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Properties

Label 16.0.84616293477...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 5^{8}\cdot 29^{6}\cdot 89^{8}\cdot 109^{4}$
Root discriminant $481.91$
Ramified primes $2, 5, 29, 89, 109$
Class number $32367411712$ (GRH)
Class group $[2, 2, 2, 2, 2, 2, 2, 252870404]$ (GRH)
Galois group $C_2^2.C_2^5.C_2$ (as 16T493)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![7448383238882771641, 0, 481407719903051007, 0, 11844611939032629, 0, 140021879295672, 0, 843697669633, 0, 2643188394, 0, 4288256, 0, 3366, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 3366*x^14 + 4288256*x^12 + 2643188394*x^10 + 843697669633*x^8 + 140021879295672*x^6 + 11844611939032629*x^4 + 481407719903051007*x^2 + 7448383238882771641)
 
gp: K = bnfinit(x^16 + 3366*x^14 + 4288256*x^12 + 2643188394*x^10 + 843697669633*x^8 + 140021879295672*x^6 + 11844611939032629*x^4 + 481407719903051007*x^2 + 7448383238882771641, 1)
 

Normalized defining polynomial

\( x^{16} + 3366 x^{14} + 4288256 x^{12} + 2643188394 x^{10} + 843697669633 x^{8} + 140021879295672 x^{6} + 11844611939032629 x^{4} + 481407719903051007 x^{2} + 7448383238882771641 \)

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:  \(8461629347780675963909292061418521600000000=2^{16}\cdot 5^{8}\cdot 29^{6}\cdot 89^{8}\cdot 109^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $481.91$
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:  $2, 5, 29, 89, 109$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{89} a^{8} - \frac{16}{89} a^{6} - \frac{31}{89} a^{4}$, $\frac{1}{89} a^{9} - \frac{16}{89} a^{7} - \frac{31}{89} a^{5}$, $\frac{1}{9701} a^{10} - \frac{13}{9701} a^{8} - \frac{3728}{9701} a^{6} + \frac{3467}{9701} a^{4} + \frac{22}{109} a^{2}$, $\frac{1}{9701} a^{11} - \frac{13}{9701} a^{9} - \frac{3728}{9701} a^{7} + \frac{3467}{9701} a^{5} + \frac{22}{109} a^{3}$, $\frac{1}{12578756647061} a^{12} - \frac{28413923}{1143523331551} a^{10} - \frac{63241846892}{12578756647061} a^{8} - \frac{3310350536}{12848576759} a^{6} + \frac{30623014287}{141334344349} a^{4} + \frac{5097}{45671} a^{2} + \frac{1245}{4609}$, $\frac{1}{12578756647061} a^{13} - \frac{28413923}{1143523331551} a^{11} - \frac{63241846892}{12578756647061} a^{9} - \frac{3310350536}{12848576759} a^{7} + \frac{30623014287}{141334344349} a^{5} + \frac{5097}{45671} a^{3} + \frac{1245}{4609} a$, $\frac{1}{5498799994386736752503430604953061297605559631} a^{14} - \frac{192388194260980831433511733945219}{5498799994386736752503430604953061297605559631} a^{12} - \frac{31331352853150522554104233752917426105425}{5498799994386736752503430604953061297605559631} a^{10} + \frac{37510683365817290183116905266055094932278}{61784269599850974747229557359023160647253479} a^{8} + \frac{2629595087470134381034175891045712144948113}{61784269599850974747229557359023160647253479} a^{6} - \frac{1405100447874280641931183879246339831374}{6368855746814861843854196202352660617179} a^{4} - \frac{837390714726755249644706679900380992}{2014823077132192927508445492677209939} a^{2} - \frac{58172201724594891267269276756838}{207692307713863820998705854311639}$, $\frac{1}{5498799994386736752503430604953061297605559631} a^{15} - \frac{192388194260980831433511733945219}{5498799994386736752503430604953061297605559631} a^{13} - \frac{31331352853150522554104233752917426105425}{5498799994386736752503430604953061297605559631} a^{11} + \frac{37510683365817290183116905266055094932278}{61784269599850974747229557359023160647253479} a^{9} + \frac{2629595087470134381034175891045712144948113}{61784269599850974747229557359023160647253479} a^{7} - \frac{1405100447874280641931183879246339831374}{6368855746814861843854196202352660617179} a^{5} - \frac{837390714726755249644706679900380992}{2014823077132192927508445492677209939} a^{3} - \frac{58172201724594891267269276756838}{207692307713863820998705854311639} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{252870404}$, which has order $32367411712$ (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:  \( 13345.9350553 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2.C_2^5.C_2$ (as 16T493):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 34 conjugacy class representatives for $C_2^2.C_2^5.C_2$
Character table for $C_2^2.C_2^5.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.725.1, 4.4.64525.2, 4.4.2225.1, 8.8.4163475625.2

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 ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$2.8.8.5$x^{8} + 2 x^{7} + 8 x^{2} + 16$$2$$4$$8$$C_8:C_2$$[2, 2]^{4}$
2.8.8.5$x^{8} + 2 x^{7} + 8 x^{2} + 16$$2$$4$$8$$C_8:C_2$$[2, 2]^{4}$
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$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}$
29.8.6.1$x^{8} - 203 x^{4} + 68121$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$89$89.4.2.2$x^{4} - 89 x^{2} + 47526$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
89.4.0.1$x^{4} - x + 27$$1$$4$$0$$C_4$$[\ ]^{4}$
89.8.6.2$x^{8} + 979 x^{4} + 285156$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$109$109.2.1.2$x^{2} + 654$$2$$1$$1$$C_2$$[\ ]_{2}$
109.2.1.2$x^{2} + 654$$2$$1$$1$$C_2$$[\ ]_{2}$
109.2.1.2$x^{2} + 654$$2$$1$$1$$C_2$$[\ ]_{2}$
109.2.1.2$x^{2} + 654$$2$$1$$1$$C_2$$[\ ]_{2}$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$