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Properties

Label 16.0.16263137215...6256.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 3^{16}\cdot 7^{8}$
Root discriminant $15.87$
Ramified primes $2, 3, 7$
Class number $1$
Class group Trivial
Galois group $C_2\times S_4$ (as 16T61)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, -24, 56, -76, 112, -164, 170, -148, 139, -92, -4, 56, -29, -8, 14, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 14*x^14 - 8*x^13 - 29*x^12 + 56*x^11 - 4*x^10 - 92*x^9 + 139*x^8 - 148*x^7 + 170*x^6 - 164*x^5 + 112*x^4 - 76*x^3 + 56*x^2 - 24*x + 4)
 
gp: K = bnfinit(x^16 - 6*x^15 + 14*x^14 - 8*x^13 - 29*x^12 + 56*x^11 - 4*x^10 - 92*x^9 + 139*x^8 - 148*x^7 + 170*x^6 - 164*x^5 + 112*x^4 - 76*x^3 + 56*x^2 - 24*x + 4, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 14 x^{14} - 8 x^{13} - 29 x^{12} + 56 x^{11} - 4 x^{10} - 92 x^{9} + 139 x^{8} - 148 x^{7} + 170 x^{6} - 164 x^{5} + 112 x^{4} - 76 x^{3} + 56 x^{2} - 24 x + 4 \)

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:  \(16263137215612256256=2^{16}\cdot 3^{16}\cdot 7^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.87$
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:  $2, 3, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{14} a^{12} - \frac{3}{7} a^{11} + \frac{3}{7} a^{10} - \frac{1}{7} a^{9} + \frac{5}{14} a^{8} - \frac{2}{7} a^{5} + \frac{3}{14} a^{4} - \frac{2}{7} a^{3} + \frac{3}{7} a^{2} + \frac{1}{7} a + \frac{1}{7}$, $\frac{1}{14} a^{13} - \frac{1}{7} a^{11} + \frac{3}{7} a^{10} - \frac{1}{2} a^{9} + \frac{1}{7} a^{8} - \frac{2}{7} a^{6} - \frac{1}{2} a^{5} - \frac{2}{7} a^{3} - \frac{2}{7} a^{2} - \frac{1}{7}$, $\frac{1}{28} a^{14} - \frac{3}{14} a^{11} + \frac{5}{28} a^{10} - \frac{1}{14} a^{9} + \frac{5}{14} a^{8} + \frac{5}{14} a^{7} - \frac{1}{4} a^{6} + \frac{3}{14} a^{5} - \frac{3}{7} a^{4} + \frac{1}{14} a^{3} - \frac{1}{14} a^{2} - \frac{3}{7} a + \frac{1}{7}$, $\frac{1}{1126300} a^{15} - \frac{3387}{1126300} a^{14} - \frac{3166}{281575} a^{13} + \frac{4394}{281575} a^{12} + \frac{53883}{225260} a^{11} + \frac{365091}{1126300} a^{10} + \frac{1728}{11263} a^{9} - \frac{205621}{563150} a^{8} - \frac{329809}{1126300} a^{7} + \frac{288431}{1126300} a^{6} + \frac{236167}{563150} a^{5} + \frac{12883}{281575} a^{4} + \frac{25391}{56315} a^{3} + \frac{132877}{563150} a^{2} - \frac{112717}{281575} a - \frac{116154}{281575}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( \frac{387}{700} a^{15} - \frac{2169}{700} a^{14} + \frac{1133}{175} a^{13} - \frac{297}{175} a^{12} - \frac{2379}{140} a^{11} + \frac{16917}{700} a^{10} + \frac{58}{7} a^{9} - \frac{17027}{350} a^{8} + \frac{39917}{700} a^{7} - \frac{5629}{100} a^{6} + \frac{24029}{350} a^{5} - \frac{10754}{175} a^{4} + \frac{181}{5} a^{3} - \frac{9301}{350} a^{2} + \frac{3471}{175} a - \frac{823}{175} \) (order $6$)
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:  \( 4047.99234926 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times S_4$ (as 16T61):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 48
The 10 conjugacy class representatives for $C_2\times S_4$
Character table for $C_2\times S_4$

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{21}) \), 4.0.3024.1, \(\Q(\sqrt{-3}, \sqrt{-7})\), 8.0.4032758016.2, 8.0.82301184.1, 8.0.448084224.9

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

Sibling fields

Degree 6 siblings: 6.4.6858432.1, 6.4.16003008.3
Degree 8 siblings: 8.0.448084224.9, 8.0.82301184.1
Degree 12 siblings: 12.4.439022168653824.1, 12.0.47038089498624.1, 12.0.2304866385432576.1, 12.8.2304866385432576.1, 12.0.2304866385432576.5, 12.0.256096265048064.2
Degree 24 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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$2.8.8.11$x^{8} + 20 x^{2} + 4$$4$$2$$8$$S_4$$[4/3, 4/3]_{3}^{2}$
2.8.8.11$x^{8} + 20 x^{2} + 4$$4$$2$$8$$S_4$$[4/3, 4/3]_{3}^{2}$
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.12.14.11$x^{12} + 6 x^{11} + 21 x^{10} + 36 x^{9} + 30 x^{8} + 36 x^{7} + 3 x^{6} + 36 x^{5} + 27 x^{4} - 9 x^{2} + 36$$6$$2$$14$$D_6$$[3/2]_{2}^{2}$
$7$7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$