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Properties

Label 16.0.61010060154...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 5^{8}\cdot 29^{6}\cdot 139^{4}\cdot 181^{4}$
Root discriminant $199.11$
Ramified primes $2, 5, 29, 139, 181$
Class number $13910272$ (GRH)
Class group $[2, 2, 2, 2, 2, 434696]$ (GRH)
Galois group 16T839

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![113334569348641, 0, 20129082334803, 0, 1466223933180, 0, 57136307508, 0, 1303778149, 0, 17822352, 0, 141940, 0, 597, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 597*x^14 + 141940*x^12 + 17822352*x^10 + 1303778149*x^8 + 57136307508*x^6 + 1466223933180*x^4 + 20129082334803*x^2 + 113334569348641)
 
gp: K = bnfinit(x^16 + 597*x^14 + 141940*x^12 + 17822352*x^10 + 1303778149*x^8 + 57136307508*x^6 + 1466223933180*x^4 + 20129082334803*x^2 + 113334569348641, 1)
 

Normalized defining polynomial

\( x^{16} + 597 x^{14} + 141940 x^{12} + 17822352 x^{10} + 1303778149 x^{8} + 57136307508 x^{6} + 1466223933180 x^{4} + 20129082334803 x^{2} + 113334569348641 \)

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:  \(6101006015475987923845106713600000000=2^{16}\cdot 5^{8}\cdot 29^{6}\cdot 139^{4}\cdot 181^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $199.11$
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:  $2, 5, 29, 139, 181$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8} + \frac{1}{5} a^{6} + \frac{2}{5} a^{4} - \frac{1}{5} a^{2} + \frac{1}{5}$, $\frac{1}{5} a^{9} + \frac{1}{5} a^{7} + \frac{2}{5} a^{5} - \frac{1}{5} a^{3} + \frac{1}{5} a$, $\frac{1}{695} a^{10} + \frac{41}{695} a^{8} - \frac{118}{695} a^{6} + \frac{189}{695} a^{4} + \frac{266}{695} a^{2}$, $\frac{1}{695} a^{11} + \frac{41}{695} a^{9} - \frac{118}{695} a^{7} + \frac{189}{695} a^{5} + \frac{266}{695} a^{3}$, $\frac{1}{193210} a^{12} - \frac{49}{96605} a^{10} - \frac{2481}{193210} a^{8} + \frac{32437}{193210} a^{6} - \frac{76323}{193210} a^{4} - \frac{67}{139} a^{2} + \frac{1}{10}$, $\frac{1}{3670990} a^{13} + \frac{229}{1835495} a^{11} - \frac{288821}{3670990} a^{9} - \frac{1501567}{3670990} a^{7} + \frac{1149379}{3670990} a^{5} + \frac{3394}{13205} a^{3} + \frac{11}{38} a$, $\frac{1}{6255613923813612264021025950290} a^{14} + \frac{249143296238344866953303}{625561392381361226402102595029} a^{12} - \frac{2876279428778703914802588991}{6255613923813612264021025950290} a^{10} + \frac{102896809527538526811883413673}{6255613923813612264021025950290} a^{8} + \frac{193182815181169190551983236871}{6255613923813612264021025950290} a^{6} + \frac{7741879840042095790813715784}{22502208359041770733888582555} a^{4} - \frac{13749546095136159633910573}{64754556428897181968024698} a^{2} - \frac{207260138367083459624}{2113702895614813549205}$, $\frac{1}{6255613923813612264021025950290} a^{15} + \frac{787365687938785986163959}{6255613923813612264021025950290} a^{13} - \frac{3656742240474359423785623509}{6255613923813612264021025950290} a^{11} + \frac{297533611899960223842610934482}{3127806961906806132010512975145} a^{9} + \frac{1375977000133609001015214535564}{3127806961906806132010512975145} a^{7} + \frac{1392974501422810643871608137}{45004416718083541467777165110} a^{5} - \frac{151965058957194253527173397}{323772782144485909840123490} a^{3} - \frac{31126617109712120506967}{80320710033362914869790} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{434696}$, which has order $13910272$ (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:  \( 29196.3261178 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T839:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 53 conjugacy class representatives for t16n839 are not computed
Character table for t16n839 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.4525.1, 4.4.131225.1, 4.4.725.1, 8.8.17220000625.2

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 ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.8.8.6$x^{8} + 2 x^{7} + 2 x^{6} + 16 x^{2} + 16$$2$$4$$8$$(C_8:C_2):C_2$$[2, 2, 2]^{4}$
2.8.8.6$x^{8} + 2 x^{7} + 2 x^{6} + 16 x^{2} + 16$$2$$4$$8$$(C_8:C_2):C_2$$[2, 2, 2]^{4}$
$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$29.4.2.2$x^{4} - 29 x^{2} + 2523$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.8.4.1$x^{8} + 31958 x^{4} - 24389 x^{2} + 255328441$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$139$$\Q_{139}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{139}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{139}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{139}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{139}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{139}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{139}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{139}$$x + 4$$1$$1$$0$Trivial$[\ ]$
139.2.1.1$x^{2} - 139$$2$$1$$1$$C_2$$[\ ]_{2}$
139.2.1.1$x^{2} - 139$$2$$1$$1$$C_2$$[\ ]_{2}$
139.2.1.1$x^{2} - 139$$2$$1$$1$$C_2$$[\ ]_{2}$
139.2.1.1$x^{2} - 139$$2$$1$$1$$C_2$$[\ ]_{2}$
$181$181.4.0.1$x^{4} - x + 54$$1$$4$$0$$C_4$$[\ ]^{4}$
181.4.0.1$x^{4} - x + 54$$1$$4$$0$$C_4$$[\ ]^{4}$
181.8.4.1$x^{8} + 3538188 x^{4} - 5929741 x^{2} + 3129693580836$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$