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Properties

Label 16.8.34713005539...3264.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{56}\cdot 3^{8}\cdot 7^{6}\cdot 79^{2}$
Root discriminant $70.19$
Ramified primes $2, 3, 7, 79$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T799

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-191543, -211552, 131136, 873856, -299644, -777264, 482632, 23096, -85132, 38224, -8736, -400, 676, -336, 80, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 80*x^14 - 336*x^13 + 676*x^12 - 400*x^11 - 8736*x^10 + 38224*x^9 - 85132*x^8 + 23096*x^7 + 482632*x^6 - 777264*x^5 - 299644*x^4 + 873856*x^3 + 131136*x^2 - 211552*x - 191543)
 
gp: K = bnfinit(x^16 - 8*x^15 + 80*x^14 - 336*x^13 + 676*x^12 - 400*x^11 - 8736*x^10 + 38224*x^9 - 85132*x^8 + 23096*x^7 + 482632*x^6 - 777264*x^5 - 299644*x^4 + 873856*x^3 + 131136*x^2 - 211552*x - 191543, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 80 x^{14} - 336 x^{13} + 676 x^{12} - 400 x^{11} - 8736 x^{10} + 38224 x^{9} - 85132 x^{8} + 23096 x^{7} + 482632 x^{6} - 777264 x^{5} - 299644 x^{4} + 873856 x^{3} + 131136 x^{2} - 211552 x - 191543 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(347130055392284294307605643264=2^{56}\cdot 3^{8}\cdot 7^{6}\cdot 79^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $70.19$
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:  $2, 3, 7, 79$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{9353742673} a^{14} - \frac{2141617512}{9353742673} a^{13} - \frac{519058973}{9353742673} a^{12} + \frac{1653578299}{9353742673} a^{11} - \frac{2057383638}{9353742673} a^{10} + \frac{4016218452}{9353742673} a^{9} + \frac{4280889595}{9353742673} a^{8} + \frac{1769473216}{9353742673} a^{7} - \frac{2294522088}{9353742673} a^{6} - \frac{77902562}{850340243} a^{5} + \frac{1925835292}{9353742673} a^{4} - \frac{2936668602}{9353742673} a^{3} + \frac{3414116478}{9353742673} a^{2} - \frac{3194168820}{9353742673} a - \frac{30788338}{850340243}$, $\frac{1}{117804068281486906549508920563850517215} a^{15} - \frac{794652730047305577428669377}{23560813656297381309901784112770103443} a^{14} - \frac{1791206907329700469484611752140760487}{23560813656297381309901784112770103443} a^{13} + \frac{14116824362897110887946702494018747404}{117804068281486906549508920563850517215} a^{12} + \frac{53007879114577677492835602281913825388}{117804068281486906549508920563850517215} a^{11} - \frac{21327901366883751068685662720844451806}{117804068281486906549508920563850517215} a^{10} - \frac{4886254912815054577786889370167331294}{10709460752862446049955356414895501565} a^{9} - \frac{1140009690177320524532569616991191194}{6929651075381582738206407091991206895} a^{8} - \frac{48742460703387485457634006912932553601}{117804068281486906549508920563850517215} a^{7} + \frac{25487637773128156629369902262063763933}{117804068281486906549508920563850517215} a^{6} - \frac{1615000346243787716223163584122361839}{117804068281486906549508920563850517215} a^{5} - \frac{33119643473838583084955688273242698216}{117804068281486906549508920563850517215} a^{4} - \frac{40953167306699805033488724098885162757}{117804068281486906549508920563850517215} a^{3} - \frac{1709411870464883698963339318931579688}{23560813656297381309901784112770103443} a^{2} + \frac{15714341047901230565331539867332601181}{117804068281486906549508920563850517215} a + \frac{1952379155336963227886076141973206261}{10709460752862446049955356414895501565}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $11$
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:  \( 594478331.246 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T799:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 62 conjugacy class representatives for t16n799 are not computed
Character table for t16n799 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{3}) \), 4.4.129024.1, 4.4.14336.1, \(\Q(\sqrt{2}, \sqrt{3})\), 8.8.66588770304.1

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 R ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$7$7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$79$79.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
79.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
79.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
79.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
79.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
79.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
79.4.2.1$x^{4} + 395 x^{2} + 56169$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$