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Properties

Label 16.8.13657651539...2656.2
Degree $16$
Signature $[8, 4]$
Discriminant $2^{56}\cdot 3^{8}\cdot 7^{6}\cdot 1567^{2}$
Root discriminant $101.97$
Ramified primes $2, 3, 7, 1567$
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![-8134226, -13201088, 9577888, 50197344, -37227492, 1628176, 5805312, -2982336, 547102, 41032, -33320, 9608, -1464, -24, 32, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 32*x^14 - 24*x^13 - 1464*x^12 + 9608*x^11 - 33320*x^10 + 41032*x^9 + 547102*x^8 - 2982336*x^7 + 5805312*x^6 + 1628176*x^5 - 37227492*x^4 + 50197344*x^3 + 9577888*x^2 - 13201088*x - 8134226)
 
gp: K = bnfinit(x^16 - 8*x^15 + 32*x^14 - 24*x^13 - 1464*x^12 + 9608*x^11 - 33320*x^10 + 41032*x^9 + 547102*x^8 - 2982336*x^7 + 5805312*x^6 + 1628176*x^5 - 37227492*x^4 + 50197344*x^3 + 9577888*x^2 - 13201088*x - 8134226, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 32 x^{14} - 24 x^{13} - 1464 x^{12} + 9608 x^{11} - 33320 x^{10} + 41032 x^{9} + 547102 x^{8} - 2982336 x^{7} + 5805312 x^{6} + 1628176 x^{5} - 37227492 x^{4} + 50197344 x^{3} + 9577888 x^{2} - 13201088 x - 8134226 \)

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:  \(136576515395793105198700252102656=2^{56}\cdot 3^{8}\cdot 7^{6}\cdot 1567^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $101.97$
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:  $2, 3, 7, 1567$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{9} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{7} + \frac{1}{3} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{1578556676307} a^{14} - \frac{141211239479}{1578556676307} a^{13} - \frac{145654990642}{1578556676307} a^{12} - \frac{50579487597}{526185558769} a^{11} - \frac{248481658910}{1578556676307} a^{10} - \frac{131688727400}{1578556676307} a^{9} - \frac{57757450259}{526185558769} a^{8} + \frac{543528990214}{1578556676307} a^{7} - \frac{25045620982}{1578556676307} a^{6} + \frac{73122917261}{526185558769} a^{5} - \frac{37874809485}{526185558769} a^{4} + \frac{461088450574}{1578556676307} a^{3} + \frac{567659642383}{1578556676307} a^{2} + \frac{256390013128}{526185558769} a + \frac{251709175161}{526185558769}$, $\frac{1}{12711744580953891006021030950983419310451971155} a^{15} - \frac{3290820058640000000689264409051261}{12711744580953891006021030950983419310451971155} a^{14} - \frac{78018928554664061518063007392543703586340489}{847449638730259400401402063398894620696798077} a^{13} - \frac{119622829838983679550157869624134330537327978}{977826506227222385078540842383339946957843935} a^{12} - \frac{602880725573938931740479672977097549443139719}{4237248193651297002007010316994473103483990385} a^{11} + \frac{390865685687925439850763187954653822638885594}{12711744580953891006021030950983419310451971155} a^{10} + \frac{782353607751304437997847268311905709245656278}{12711744580953891006021030950983419310451971155} a^{9} + \frac{639263622680066911539940870183604275395077436}{4237248193651297002007010316994473103483990385} a^{8} - \frac{356558492882357233737235939160022507487503647}{12711744580953891006021030950983419310451971155} a^{7} - \frac{4674094099869555727027429641485127834735091}{2542348916190778201204206190196683862090394231} a^{6} - \frac{4112170219095868733479922752768683507300348923}{12711744580953891006021030950983419310451971155} a^{5} - \frac{16451107335704112863348630735865694947491287}{59124393399785539562888516051085671211404517} a^{4} - \frac{1496683239870309453499289535464130144222367024}{4237248193651297002007010316994473103483990385} a^{3} + \frac{106298091428537590568614626310729121638110574}{847449638730259400401402063398894620696798077} a^{2} - \frac{5202866477667478165726160306037011808088037557}{12711744580953891006021030950983419310451971155} a - \frac{14225474666010436456623042618939180972469034}{174133487410327274055082615766896154937698235}$

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:  \( 12528400626.5 \) (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

\(\Q(\sqrt{6}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{3}) \), 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 ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$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}$
1567Data not computed