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Properties

Label 16.0.13345917567...7041.1
Degree $16$
Signature $[0, 8]$
Discriminant $11^{8}\cdot 53^{8}$
Root discriminant $24.15$
Ramified primes $11, 53$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $D_{8}$ (as 16T13)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![18225, 810, -7551, -168, 1091, -3224, 809, 1736, -532, -312, 165, -18, -17, 17, -5, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 5*x^14 + 17*x^13 - 17*x^12 - 18*x^11 + 165*x^10 - 312*x^9 - 532*x^8 + 1736*x^7 + 809*x^6 - 3224*x^5 + 1091*x^4 - 168*x^3 - 7551*x^2 + 810*x + 18225)
 
gp: K = bnfinit(x^16 - 2*x^15 - 5*x^14 + 17*x^13 - 17*x^12 - 18*x^11 + 165*x^10 - 312*x^9 - 532*x^8 + 1736*x^7 + 809*x^6 - 3224*x^5 + 1091*x^4 - 168*x^3 - 7551*x^2 + 810*x + 18225, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 5 x^{14} + 17 x^{13} - 17 x^{12} - 18 x^{11} + 165 x^{10} - 312 x^{9} - 532 x^{8} + 1736 x^{7} + 809 x^{6} - 3224 x^{5} + 1091 x^{4} - 168 x^{3} - 7551 x^{2} + 810 x + 18225 \)

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:  \(13345917567985773647041=11^{8}\cdot 53^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.15$
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:  $11, 53$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{4}$, $\frac{1}{9} a^{13} - \frac{1}{9} a^{12} - \frac{1}{9} a^{10} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{4}{9} a^{5} + \frac{4}{9} a^{4} - \frac{1}{3} a^{3} + \frac{4}{9} a^{2} - \frac{1}{3} a$, $\frac{1}{315} a^{14} - \frac{2}{105} a^{13} - \frac{46}{315} a^{12} - \frac{4}{315} a^{11} + \frac{2}{45} a^{10} - \frac{8}{105} a^{9} + \frac{52}{105} a^{8} + \frac{11}{35} a^{7} - \frac{43}{315} a^{6} - \frac{4}{105} a^{5} + \frac{82}{315} a^{4} - \frac{11}{45} a^{3} + \frac{94}{315} a^{2} + \frac{52}{105} a + \frac{3}{7}$, $\frac{1}{5839755134352304093672275} a^{15} + \frac{940522111528471781779}{834250733478900584810325} a^{14} + \frac{45517165261460994281102}{1167951026870460818734455} a^{13} - \frac{55703874540324843513094}{834250733478900584810325} a^{12} - \frac{883247673811930408179782}{5839755134352304093672275} a^{11} + \frac{79031745224564689616033}{648861681594700454852475} a^{10} - \frac{2254574253072278158097}{77863401791364054582297} a^{9} + \frac{374340865711587381466771}{1946585044784101364557425} a^{8} - \frac{2850530445523588570213942}{5839755134352304093672275} a^{7} + \frac{638603735425645983004826}{5839755134352304093672275} a^{6} - \frac{2145974918149963845042886}{5839755134352304093672275} a^{5} + \frac{163052098137636704987146}{5839755134352304093672275} a^{4} + \frac{173277610291963252745246}{5839755134352304093672275} a^{3} - \frac{457492400899308989545721}{1946585044784101364557425} a^{2} + \frac{24348694826200051223518}{92694525942100064978925} a + \frac{6236500119336681808441}{14419148479882232330055}$

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:  $7$
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:  \( 44918.3429831 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_8$ (as 16T13):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 7 conjugacy class representatives for $D_{8}$
Character table for $D_{8}$

Intermediate fields

\(\Q(\sqrt{-583}) \), \(\Q(\sqrt{53}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{-11}, \sqrt{53})\), 4.2.30899.1 x2, 4.0.6413.1 x2, 8.0.115524532321.1, 8.2.10502230211.1 x4, 8.0.2179708157.1 x4

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$11$11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$53$53.4.2.1$x^{4} + 477 x^{2} + 70225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
53.4.2.1$x^{4} + 477 x^{2} + 70225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
53.4.2.1$x^{4} + 477 x^{2} + 70225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
53.4.2.1$x^{4} + 477 x^{2} + 70225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$