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Properties

Label 16.0.70292060829...5616.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 18097^{4}$
Root discriminant $23.20$
Ramified primes $2, 18097$
Class number $2$
Class group $[2]$
Galois group $C_2\times C_2^3:S_4$ (as 16T747)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![25, -48, 178, -452, 833, -1068, 852, -216, -362, 512, -318, 76, 48, -56, 28, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 28*x^14 - 56*x^13 + 48*x^12 + 76*x^11 - 318*x^10 + 512*x^9 - 362*x^8 - 216*x^7 + 852*x^6 - 1068*x^5 + 833*x^4 - 452*x^3 + 178*x^2 - 48*x + 25)
 
gp: K = bnfinit(x^16 - 8*x^15 + 28*x^14 - 56*x^13 + 48*x^12 + 76*x^11 - 318*x^10 + 512*x^9 - 362*x^8 - 216*x^7 + 852*x^6 - 1068*x^5 + 833*x^4 - 452*x^3 + 178*x^2 - 48*x + 25, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 28 x^{14} - 56 x^{13} + 48 x^{12} + 76 x^{11} - 318 x^{10} + 512 x^{9} - 362 x^{8} - 216 x^{7} + 852 x^{6} - 1068 x^{5} + 833 x^{4} - 452 x^{3} + 178 x^{2} - 48 x + 25 \)

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:  \(7029206082976827375616=2^{16}\cdot 18097^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.20$
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:  $2, 18097$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{4} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4}$, $\frac{1}{4} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a$, $\frac{1}{4} a^{10} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{11} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{2} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{19532} a^{14} - \frac{7}{19532} a^{13} - \frac{1875}{19532} a^{12} + \frac{1575}{19532} a^{11} + \frac{7}{76} a^{10} - \frac{203}{4883} a^{9} + \frac{1183}{19532} a^{8} + \frac{7985}{19532} a^{7} + \frac{1488}{4883} a^{6} - \frac{3145}{9766} a^{5} - \frac{467}{9766} a^{4} - \frac{5331}{19532} a^{3} - \frac{6163}{19532} a^{2} + \frac{1950}{4883} a + \frac{781}{4883}$, $\frac{1}{21582860} a^{15} + \frac{109}{4316572} a^{14} - \frac{425677}{21582860} a^{13} + \frac{65613}{830110} a^{12} - \frac{672127}{5395715} a^{11} + \frac{598661}{10791430} a^{10} + \frac{784819}{10791430} a^{9} + \frac{391089}{5395715} a^{8} - \frac{2925197}{10791430} a^{7} + \frac{7532307}{21582860} a^{6} + \frac{10389823}{21582860} a^{5} + \frac{3877893}{10791430} a^{4} + \frac{3994751}{21582860} a^{3} - \frac{88683}{1660220} a^{2} + \frac{1764681}{21582860} a - \frac{232281}{4316572}$

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:  \( -\frac{237}{9766} a^{14} + \frac{1659}{9766} a^{13} - \frac{4861}{9766} a^{12} + \frac{7599}{9766} a^{11} - \frac{3}{19} a^{10} - \frac{11204}{4883} a^{9} + \frac{51673}{9766} a^{8} - \frac{56437}{9766} a^{7} + \frac{2723}{4883} a^{6} + \frac{37330}{4883} a^{5} - \frac{115569}{9766} a^{4} + \frac{91527}{9766} a^{3} - \frac{28991}{4883} a^{2} + \frac{13236}{4883} a - \frac{3969}{4883} \) (order $4$)
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:  \( 49808.1821928 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^3:S_4$ (as 16T747):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 384
The 26 conjugacy class representatives for $C_2\times C_2^3:S_4$
Character table for $C_2\times C_2^3:S_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), 4.4.289552.2, 8.0.83840360704.2, 8.8.83840360704.1, 8.0.83840360704.4

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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
18097Data not computed