↑

Properties

Label 16.0.11953891353...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 3^{4}\cdot 5^{8}\cdot 7^{8}$
Root discriminant $15.57$
Ramified primes $2, 3, 5, 7$
Class number $1$
Class group Trivial
Galois group $Q_8 : C_2^2$ (as 16T23)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, -28, 86, -150, 145, -26, -124, 138, 76, -378, 542, -478, 292, -126, 38, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 38*x^14 - 126*x^13 + 292*x^12 - 478*x^11 + 542*x^10 - 378*x^9 + 76*x^8 + 138*x^7 - 124*x^6 - 26*x^5 + 145*x^4 - 150*x^3 + 86*x^2 - 28*x + 4)
 
gp: K = bnfinit(x^16 - 8*x^15 + 38*x^14 - 126*x^13 + 292*x^12 - 478*x^11 + 542*x^10 - 378*x^9 + 76*x^8 + 138*x^7 - 124*x^6 - 26*x^5 + 145*x^4 - 150*x^3 + 86*x^2 - 28*x + 4, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 38 x^{14} - 126 x^{13} + 292 x^{12} - 478 x^{11} + 542 x^{10} - 378 x^{9} + 76 x^{8} + 138 x^{7} - 124 x^{6} - 26 x^{5} + 145 x^{4} - 150 x^{3} + 86 x^{2} - 28 x + 4 \)

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:  \(11953891353600000000=2^{16}\cdot 3^{4}\cdot 5^{8}\cdot 7^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.57$
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:  $2, 3, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5}$, $\frac{1}{16} a^{12} + \frac{1}{8} a^{11} + \frac{1}{16} a^{10} + \frac{1}{8} a^{9} + \frac{1}{8} a^{8} - \frac{3}{8} a^{7} - \frac{5}{16} a^{6} - \frac{1}{2} a^{5} + \frac{7}{16} a^{4} - \frac{1}{8} a^{3} + \frac{1}{8} a^{2} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{16} a^{13} - \frac{3}{16} a^{11} - \frac{1}{8} a^{9} - \frac{1}{8} a^{8} + \frac{7}{16} a^{7} + \frac{1}{8} a^{6} + \frac{7}{16} a^{5} + \frac{3}{8} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{368} a^{14} - \frac{7}{368} a^{13} + \frac{5}{184} a^{12} + \frac{31}{368} a^{11} + \frac{67}{368} a^{10} - \frac{25}{184} a^{9} - \frac{41}{368} a^{8} - \frac{13}{368} a^{7} - \frac{11}{46} a^{6} - \frac{137}{368} a^{5} - \frac{1}{16} a^{4} - \frac{29}{92} a^{3} + \frac{3}{184} a^{2} - \frac{1}{46} a + \frac{11}{92}$, $\frac{1}{21712} a^{15} + \frac{11}{10856} a^{14} - \frac{423}{21712} a^{13} + \frac{459}{21712} a^{12} + \frac{31}{236} a^{11} + \frac{1479}{21712} a^{10} + \frac{2373}{21712} a^{9} - \frac{1705}{10856} a^{8} + \frac{8873}{21712} a^{7} - \frac{6783}{21712} a^{6} - \frac{3631}{10856} a^{5} + \frac{6255}{21712} a^{4} + \frac{159}{472} a^{3} + \frac{957}{10856} a^{2} - \frac{875}{5428} a - \frac{2303}{5428}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( -\frac{1774}{1357} a^{15} + \frac{13305}{1357} a^{14} - \frac{61025}{1357} a^{13} + \frac{194870}{1357} a^{12} - \frac{428656}{1357} a^{11} + \frac{1315963}{2714} a^{10} - \frac{1362355}{2714} a^{9} + \frac{394485}{1357} a^{8} + \frac{10882}{1357} a^{7} - \frac{228456}{1357} a^{6} + \frac{127755}{1357} a^{5} + \frac{177405}{2714} a^{4} - \frac{433221}{2714} a^{3} + \frac{178076}{1357} a^{2} - \frac{82760}{1357} a + \frac{17201}{1357} \) (order $4$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 3578.38600842 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$Q_8:C_2^2$ (as 16T23):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 17 conjugacy class representatives for $Q_8 : C_2^2$
Character table for $Q_8 : C_2^2$

Intermediate fields

\(\Q(\sqrt{-35}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{35}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-5}, \sqrt{-7})\), \(\Q(\sqrt{-5}, \sqrt{7})\), \(\Q(i, \sqrt{5})\), \(\Q(i, \sqrt{35})\), \(\Q(\sqrt{5}, \sqrt{7})\), \(\Q(i, \sqrt{7})\), \(\Q(\sqrt{5}, \sqrt{-7})\), 8.0.384160000.1, 8.4.3457440000.1 x2, 8.0.138297600.1 x2

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 R R R R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
$3$3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$7$7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$