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Properties

Label 16.0.26873856000...0000.3
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 3^{8}\cdot 5^{12}$
Root discriminant $16.38$
Ramified primes $2, 3, 5$
Class number $2$
Class group $[2]$
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![1, -18, 143, -630, 1595, -2292, 2244, -1620, 824, -240, 36, -24, 35, -30, 17, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 17*x^14 - 30*x^13 + 35*x^12 - 24*x^11 + 36*x^10 - 240*x^9 + 824*x^8 - 1620*x^7 + 2244*x^6 - 2292*x^5 + 1595*x^4 - 630*x^3 + 143*x^2 - 18*x + 1)
 
gp: K = bnfinit(x^16 - 6*x^15 + 17*x^14 - 30*x^13 + 35*x^12 - 24*x^11 + 36*x^10 - 240*x^9 + 824*x^8 - 1620*x^7 + 2244*x^6 - 2292*x^5 + 1595*x^4 - 630*x^3 + 143*x^2 - 18*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 17 x^{14} - 30 x^{13} + 35 x^{12} - 24 x^{11} + 36 x^{10} - 240 x^{9} + 824 x^{8} - 1620 x^{7} + 2244 x^{6} - 2292 x^{5} + 1595 x^{4} - 630 x^{3} + 143 x^{2} - 18 x + 1 \)

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:  \(26873856000000000000=2^{24}\cdot 3^{8}\cdot 5^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.38$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5}$, $\frac{1}{12} a^{12} - \frac{1}{6} a^{10} + \frac{1}{6} a^{6} - \frac{1}{2} a^{4} + \frac{1}{3} a^{2} + \frac{1}{12}$, $\frac{1}{12} a^{13} - \frac{1}{6} a^{11} + \frac{1}{6} a^{7} - \frac{1}{2} a^{5} + \frac{1}{3} a^{3} + \frac{1}{12} a$, $\frac{1}{50616} a^{14} + \frac{1433}{50616} a^{13} + \frac{101}{50616} a^{12} + \frac{2005}{25308} a^{11} - \frac{1285}{25308} a^{10} - \frac{463}{4218} a^{9} + \frac{3319}{25308} a^{8} - \frac{985}{25308} a^{7} + \frac{412}{6327} a^{6} - \frac{1189}{8436} a^{5} + \frac{5873}{25308} a^{4} - \frac{439}{12654} a^{3} - \frac{19567}{50616} a^{2} - \frac{3955}{50616} a + \frac{12823}{50616}$, $\frac{1}{96625944} a^{15} + \frac{11}{12078243} a^{14} + \frac{935}{211899} a^{13} + \frac{339195}{10736216} a^{12} + \frac{5800505}{24156486} a^{11} + \frac{4260565}{48312972} a^{10} + \frac{158647}{48312972} a^{9} - \frac{5566849}{24156486} a^{8} + \frac{2819645}{48312972} a^{7} - \frac{9867361}{48312972} a^{6} - \frac{88441}{326439} a^{5} + \frac{8031323}{48312972} a^{4} - \frac{1690997}{32208648} a^{3} - \frac{39841}{108813} a^{2} + \frac{12346459}{48312972} a + \frac{31542671}{96625944}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( \frac{993509}{652878} a^{15} - \frac{5755225}{652878} a^{14} + \frac{1744735}{72542} a^{13} - \frac{932855}{22908} a^{12} + \frac{14680940}{326439} a^{11} - \frac{17924615}{652878} a^{10} + \frac{16115495}{326439} a^{9} - \frac{231869225}{652878} a^{8} + \frac{385390030}{326439} a^{7} - \frac{1451143885}{652878} a^{6} + \frac{966762505}{326439} a^{5} - \frac{1885549915}{652878} a^{4} + \frac{133961985}{72542} a^{3} - \frac{64600495}{108813} a^{2} + \frac{68940295}{652878} a - \frac{10273687}{1305756} \) (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:  \( 4471.3211085 \)
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

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{-30}) \), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\sqrt{-3}, \sqrt{10})\), \(\Q(\sqrt{-6}, \sqrt{10})\), \(\Q(\sqrt{5}, \sqrt{-6})\), \(\Q(\sqrt{2}, \sqrt{-15})\), \(\Q(\sqrt{2}, \sqrt{5})\), 4.0.9000.2 x2, 4.0.9000.1 x2, 4.2.24000.2 x2, 4.2.24000.1 x2, 8.0.207360000.1, 8.0.81000000.1 x2, 8.0.5184000000.13 x2, 8.0.5184000000.10, 8.0.5184000000.12, 8.0.5184000000.11 x2, 8.4.576000000.1 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 R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.1$x^{4} - 6 x^{2} + 4$$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}$
$5$5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$