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Properties

Label 16.0.77854178285...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 3^{8}\cdot 5^{8}\cdot 29^{4}$
Root discriminant $35.95$
Ramified primes $2, 3, 5, 29$
Class number $80$ (GRH)
Class group $[2, 40]$ (GRH)
Galois group $C_2^2 \times D_4$ (as 16T25)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![181476, 0, 202728, 0, 99328, 0, 26220, 0, 5541, 0, 1040, 0, 138, 0, 12, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 12*x^14 + 138*x^12 + 1040*x^10 + 5541*x^8 + 26220*x^6 + 99328*x^4 + 202728*x^2 + 181476)
 
gp: K = bnfinit(x^16 + 12*x^14 + 138*x^12 + 1040*x^10 + 5541*x^8 + 26220*x^6 + 99328*x^4 + 202728*x^2 + 181476, 1)
 

Normalized defining polynomial

\( x^{16} + 12 x^{14} + 138 x^{12} + 1040 x^{10} + 5541 x^{8} + 26220 x^{6} + 99328 x^{4} + 202728 x^{2} + 181476 \)

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:  \(7785417828571545600000000=2^{32}\cdot 3^{8}\cdot 5^{8}\cdot 29^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $35.95$
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:  $2, 3, 5, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3}$, $\frac{1}{24} a^{8} - \frac{1}{6} a^{7} - \frac{1}{12} a^{6} + \frac{1}{6} a^{5} + \frac{11}{24} a^{4} - \frac{1}{6} a^{3} - \frac{5}{12} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{72} a^{9} + \frac{1}{12} a^{7} - \frac{1}{6} a^{6} + \frac{3}{8} a^{5} - \frac{1}{3} a^{4} - \frac{13}{36} a^{3} - \frac{1}{6} a^{2} - \frac{5}{12} a$, $\frac{1}{72} a^{10} - \frac{1}{6} a^{7} - \frac{1}{8} a^{6} - \frac{1}{3} a^{5} + \frac{7}{18} a^{4} - \frac{1}{6} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{144} a^{11} + \frac{5}{48} a^{7} - \frac{1}{6} a^{6} - \frac{17}{36} a^{5} + \frac{1}{6} a^{4} + \frac{1}{24} a^{3} - \frac{1}{6} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{25920} a^{12} - \frac{23}{6480} a^{10} - \frac{61}{8640} a^{8} - \frac{163}{1296} a^{6} - \frac{1}{2} a^{5} + \frac{3157}{6480} a^{4} - \frac{1}{2} a^{3} + \frac{377}{1080} a^{2} - \frac{1}{2} a - \frac{131}{720}$, $\frac{1}{25920} a^{13} + \frac{11}{3240} a^{11} + \frac{59}{8640} a^{9} + \frac{5}{81} a^{7} - \frac{1}{6} a^{6} + \frac{2527}{6480} a^{5} - \frac{1}{3} a^{4} + \frac{4}{135} a^{3} - \frac{1}{6} a^{2} - \frac{251}{720} a - \frac{1}{2}$, $\frac{1}{1311163200} a^{14} - \frac{463}{262232640} a^{12} + \frac{952631}{437054400} a^{10} - \frac{25704371}{1311163200} a^{8} + \frac{282097}{20486925} a^{6} + \frac{31509517}{109263600} a^{4} - \frac{1}{2} a^{3} - \frac{3373501}{7284240} a^{2} - \frac{1}{2} a + \frac{273637}{4046800}$, $\frac{1}{93092587200} a^{15} - \frac{223037}{18618517440} a^{13} - \frac{277180667}{93092587200} a^{11} + \frac{560373439}{93092587200} a^{9} + \frac{6381469}{2909143350} a^{7} - \frac{1}{6} a^{6} - \frac{3842198489}{23273146800} a^{5} + \frac{1}{6} a^{4} - \frac{556721779}{1551543120} a^{3} + \frac{1}{3} a^{2} - \frac{6541397}{2585905200} a$

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

Class group and class number

$C_{2}\times C_{40}$, which has order $80$ (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:  \( 15197.4244561 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times D_4$ (as 16T25):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_2^2 \times D_4$
Character table for $C_2^2 \times D_4$

Intermediate fields

\(\Q(\sqrt{6}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{30}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{10}) \), 4.0.83520.4, 4.0.334080.4, 4.0.83520.3, 4.0.334080.7, \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\sqrt{2}, \sqrt{15})\), \(\Q(\sqrt{5}, \sqrt{6})\), \(\Q(\sqrt{3}, \sqrt{5})\), \(\Q(\sqrt{6}, \sqrt{10})\), \(\Q(\sqrt{3}, \sqrt{10})\), 8.8.3317760000.1, 8.0.111609446400.25, 8.0.111609446400.35, 8.0.174389760000.18, 8.0.2790236160000.39, 8.0.2790236160000.16, 8.0.2790236160000.43

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 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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[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$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$