5} + \frac{69103760006823765767592478}{189773862245392752300773107} a^{4} + \frac{45102228229169250827422550}{189773862245392752300773107} a^{3} + \frac{63917658460032539368018033}{189773862245392752300773107} a^{2} + \frac{1663836527937566359089202}{27110551749341821757253301} a + \frac{6017392040481336473383619}{17252169295035704754615737}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( 110152.755114 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4.D_4:C_4$ (as 16T289):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 44 conjugacy class representatives for $C_4.D_4:C_4$
Character table for $C_4.D_4:C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-83}) \), 4.0.117113.1, 8.0.233162731073.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ $16$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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$17.8.0.1$x^{8} + x^{2} - 3 x + 3$$1$$8$$0$$C_8$$[\ ]^{8}$
17.8.7.6$x^{8} + 37179$$8$$1$$7$$C_8$$[\ ]_{8}$
83Data not computed