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Properties

Label 16.0.58102542838...0000.3
Degree $16$
Signature $[0, 8]$
Discriminant $2^{28}\cdot 5^{10}\cdot 53^{6}$
Root discriminant $40.76$
Ramified primes $2, 5, 53$
Class number $8$ (GRH)
Class group $[8]$ (GRH)
Galois group 16T1048

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![584, -2176, 5824, -10400, 15792, -17600, 13840, -8728, 4556, -1992, 664, -128, 38, -4, -4, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 4*x^14 - 4*x^13 + 38*x^12 - 128*x^11 + 664*x^10 - 1992*x^9 + 4556*x^8 - 8728*x^7 + 13840*x^6 - 17600*x^5 + 15792*x^4 - 10400*x^3 + 5824*x^2 - 2176*x + 584)
 
gp: K = bnfinit(x^16 - 2*x^15 - 4*x^14 - 4*x^13 + 38*x^12 - 128*x^11 + 664*x^10 - 1992*x^9 + 4556*x^8 - 8728*x^7 + 13840*x^6 - 17600*x^5 + 15792*x^4 - 10400*x^3 + 5824*x^2 - 2176*x + 584, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 4 x^{14} - 4 x^{13} + 38 x^{12} - 128 x^{11} + 664 x^{10} - 1992 x^{9} + 4556 x^{8} - 8728 x^{7} + 13840 x^{6} - 17600 x^{5} + 15792 x^{4} - 10400 x^{3} + 5824 x^{2} - 2176 x + 584 \)

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:  \(58102542838005760000000000=2^{28}\cdot 5^{10}\cdot 53^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $40.76$
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:  $2, 5, 53$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{4} a^{11}$, $\frac{1}{4} a^{12}$, $\frac{1}{20} a^{13} - \frac{1}{10} a^{11} - \frac{1}{5} a^{10} - \frac{1}{5} a^{9} - \frac{1}{10} a^{8} - \frac{1}{5} a^{7} - \frac{1}{5} a^{4} - \frac{2}{5} a^{3} - \frac{1}{5} a^{2} - \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{20} a^{14} - \frac{1}{10} a^{12} + \frac{1}{20} a^{11} - \frac{1}{5} a^{10} - \frac{1}{10} a^{9} - \frac{1}{5} a^{8} - \frac{1}{5} a^{5} - \frac{2}{5} a^{4} - \frac{1}{5} a^{3} - \frac{2}{5} a^{2} + \frac{1}{5} a$, $\frac{1}{379926558503594993140} a^{15} - \frac{6292630172287608519}{379926558503594993140} a^{14} + \frac{4324021239016131691}{189963279251797496570} a^{13} - \frac{12085423813437162721}{379926558503594993140} a^{12} - \frac{2472812912872976914}{94981639625898748285} a^{11} + \frac{10325161747147947387}{94981639625898748285} a^{10} + \frac{2026089452339307777}{94981639625898748285} a^{9} + \frac{35530593807557750109}{189963279251797496570} a^{8} - \frac{21897441470439225013}{189963279251797496570} a^{7} - \frac{20804438820135348256}{94981639625898748285} a^{6} - \frac{22560420491161331513}{94981639625898748285} a^{5} - \frac{23490389658786798762}{94981639625898748285} a^{4} - \frac{20515944775924967076}{94981639625898748285} a^{3} + \frac{651943519789437691}{18996327925179749657} a^{2} - \frac{17070681458313069142}{94981639625898748285} a + \frac{2601775464814351454}{94981639625898748285}$

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

Class group and class number

$C_{8}$, which has order $8$ (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:  \( \frac{12661599621}{435679683812} a^{15} - \frac{47038336417}{2178398419060} a^{14} - \frac{153932589067}{1089199209530} a^{13} - \frac{159835745294}{544599604765} a^{12} + \frac{788166078383}{1089199209530} a^{11} - \frac{3094267047633}{1089199209530} a^{10} + \frac{1716702237891}{108919920953} a^{9} - \frac{41637700049827}{1089199209530} a^{8} + \frac{46438311429714}{544599604765} a^{7} - \frac{16132923104904}{108919920953} a^{6} + \frac{119517919306022}{544599604765} a^{5} - \frac{130958735871167}{544599604765} a^{4} + \frac{17720359132852}{108919920953} a^{3} - \frac{55053426271362}{544599604765} a^{2} + \frac{21686316803336}{544599604765} a - \frac{6941423172219}{544599604765} \) (order $4$)
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:  \( 1523144.19832 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1048:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 768
The 31 conjugacy class representatives for t16n1048
Character table for t16n1048 is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), 4.4.21200.1, 8.0.7191040000.3

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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$5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.3.2.1$x^{3} - 5$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
5.3.2.1$x^{3} - 5$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
5.6.5.1$x^{6} - 5$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$
53Data not computed