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Properties

Label 16.0.58432555897...0000.3
Degree $16$
Signature $[0, 8]$
Discriminant $2^{48}\cdot 3^{12}\cdot 5^{8}$
Root discriminant $40.78$
Ramified primes $2, 3, 5$
Class number $144$ (GRH)
Class group $[6, 24]$ (GRH)
Galois group $D_8$ (as 16T7)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![27646, 32616, 53260, 39336, 32086, 20184, 11128, 3840, -113, -1320, -200, 408, 178, -24, -20, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 20*x^14 - 24*x^13 + 178*x^12 + 408*x^11 - 200*x^10 - 1320*x^9 - 113*x^8 + 3840*x^7 + 11128*x^6 + 20184*x^5 + 32086*x^4 + 39336*x^3 + 53260*x^2 + 32616*x + 27646)
 
gp: K = bnfinit(x^16 - 20*x^14 - 24*x^13 + 178*x^12 + 408*x^11 - 200*x^10 - 1320*x^9 - 113*x^8 + 3840*x^7 + 11128*x^6 + 20184*x^5 + 32086*x^4 + 39336*x^3 + 53260*x^2 + 32616*x + 27646, 1)
 

Normalized defining polynomial

\( x^{16} - 20 x^{14} - 24 x^{13} + 178 x^{12} + 408 x^{11} - 200 x^{10} - 1320 x^{9} - 113 x^{8} + 3840 x^{7} + 11128 x^{6} + 20184 x^{5} + 32086 x^{4} + 39336 x^{3} + 53260 x^{2} + 32616 x + 27646 \)

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:  \(58432555897690521600000000=2^{48}\cdot 3^{12}\cdot 5^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $40.78$
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:  $2, 3, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{6} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a$, $\frac{1}{69} a^{12} + \frac{1}{23} a^{11} - \frac{11}{69} a^{10} + \frac{7}{69} a^{9} + \frac{1}{23} a^{8} - \frac{25}{69} a^{7} - \frac{7}{23} a^{6} + \frac{1}{23} a^{5} - \frac{16}{69} a^{4} - \frac{19}{69} a^{3} - \frac{8}{23} a^{2} + \frac{7}{69} a + \frac{1}{3}$, $\frac{1}{69} a^{13} + \frac{1}{23} a^{11} - \frac{2}{23} a^{10} + \frac{5}{69} a^{9} - \frac{11}{69} a^{8} + \frac{8}{69} a^{7} + \frac{20}{69} a^{6} + \frac{7}{23} a^{5} + \frac{2}{23} a^{4} + \frac{10}{69} a^{3} - \frac{13}{69} a^{2} - \frac{7}{23} a - \frac{1}{3}$, $\frac{1}{26427} a^{14} - \frac{53}{8809} a^{13} + \frac{19}{26427} a^{12} - \frac{343}{26427} a^{11} + \frac{1013}{26427} a^{10} - \frac{2189}{26427} a^{9} - \frac{2128}{26427} a^{8} - \frac{865}{8809} a^{7} + \frac{8419}{26427} a^{6} + \frac{11113}{26427} a^{5} - \frac{1568}{26427} a^{4} + \frac{10421}{26427} a^{3} - \frac{2275}{8809} a^{2} - \frac{3373}{8809} a - \frac{310}{1149}$, $\frac{1}{186606439817510114292027} a^{15} + \frac{801229691612260037}{186606439817510114292027} a^{14} + \frac{51283238722819264383}{62202146605836704764009} a^{13} + \frac{376359236740182395729}{186606439817510114292027} a^{12} - \frac{3386183242855843527646}{62202146605836704764009} a^{11} - \frac{802209498537335990956}{62202146605836704764009} a^{10} + \frac{17008997562650639604079}{186606439817510114292027} a^{9} + \frac{15454115887311182390863}{186606439817510114292027} a^{8} - \frac{3613423635700702751057}{62202146605836704764009} a^{7} - \frac{20127889560048146550530}{62202146605836704764009} a^{6} + \frac{19163081775292674411934}{62202146605836704764009} a^{5} + \frac{30380175198581915158170}{62202146605836704764009} a^{4} + \frac{1757415632986298094992}{8113323470326526708349} a^{3} + \frac{13962030639811094794265}{62202146605836704764009} a^{2} - \frac{6722565334430047044564}{62202146605836704764009} a - \frac{362941406181619191721}{8113323470326526708349}$

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

Class group and class number

$C_{6}\times C_{24}$, which has order $144$ (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:  \( 45687.5845647 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times Q_8$ (as 16T7):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $D_8$
Character table for $D_8$

Intermediate fields

\(\Q(\sqrt{-10}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-30}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{2}, \sqrt{-5})\), \(\Q(\sqrt{6}, \sqrt{-10})\), \(\Q(\sqrt{3}, \sqrt{-10})\), \(\Q(\sqrt{-5}, \sqrt{6})\), \(\Q(\sqrt{3}, \sqrt{-5})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\sqrt{2}, \sqrt{-15})\), 8.0.3317760000.6, 8.0.7644119040000.1, 8.8.12230590464.1

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

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.8.24.4$x^{8} + 14 x^{4} + 8 x^{3} + 12 x^{2} + 8 x + 14$$8$$1$$24$$Q_8$$[2, 3, 4]$
2.8.24.4$x^{8} + 14 x^{4} + 8 x^{3} + 12 x^{2} + 8 x + 14$$8$$1$$24$$Q_8$$[2, 3, 4]$
$3$3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{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}$