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Properties

Label 16.0.27850097600...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{40}\cdot 5^{10}\cdot 11^{10}$
Root discriminant $69.23$
Ramified primes $2, 5, 11$
Class number $2352$ (GRH)
Class group $[2, 2, 588]$ (GRH)
Galois group $C_4.C_2^2:D_4$ (as 16T305)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![75625, 0, 453750, 0, 609730, 0, 347270, 0, 99214, 0, 15434, 0, 1322, 0, 58, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 58*x^14 + 1322*x^12 + 15434*x^10 + 99214*x^8 + 347270*x^6 + 609730*x^4 + 453750*x^2 + 75625)
 
gp: K = bnfinit(x^16 + 58*x^14 + 1322*x^12 + 15434*x^10 + 99214*x^8 + 347270*x^6 + 609730*x^4 + 453750*x^2 + 75625, 1)
 

Normalized defining polynomial

\( x^{16} + 58 x^{14} + 1322 x^{12} + 15434 x^{10} + 99214 x^{8} + 347270 x^{6} + 609730 x^{4} + 453750 x^{2} + 75625 \)

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:  \(278500976009402122240000000000=2^{40}\cdot 5^{10}\cdot 11^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $69.23$
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:  $2, 5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{22} a^{8} - \frac{2}{11} a^{6} + \frac{2}{11} a^{4} - \frac{1}{2}$, $\frac{1}{22} a^{9} - \frac{2}{11} a^{7} + \frac{2}{11} a^{5} - \frac{1}{2} a$, $\frac{1}{22} a^{10} + \frac{5}{11} a^{6} - \frac{3}{11} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{22} a^{11} + \frac{5}{11} a^{7} - \frac{3}{11} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{110} a^{12} - \frac{1}{55} a^{10} + \frac{1}{55} a^{8} - \frac{8}{55} a^{6} - \frac{31}{110} a^{4}$, $\frac{1}{110} a^{13} - \frac{1}{55} a^{11} + \frac{1}{55} a^{9} - \frac{8}{55} a^{7} - \frac{31}{110} a^{5}$, $\frac{1}{869113850} a^{14} - \frac{3928687}{869113850} a^{12} + \frac{864596}{39505175} a^{10} - \frac{1737053}{434556925} a^{8} + \frac{273270609}{869113850} a^{6} + \frac{6548203}{173822770} a^{4} + \frac{2184178}{7901035} a^{2} - \frac{262007}{1580207}$, $\frac{1}{4345569250} a^{15} + \frac{11873383}{4345569250} a^{13} - \frac{52088203}{4345569250} a^{11} + \frac{67635209}{4345569250} a^{9} + \frac{336478889}{4345569250} a^{7} + \frac{14673499}{79010350} a^{5} + \frac{28071461}{79010350} a^{3} - \frac{1052927}{3160414} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{588}$, which has order $2352$ (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:  \( 89318.5652847 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4.C_2^2:D_4$ (as 16T305):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 29 conjugacy class representatives for $C_4.C_2^2:D_4$
Character table for $C_4.C_2^2:D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{11}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{55}) \), 4.4.38720.1 x2, 4.4.4400.1 x2, \(\Q(\sqrt{5}, \sqrt{11})\), 8.8.37480960000.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 16 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$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$$[\ ]_{2}^{2}$
11.8.6.2$x^{8} - 781 x^{4} + 290521$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$