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Properties

Label 16.0.11033416203...5625.2
Degree $16$
Signature $[0, 8]$
Discriminant $5^{6}\cdot 29^{8}\cdot 109^{4}$
Root discriminant $31.82$
Ramified primes $5, 29, 109$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $D_4^2.C_2$ (as 16T390)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![169, -1105, 3628, -7225, 9757, -9787, 7916, -5822, 4427, -3172, 1865, -888, 345, -110, 32, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 32*x^14 - 110*x^13 + 345*x^12 - 888*x^11 + 1865*x^10 - 3172*x^9 + 4427*x^8 - 5822*x^7 + 7916*x^6 - 9787*x^5 + 9757*x^4 - 7225*x^3 + 3628*x^2 - 1105*x + 169)
 
gp: K = bnfinit(x^16 - 6*x^15 + 32*x^14 - 110*x^13 + 345*x^12 - 888*x^11 + 1865*x^10 - 3172*x^9 + 4427*x^8 - 5822*x^7 + 7916*x^6 - 9787*x^5 + 9757*x^4 - 7225*x^3 + 3628*x^2 - 1105*x + 169, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 32 x^{14} - 110 x^{13} + 345 x^{12} - 888 x^{11} + 1865 x^{10} - 3172 x^{9} + 4427 x^{8} - 5822 x^{7} + 7916 x^{6} - 9787 x^{5} + 9757 x^{4} - 7225 x^{3} + 3628 x^{2} - 1105 x + 169 \)

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:  \(1103341620319083198765625=5^{6}\cdot 29^{8}\cdot 109^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.82$
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:  $5, 29, 109$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{11} + \frac{1}{3} a^{8} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{15} a^{13} - \frac{1}{15} a^{12} + \frac{1}{15} a^{11} + \frac{1}{5} a^{10} + \frac{7}{15} a^{9} + \frac{1}{15} a^{8} - \frac{1}{15} a^{7} - \frac{4}{15} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{5} a^{3} - \frac{2}{15} a^{2} - \frac{1}{15} a - \frac{2}{15}$, $\frac{1}{15} a^{14} + \frac{4}{15} a^{11} - \frac{1}{3} a^{10} - \frac{7}{15} a^{9} - \frac{1}{3} a^{7} + \frac{2}{5} a^{6} + \frac{1}{3} a^{5} - \frac{2}{15} a^{4} + \frac{1}{15} a^{3} - \frac{1}{5} a^{2} - \frac{1}{5} a - \frac{2}{15}$, $\frac{1}{474202587311769174495} a^{15} - \frac{620596205735437291}{94840517462353834899} a^{14} - \frac{1812783489906201352}{474202587311769174495} a^{13} - \frac{8381312984839901879}{474202587311769174495} a^{12} - \frac{38802113212118642934}{158067529103923058165} a^{11} - \frac{18079866842649425616}{158067529103923058165} a^{10} - \frac{158032201587343839464}{474202587311769174495} a^{9} - \frac{4858074505513912189}{36477122100905321115} a^{8} + \frac{101441734296900255668}{474202587311769174495} a^{7} - \frac{168640294734907708607}{474202587311769174495} a^{6} - \frac{179892641923312145257}{474202587311769174495} a^{5} - \frac{42115756884292144308}{158067529103923058165} a^{4} - \frac{17173611798031133204}{474202587311769174495} a^{3} - \frac{31686504376410526117}{67743226758824167785} a^{2} - \frac{21264717782150372860}{94840517462353834899} a + \frac{4852139133128129153}{36477122100905321115}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 152645.794684 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_4^2.C_2$ (as 16T390):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 20 conjugacy class representatives for $D_4^2.C_2$
Character table for $D_4^2.C_2$

Intermediate fields

\(\Q(\sqrt{29}) \), 4.4.91669.1, 4.4.458345.2, 4.4.4205.1, 8.0.42016027805.1, 8.0.1050400695125.1, 8.8.210080139025.2

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

Sibling fields

Degree 8 siblings: data not computed
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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$5$5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$29$29.8.4.1$x^{8} + 31958 x^{4} - 24389 x^{2} + 255328441$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
29.8.4.1$x^{8} + 31958 x^{4} - 24389 x^{2} + 255328441$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$109$109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.2.1.2$x^{2} + 654$$2$$1$$1$$C_2$$[\ ]_{2}$
109.2.1.2$x^{2} + 654$$2$$1$$1$$C_2$$[\ ]_{2}$
109.2.1.2$x^{2} + 654$$2$$1$$1$$C_2$$[\ ]_{2}$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.2.1.2$x^{2} + 654$$2$$1$$1$$C_2$$[\ ]_{2}$