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Properties

Label 16.0.32200233975...0000.3
Degree $16$
Signature $[0, 8]$
Discriminant $2^{8}\cdot 5^{12}\cdot 61^{6}$
Root discriminant $22.09$
Ramified primes $2, 5, 61$
Class number $2$
Class group $[2]$
Galois group 16T852

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16, 64, 160, 280, 392, 488, 430, 190, -41, -146, -82, 24, 33, 9, -3, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 3*x^14 + 9*x^13 + 33*x^12 + 24*x^11 - 82*x^10 - 146*x^9 - 41*x^8 + 190*x^7 + 430*x^6 + 488*x^5 + 392*x^4 + 280*x^3 + 160*x^2 + 64*x + 16)
 
gp: K = bnfinit(x^16 - 4*x^15 - 3*x^14 + 9*x^13 + 33*x^12 + 24*x^11 - 82*x^10 - 146*x^9 - 41*x^8 + 190*x^7 + 430*x^6 + 488*x^5 + 392*x^4 + 280*x^3 + 160*x^2 + 64*x + 16, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 3 x^{14} + 9 x^{13} + 33 x^{12} + 24 x^{11} - 82 x^{10} - 146 x^{9} - 41 x^{8} + 190 x^{7} + 430 x^{6} + 488 x^{5} + 392 x^{4} + 280 x^{3} + 160 x^{2} + 64 x + 16 \)

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:  \(3220023397562500000000=2^{8}\cdot 5^{12}\cdot 61^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.09$
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:  $2, 5, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{10} + \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{2} a^{7} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{14} - \frac{1}{8} a^{12} + \frac{1}{8} a^{11} - \frac{1}{8} a^{10} + \frac{1}{4} a^{9} - \frac{1}{2} a^{8} + \frac{1}{4} a^{7} + \frac{3}{8} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4545668888} a^{15} + \frac{139178377}{2272834444} a^{14} + \frac{523289389}{4545668888} a^{13} + \frac{120153817}{4545668888} a^{12} - \frac{104772289}{4545668888} a^{11} - \frac{163269797}{1136417222} a^{10} + \frac{207523030}{568208611} a^{9} + \frac{241164833}{568208611} a^{8} + \frac{1755649479}{4545668888} a^{7} - \frac{122269869}{568208611} a^{6} + \frac{1108180007}{2272834444} a^{5} - \frac{702349335}{2272834444} a^{4} + \frac{323476563}{1136417222} a^{3} - \frac{218607963}{568208611} a^{2} + \frac{121122153}{568208611} a - \frac{58583529}{568208611}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( \frac{2091476009}{4545668888} a^{15} - \frac{6302035257}{2272834444} a^{14} + \frac{16694778081}{4545668888} a^{13} - \frac{18910305}{4545668888} a^{12} + \frac{51485043347}{4545668888} a^{11} - \frac{28732915045}{2272834444} a^{10} - \frac{59728454723}{2272834444} a^{9} - \frac{2247027465}{2272834444} a^{8} + \frac{80820221195}{4545668888} a^{7} + \frac{33872116462}{568208611} a^{6} + \frac{128153296015}{2272834444} a^{5} + \frac{18778031300}{568208611} a^{4} + \frac{34616016545}{1136417222} a^{3} + \frac{11170199279}{1136417222} a^{2} + \frac{2019603138}{568208611} a - \frac{893301391}{568208611} \) (order $10$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 35852.7433885 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T852:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 41 conjugacy class representatives for t16n852
Character table for t16n852 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 8.0.58140625.1

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 ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$2$2.8.8.4$x^{8} + 2 x^{7} + 2 x^{6} + 8 x^{3} + 48$$2$$4$$8$$C_8$$[2]^{4}$
2.8.0.1$x^{8} + x^{4} + x^{3} + x + 1$$1$$8$$0$$C_8$$[\ ]^{8}$
5Data not computed
$61$$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
61.4.3.2$x^{4} - 244$$4$$1$$3$$C_4$$[\ ]_{4}$
61.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
61.4.3.3$x^{4} + 122$$4$$1$$3$$C_4$$[\ ]_{4}$