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Properties

Label 16.0.85759734472...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 5^{8}\cdot 29^{4}\cdot 41^{6}\cdot 79^{4}$
Root discriminant $176.13$
Ramified primes $2, 5, 29, 41, 79$
Class number $64$ (GRH)
Class group $[2, 2, 16]$ (GRH)
Galois group 16T839

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![167182084, -326150416, 428908704, -335855668, 157876202, -46015620, 10793082, -3746254, 1449261, -327928, 29018, 36, 979, -290, 18, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 18*x^14 - 290*x^13 + 979*x^12 + 36*x^11 + 29018*x^10 - 327928*x^9 + 1449261*x^8 - 3746254*x^7 + 10793082*x^6 - 46015620*x^5 + 157876202*x^4 - 335855668*x^3 + 428908704*x^2 - 326150416*x + 167182084)
 
gp: K = bnfinit(x^16 + 18*x^14 - 290*x^13 + 979*x^12 + 36*x^11 + 29018*x^10 - 327928*x^9 + 1449261*x^8 - 3746254*x^7 + 10793082*x^6 - 46015620*x^5 + 157876202*x^4 - 335855668*x^3 + 428908704*x^2 - 326150416*x + 167182084, 1)
 

Normalized defining polynomial

\( x^{16} + 18 x^{14} - 290 x^{13} + 979 x^{12} + 36 x^{11} + 29018 x^{10} - 327928 x^{9} + 1449261 x^{8} - 3746254 x^{7} + 10793082 x^{6} - 46015620 x^{5} + 157876202 x^{4} - 335855668 x^{3} + 428908704 x^{2} - 326150416 x + 167182084 \)

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:  \(857597344729811353117661593600000000=2^{24}\cdot 5^{8}\cdot 29^{4}\cdot 41^{6}\cdot 79^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $176.13$
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:  $2, 5, 29, 41, 79$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{10} - \frac{1}{2} a^{6}$, $\frac{1}{47421129430686521892213373743170371197435571465859071007918} a^{15} + \frac{221098768427934825980794206463092123357937414076064167098}{23710564715343260946106686871585185598717785732929535503959} a^{14} + \frac{2097964070085304273583334490908471647813204608098057298883}{23710564715343260946106686871585185598717785732929535503959} a^{13} + \frac{3437762396620904035090888455989928443978955027014881601777}{47421129430686521892213373743170371197435571465859071007918} a^{12} + \frac{14600486541606654420148867196387522510002452720482831519837}{47421129430686521892213373743170371197435571465859071007918} a^{11} + \frac{11046579111600026441726578273566582448824014276102496683478}{23710564715343260946106686871585185598717785732929535503959} a^{10} - \frac{1266308598807794432534005904287888361794212715800176688408}{23710564715343260946106686871585185598717785732929535503959} a^{9} - \frac{17971986617950179357018157487636129542243279942991181759959}{47421129430686521892213373743170371197435571465859071007918} a^{8} - \frac{4327517239078535033028081091629241389748372834497594870169}{47421129430686521892213373743170371197435571465859071007918} a^{7} + \frac{546371976028871585399163378283642873696620938850373336709}{23710564715343260946106686871585185598717785732929535503959} a^{6} + \frac{3978879106326310007466972597894547704088338209984842986865}{23710564715343260946106686871585185598717785732929535503959} a^{5} - \frac{9304178952227158261579523231941291553554621216574341258133}{47421129430686521892213373743170371197435571465859071007918} a^{4} - \frac{1980440417025318651246781540836706268084765297697975181387}{23710564715343260946106686871585185598717785732929535503959} a^{3} - \frac{5597397434605193735069831739062406559305185281658391776908}{23710564715343260946106686871585185598717785732929535503959} a^{2} + \frac{5382315235575299757197135042347849404879438715953350362826}{23710564715343260946106686871585185598717785732929535503959} a - \frac{1652039848072812552642024532829308585565143595653669125711}{23710564715343260946106686871585185598717785732929535503959}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{16}$, which has order $64$ (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:  \( 4924308520.35 \) (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

\(\Q(\sqrt{5}) \), 4.4.31600.1, 4.0.1025.1, 4.0.1295600.1, 8.0.1678579360000.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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$2.8.12.21$x^{8} + 12 x^{6} + 12 x^{4} + 80$$4$$2$$12$$(((C_4 \times C_2): C_2):C_2):C_2$$[2, 2, 2, 2]^{4}$
2.8.12.21$x^{8} + 12 x^{6} + 12 x^{4} + 80$$4$$2$$12$$(((C_4 \times C_2): C_2):C_2):C_2$$[2, 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}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.2.2$x^{4} - 29 x^{2} + 2523$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$41$41.4.2.2$x^{4} - 41 x^{2} + 20172$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
$79$$\Q_{79}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{79}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{79}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{79}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{79}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{79}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{79}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{79}$$x + 2$$1$$1$$0$Trivial$[\ ]$
79.8.4.1$x^{8} + 37446 x^{4} - 493039 x^{2} + 350550729$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$