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Properties

Label 16.0.23595621172...7056.3
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 3^{8}\cdot 11^{8}$
Root discriminant $16.25$
Ramified primes $2, 3, 11$
Class number $1$
Class group Trivial
Galois group $D_4\times C_2$ (as 16T9)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9, -18, -33, 42, 277, -444, 294, -112, 62, -88, 62, -4, -27, 18, -1, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - x^14 + 18*x^13 - 27*x^12 - 4*x^11 + 62*x^10 - 88*x^9 + 62*x^8 - 112*x^7 + 294*x^6 - 444*x^5 + 277*x^4 + 42*x^3 - 33*x^2 - 18*x + 9)
 
gp: K = bnfinit(x^16 - 2*x^15 - x^14 + 18*x^13 - 27*x^12 - 4*x^11 + 62*x^10 - 88*x^9 + 62*x^8 - 112*x^7 + 294*x^6 - 444*x^5 + 277*x^4 + 42*x^3 - 33*x^2 - 18*x + 9, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - x^{14} + 18 x^{13} - 27 x^{12} - 4 x^{11} + 62 x^{10} - 88 x^{9} + 62 x^{8} - 112 x^{7} + 294 x^{6} - 444 x^{5} + 277 x^{4} + 42 x^{3} - 33 x^{2} - 18 x + 9 \)

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:  \(23595621172490797056=2^{24}\cdot 3^{8}\cdot 11^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.25$
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:  $2, 3, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{12} a^{12} + \frac{1}{12} a^{10} + \frac{1}{6} a^{9} - \frac{1}{4} a^{8} - \frac{1}{12} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{1}{3} a^{3} - \frac{5}{12} a^{2} - \frac{1}{2} a$, $\frac{1}{2412} a^{13} - \frac{2}{67} a^{12} + \frac{247}{2412} a^{11} + \frac{113}{2412} a^{10} + \frac{71}{804} a^{9} + \frac{49}{804} a^{8} - \frac{409}{2412} a^{7} + \frac{125}{804} a^{6} - \frac{89}{268} a^{5} - \frac{1123}{2412} a^{4} + \frac{853}{2412} a^{3} + \frac{121}{804} a^{2} + \frac{169}{402} a + \frac{15}{268}$, $\frac{1}{9648} a^{14} + \frac{11}{1206} a^{12} + \frac{205}{4824} a^{11} - \frac{55}{1072} a^{10} - \frac{11}{268} a^{9} - \frac{1885}{9648} a^{8} - \frac{323}{1608} a^{7} - \frac{781}{3216} a^{6} + \frac{527}{2412} a^{5} + \frac{3211}{9648} a^{4} - \frac{63}{536} a^{3} - \frac{32}{67} a^{2} - \frac{31}{134} a + \frac{75}{1072}$, $\frac{1}{174956832} a^{15} + \frac{6497}{174956832} a^{14} + \frac{1091}{7289868} a^{13} + \frac{2684021}{87478416} a^{12} + \frac{4808345}{58318944} a^{11} + \frac{6221521}{174956832} a^{10} - \frac{19644025}{174956832} a^{9} + \frac{15410653}{174956832} a^{8} + \frac{21712183}{174956832} a^{7} - \frac{6342079}{174956832} a^{6} - \frac{7710529}{174956832} a^{5} - \frac{5160293}{58318944} a^{4} + \frac{26190709}{87478416} a^{3} + \frac{1034923}{4859912} a^{2} - \frac{13159199}{58318944} a - \frac{104041}{19439648}$

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:  \( \frac{909313}{19439648} a^{15} - \frac{3834697}{58318944} a^{14} - \frac{116787}{1214978} a^{13} + \frac{7819993}{9719824} a^{12} - \frac{44980343}{58318944} a^{11} - \frac{48183353}{58318944} a^{10} + \frac{155472245}{58318944} a^{9} - \frac{142707677}{58318944} a^{8} + \frac{16766399}{19439648} a^{7} - \frac{228814249}{58318944} a^{6} + \frac{636283309}{58318944} a^{5} - \frac{763800625}{58318944} a^{4} + \frac{62988187}{29159472} a^{3} + \frac{109789405}{14579736} a^{2} + \frac{987131}{19439648} a - \frac{8652165}{19439648} \) (order $6$)
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:  \( 8400.0296703 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times D_4$ (as 16T9):

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 $D_4\times C_2$
Character table for $D_4\times C_2$

Intermediate fields

\(\Q(\sqrt{-66}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{33}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{22}) \), \(\Q(\sqrt{6}, \sqrt{-11})\), \(\Q(\sqrt{-2}, \sqrt{33})\), \(\Q(\sqrt{-3}, \sqrt{22})\), \(\Q(\sqrt{-3}, \sqrt{-11})\), \(\Q(\sqrt{-2}, \sqrt{-11})\), \(\Q(\sqrt{6}, \sqrt{22})\), \(\Q(\sqrt{-2}, \sqrt{-3})\), 4.2.8712.1 x2, 4.2.8712.2 x2, 4.0.2112.2 x2, 4.0.2112.1 x2, 8.0.4857532416.2, 8.0.4857532416.6, 8.0.4857532416.8, 8.0.75898944.1 x2, 8.0.539725824.3 x2, 8.4.4857532416.2 x2, 8.0.40144896.1 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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$2.4.6.2$x^{4} - 2 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.2$x^{4} - 2 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.2$x^{4} - 2 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.2$x^{4} - 2 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$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}$