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Properties

Label 16.0.28179280429...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{40}\cdot 3^{8}\cdot 5^{8}$
Root discriminant $21.91$
Ramified primes $2, 3, 5$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $Q_8 : C_2$ (as 16T11)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -100, 3936, 6780, -2996, -7848, -1100, 1912, 2367, 288, -908, -240, 156, 52, -16, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 16*x^14 + 52*x^13 + 156*x^12 - 240*x^11 - 908*x^10 + 288*x^9 + 2367*x^8 + 1912*x^7 - 1100*x^6 - 7848*x^5 - 2996*x^4 + 6780*x^3 + 3936*x^2 - 100*x + 1)
 
gp: K = bnfinit(x^16 - 4*x^15 - 16*x^14 + 52*x^13 + 156*x^12 - 240*x^11 - 908*x^10 + 288*x^9 + 2367*x^8 + 1912*x^7 - 1100*x^6 - 7848*x^5 - 2996*x^4 + 6780*x^3 + 3936*x^2 - 100*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 16 x^{14} + 52 x^{13} + 156 x^{12} - 240 x^{11} - 908 x^{10} + 288 x^{9} + 2367 x^{8} + 1912 x^{7} - 1100 x^{6} - 7848 x^{5} - 2996 x^{4} + 6780 x^{3} + 3936 x^{2} - 100 x + 1 \)

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:  \(2817928042905600000000=2^{40}\cdot 3^{8}\cdot 5^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.91$
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 not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{101905} a^{14} + \frac{3830}{20381} a^{13} + \frac{5674}{20381} a^{12} + \frac{20757}{101905} a^{11} - \frac{41256}{101905} a^{10} - \frac{19787}{101905} a^{9} + \frac{31263}{101905} a^{8} - \frac{24132}{101905} a^{7} + \frac{29247}{101905} a^{6} + \frac{17763}{101905} a^{5} + \frac{2094}{101905} a^{4} + \frac{28606}{101905} a^{3} + \frac{34362}{101905} a^{2} - \frac{861}{101905} a - \frac{27381}{101905}$, $\frac{1}{10743048352558333947265} a^{15} - \frac{4324797845099716}{2148609670511666789453} a^{14} + \frac{48045191078213129626}{2148609670511666789453} a^{13} + \frac{1422678082520352171802}{10743048352558333947265} a^{12} - \frac{896130866551442329276}{10743048352558333947265} a^{11} - \frac{3790519902258591630842}{10743048352558333947265} a^{10} + \frac{3721746608258173171893}{10743048352558333947265} a^{9} + \frac{1223558218542340033663}{10743048352558333947265} a^{8} - \frac{3848135687842963186928}{10743048352558333947265} a^{7} + \frac{162933505678071591241}{826388334812179534405} a^{6} - \frac{2530003774520026519866}{10743048352558333947265} a^{5} - \frac{25715431657466764737}{290352658177252268845} a^{4} + \frac{53411272865449672243}{120708408455711617385} a^{3} + \frac{102999862438229655672}{290352658177252268845} a^{2} - \frac{2163503795838640631151}{10743048352558333947265} a - \frac{394971228242274604788}{2148609670511666789453}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( -\frac{47996495270768184306}{10743048352558333947265} a^{15} + \frac{195701567381154120851}{10743048352558333947265} a^{14} + \frac{156527762504618615027}{2148609670511666789453} a^{13} - \frac{2651427262375547107212}{10743048352558333947265} a^{12} - \frac{7812563345787085290057}{10743048352558333947265} a^{11} + \frac{13170990326214308888316}{10743048352558333947265} a^{10} + \frac{9519509763107411495945}{2148609670511666789453} a^{9} - \frac{3944382281832475696950}{2148609670511666789453} a^{8} - \frac{136567831280932268718469}{10743048352558333947265} a^{7} - \frac{7236395957710651399572}{826388334812179534405} a^{6} + \frac{98986936844542044814219}{10743048352558333947265} a^{5} + \frac{12040037827571813483594}{290352658177252268845} a^{4} + \frac{173101083111823837943449}{10743048352558333947265} a^{3} - \frac{12309705773149514577736}{290352658177252268845} a^{2} - \frac{56268974577381498226484}{2148609670511666789453} a + \frac{13890687315272607294084}{10743048352558333947265} \) (order $24$)
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:  \( 31102.0401588 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_4:C_2$ (as 16T11):

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 $Q_8 : C_2$
Character table for $Q_8 : C_2$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{2}) \), \(\Q(i, \sqrt{6})\), \(\Q(\zeta_{12})\), \(\Q(\zeta_{8})\), \(\Q(\sqrt{-2}, \sqrt{3})\), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\sqrt{-2}, \sqrt{-3})\), \(\Q(\zeta_{24})\), 8.0.3317760000.16 x2, 8.0.5898240000.5 x2, 8.4.53084160000.4 x2

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 R R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$