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Properties

Label 16.0.82944000000...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 3^{4}\cdot 5^{14}$
Root discriminant $15.22$
Ramified primes $2, 3, 5$
Class number $2$
Class group $[2]$
Galois group $(C_8:C_2):C_2$ (as 16T16)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![181, -184, 778, -698, 1546, -994, 1210, -224, 199, 222, -30, 68, 26, -6, 12, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 12*x^14 - 6*x^13 + 26*x^12 + 68*x^11 - 30*x^10 + 222*x^9 + 199*x^8 - 224*x^7 + 1210*x^6 - 994*x^5 + 1546*x^4 - 698*x^3 + 778*x^2 - 184*x + 181)
 
gp: K = bnfinit(x^16 - 2*x^15 + 12*x^14 - 6*x^13 + 26*x^12 + 68*x^11 - 30*x^10 + 222*x^9 + 199*x^8 - 224*x^7 + 1210*x^6 - 994*x^5 + 1546*x^4 - 698*x^3 + 778*x^2 - 184*x + 181, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 12 x^{14} - 6 x^{13} + 26 x^{12} + 68 x^{11} - 30 x^{10} + 222 x^{9} + 199 x^{8} - 224 x^{7} + 1210 x^{6} - 994 x^{5} + 1546 x^{4} - 698 x^{3} + 778 x^{2} - 184 x + 181 \)

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:  \(8294400000000000000=2^{24}\cdot 3^{4}\cdot 5^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.22$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{10} a^{12} - \frac{1}{5} a^{10} - \frac{1}{5} a^{9} - \frac{1}{10} a^{8} - \frac{1}{5} a^{7} - \frac{2}{5} a^{6} - \frac{2}{5} a^{5} - \frac{3}{10} a^{4} - \frac{2}{5} a^{3} - \frac{2}{5} a^{2} + \frac{2}{5} a - \frac{2}{5}$, $\frac{1}{10} a^{13} - \frac{1}{5} a^{11} - \frac{1}{5} a^{10} - \frac{1}{10} a^{9} - \frac{1}{5} a^{8} - \frac{2}{5} a^{7} - \frac{2}{5} a^{6} - \frac{3}{10} a^{5} - \frac{2}{5} a^{4} - \frac{2}{5} a^{3} + \frac{2}{5} a^{2} - \frac{2}{5} a$, $\frac{1}{610} a^{14} - \frac{5}{122} a^{13} + \frac{12}{305} a^{12} - \frac{147}{610} a^{11} + \frac{41}{305} a^{10} - \frac{2}{305} a^{9} - \frac{11}{61} a^{8} - \frac{181}{610} a^{7} + \frac{84}{305} a^{6} + \frac{16}{305} a^{5} - \frac{146}{305} a^{4} - \frac{31}{122} a^{3} - \frac{73}{610} a^{2} - \frac{21}{610} a + \frac{13}{305}$, $\frac{1}{8128293310} a^{15} - \frac{3165083}{4064146655} a^{14} + \frac{195192953}{8128293310} a^{13} + \frac{184167763}{4064146655} a^{12} + \frac{1820426381}{8128293310} a^{11} - \frac{773554479}{4064146655} a^{10} - \frac{730587197}{4064146655} a^{9} + \frac{290543622}{4064146655} a^{8} - \frac{2150133481}{8128293310} a^{7} + \frac{59496933}{812829331} a^{6} - \frac{393469047}{4064146655} a^{5} - \frac{124069431}{812829331} a^{4} - \frac{1980555646}{4064146655} a^{3} + \frac{163170073}{812829331} a^{2} - \frac{2022331091}{8128293310} a - \frac{589866062}{4064146655}$

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:  \( -\frac{3186199}{8128293310} a^{15} + \frac{37306007}{4064146655} a^{14} - \frac{181010009}{8128293310} a^{13} + \frac{646166013}{8128293310} a^{12} - \frac{28735922}{812829331} a^{11} - \frac{100097744}{4064146655} a^{10} + \frac{4106833889}{8128293310} a^{9} - \frac{2257117042}{4064146655} a^{8} + \frac{836041099}{4064146655} a^{7} + \frac{1559886856}{812829331} a^{6} - \frac{31605400261}{8128293310} a^{5} + \frac{19730441766}{4064146655} a^{4} - \frac{27113715267}{8128293310} a^{3} + \frac{9383553417}{4064146655} a^{2} - \frac{1436236385}{812829331} a + \frac{11316744413}{8128293310} \) (order $20$)
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:  \( 3660.4680193 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$OD_{16}:C_2$ (as 16T16):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 20 conjugacy class representatives for $(C_8:C_2):C_2$
Character table for $(C_8:C_2):C_2$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5}) \), \(\Q(\zeta_{5})\), \(\Q(\zeta_{20})^+\), \(\Q(i, \sqrt{5})\), \(\Q(\zeta_{20})\)

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: data not computed
Degree 16 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 R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
2Data not computed
$3$3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
3.8.0.1$x^{8} - x^{3} + 2$$1$$8$$0$$C_8$$[\ ]^{8}$
$5$5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$