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Properties

Label 16.8.12383987090...8352.2
Degree $16$
Signature $[8, 4]$
Discriminant $2^{8}\cdot 13^{2}\cdot 17^{15}$
Root discriminant $27.75$
Ramified primes $2, 13, 17$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $(C_2\times C_8).D_4$ (as 16T306)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-67, 724, -1362, -513, 4509, -7967, 7914, -4313, 390, 1673, -1698, 853, -242, 28, 8, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 8*x^14 + 28*x^13 - 242*x^12 + 853*x^11 - 1698*x^10 + 1673*x^9 + 390*x^8 - 4313*x^7 + 7914*x^6 - 7967*x^5 + 4509*x^4 - 513*x^3 - 1362*x^2 + 724*x - 67)
 
gp: K = bnfinit(x^16 - 5*x^15 + 8*x^14 + 28*x^13 - 242*x^12 + 853*x^11 - 1698*x^10 + 1673*x^9 + 390*x^8 - 4313*x^7 + 7914*x^6 - 7967*x^5 + 4509*x^4 - 513*x^3 - 1362*x^2 + 724*x - 67, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 8 x^{14} + 28 x^{13} - 242 x^{12} + 853 x^{11} - 1698 x^{10} + 1673 x^{9} + 390 x^{8} - 4313 x^{7} + 7914 x^{6} - 7967 x^{5} + 4509 x^{4} - 513 x^{3} - 1362 x^{2} + 724 x - 67 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(123839870900520670468352=2^{8}\cdot 13^{2}\cdot 17^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.75$
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:  $2, 13, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{103} a^{13} + \frac{9}{103} a^{12} - \frac{13}{103} a^{11} + \frac{48}{103} a^{10} - \frac{2}{103} a^{9} + \frac{3}{103} a^{8} + \frac{47}{103} a^{7} - \frac{27}{103} a^{6} + \frac{47}{103} a^{5} - \frac{8}{103} a^{4} - \frac{9}{103} a^{3} - \frac{29}{103} a^{2} + \frac{24}{103} a + \frac{43}{103}$, $\frac{1}{103} a^{14} + \frac{9}{103} a^{12} - \frac{41}{103} a^{11} - \frac{22}{103} a^{10} + \frac{21}{103} a^{9} + \frac{20}{103} a^{8} - \frac{38}{103} a^{7} - \frac{19}{103} a^{6} - \frac{19}{103} a^{5} - \frac{40}{103} a^{4} - \frac{51}{103} a^{3} - \frac{24}{103} a^{2} + \frac{33}{103} a + \frac{25}{103}$, $\frac{1}{9328705580796459804823} a^{15} + \frac{31544531043298409118}{9328705580796459804823} a^{14} + \frac{32498717971875798994}{9328705580796459804823} a^{13} - \frac{3480533072374968723462}{9328705580796459804823} a^{12} + \frac{4409054289543578851845}{9328705580796459804823} a^{11} - \frac{3532668283846115710608}{9328705580796459804823} a^{10} - \frac{4433696521715939827469}{9328705580796459804823} a^{9} + \frac{1232339136799894057561}{9328705580796459804823} a^{8} - \frac{3528609709151762974705}{9328705580796459804823} a^{7} + \frac{981253411228128698}{9328705580796459804823} a^{6} + \frac{3120645263519641189831}{9328705580796459804823} a^{5} + \frac{2710290734714445648843}{9328705580796459804823} a^{4} - \frac{2474372655293035051687}{9328705580796459804823} a^{3} + \frac{2586116489477747165075}{9328705580796459804823} a^{2} + \frac{264409492131079050617}{9328705580796459804823} a - \frac{1752292990506656726789}{9328705580796459804823}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $11$
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:  \( 303061.735576 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times C_8).D_4$ (as 16T306):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 26 conjugacy class representatives for $(C_2\times C_8).D_4$
Character table for $(C_2\times C_8).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, \(\Q(\zeta_{17})^+\)

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 16 sibling: 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 $16$ $16$ $16$ $16$ R R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$2$2.8.8.4$x^{8} + 2 x^{7} + 2 x^{6} + 8 x^{3} + 48$$2$$4$$8$$C_8$$[2]^{4}$
2.8.0.1$x^{8} + x^{4} + x^{3} + x + 1$$1$$8$$0$$C_8$$[\ ]^{8}$
$13$13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
17Data not computed