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Properties

Label 16.0.34881684083...0625.2
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 13^{4}\cdot 29^{8}$
Root discriminant $34.19$
Ramified primes $5, 13, 29$
Class number $16$ (GRH)
Class group $[2, 8]$ (GRH)
Galois group $C_2^2:D_4$ (as 16T43)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1856, 464, 4037, -5933, 7504, -10495, 8857, -6374, 3417, -1411, 605, -182, 85, -31, 16, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 16*x^14 - 31*x^13 + 85*x^12 - 182*x^11 + 605*x^10 - 1411*x^9 + 3417*x^8 - 6374*x^7 + 8857*x^6 - 10495*x^5 + 7504*x^4 - 5933*x^3 + 4037*x^2 + 464*x + 1856)
 
gp: K = bnfinit(x^16 - 5*x^15 + 16*x^14 - 31*x^13 + 85*x^12 - 182*x^11 + 605*x^10 - 1411*x^9 + 3417*x^8 - 6374*x^7 + 8857*x^6 - 10495*x^5 + 7504*x^4 - 5933*x^3 + 4037*x^2 + 464*x + 1856, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 16 x^{14} - 31 x^{13} + 85 x^{12} - 182 x^{11} + 605 x^{10} - 1411 x^{9} + 3417 x^{8} - 6374 x^{7} + 8857 x^{6} - 10495 x^{5} + 7504 x^{4} - 5933 x^{3} + 4037 x^{2} + 464 x + 1856 \)

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:  \(3488168408344511962890625=5^{12}\cdot 13^{4}\cdot 29^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.19$
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:  $5, 13, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{4} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{7} - \frac{1}{8} a^{6} - \frac{1}{8} a^{4} - \frac{1}{8} a^{2} - \frac{1}{8} a$, $\frac{1}{8} a^{8} - \frac{1}{8} a^{6} - \frac{1}{8} a^{5} - \frac{1}{8} a^{4} - \frac{1}{8} a^{3} - \frac{1}{4} a^{2} - \frac{1}{8} a$, $\frac{1}{32} a^{9} - \frac{1}{16} a^{7} - \frac{1}{32} a^{5} - \frac{1}{4} a^{4} - \frac{1}{16} a^{3} + \frac{1}{4} a^{2} - \frac{15}{32} a + \frac{1}{4}$, $\frac{1}{32} a^{10} - \frac{1}{16} a^{8} - \frac{1}{32} a^{6} - \frac{1}{16} a^{4} + \frac{1}{32} a^{2}$, $\frac{1}{64} a^{11} - \frac{1}{64} a^{10} - \frac{1}{64} a^{9} + \frac{1}{32} a^{8} - \frac{3}{64} a^{7} - \frac{7}{64} a^{6} + \frac{5}{64} a^{5} + \frac{1}{32} a^{4} + \frac{7}{64} a^{3} + \frac{31}{64} a^{2} - \frac{23}{64} a - \frac{3}{8}$, $\frac{1}{64} a^{12} - \frac{1}{64} a^{9} + \frac{3}{64} a^{8} + \frac{1}{32} a^{7} - \frac{1}{16} a^{6} + \frac{1}{64} a^{5} - \frac{11}{64} a^{4} + \frac{1}{32} a^{3} + \frac{9}{32} a^{2} + \frac{31}{64} a + \frac{3}{8}$, $\frac{1}{128} a^{13} - \frac{1}{128} a^{11} - \frac{1}{64} a^{9} - \frac{1}{16} a^{8} - \frac{5}{128} a^{7} + \frac{7}{64} a^{5} - \frac{3}{16} a^{4} + \frac{15}{128} a^{3} + \frac{3}{8} a^{2} - \frac{47}{128} a + \frac{5}{16}$, $\frac{1}{7424} a^{14} - \frac{5}{3712} a^{13} - \frac{49}{7424} a^{12} + \frac{15}{3712} a^{11} + \frac{1}{3712} a^{10} + \frac{3}{1856} a^{9} - \frac{381}{7424} a^{8} + \frac{79}{3712} a^{7} - \frac{443}{3712} a^{6} - \frac{113}{1856} a^{5} - \frac{441}{7424} a^{4} + \frac{327}{3712} a^{3} - \frac{3515}{7424} a^{2} + \frac{63}{128} a + \frac{7}{16}$, $\frac{1}{121507643199232} a^{15} + \frac{6012114855}{121507643199232} a^{14} - \frac{332306786955}{121507643199232} a^{13} + \frac{603083273789}{121507643199232} a^{12} - \frac{160458128747}{30376910799808} a^{11} - \frac{929879946115}{60753821599616} a^{10} + \frac{15777669737}{2059451579648} a^{9} + \frac{3657124062169}{121507643199232} a^{8} + \frac{1449124823271}{30376910799808} a^{7} - \frac{5436519827443}{60753821599616} a^{6} + \frac{8560742119279}{121507643199232} a^{5} + \frac{29978582051181}{121507643199232} a^{4} + \frac{17229632551247}{121507643199232} a^{3} - \frac{20612348378841}{121507643199232} a^{2} - \frac{443545640627}{2094959365504} a + \frac{66713140181}{261869920688}$

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

Class group and class number

$C_{2}\times C_{8}$, which has order $16$ (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:  \( 266040.845401 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:D_4$ (as 16T43):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_2^2:D_4$
Character table for $C_2^2:D_4$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{145}) \), \(\Q(\sqrt{29}) \), 4.0.3625.1 x2, 4.0.105125.1 x2, 4.0.54665.1, \(\Q(\sqrt{5}, \sqrt{29})\), 4.0.54665.2, 8.0.74706555625.3, 8.8.1867663890625.2, 8.0.11051265625.4

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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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
$5$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$13$13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$29$29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$