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Properties

Label 16.0.22153345600...0000.3
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 5^{12}\cdot 61^{4}$
Root discriminant $18.69$
Ramified primes $2, 5, 61$
Class number $2$
Class group $[2]$
Galois group $C_2^4.C_2^3$ (as 16T268)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![405, 0, -1170, 480, 1316, -990, -268, 886, -296, -116, 214, -58, 7, 8, 4, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 4*x^14 + 8*x^13 + 7*x^12 - 58*x^11 + 214*x^10 - 116*x^9 - 296*x^8 + 886*x^7 - 268*x^6 - 990*x^5 + 1316*x^4 + 480*x^3 - 1170*x^2 + 405)
 
gp: K = bnfinit(x^16 - 2*x^15 + 4*x^14 + 8*x^13 + 7*x^12 - 58*x^11 + 214*x^10 - 116*x^9 - 296*x^8 + 886*x^7 - 268*x^6 - 990*x^5 + 1316*x^4 + 480*x^3 - 1170*x^2 + 405, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 4 x^{14} + 8 x^{13} + 7 x^{12} - 58 x^{11} + 214 x^{10} - 116 x^{9} - 296 x^{8} + 886 x^{7} - 268 x^{6} - 990 x^{5} + 1316 x^{4} + 480 x^{3} - 1170 x^{2} + 405 \)

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:  \(221533456000000000000=2^{16}\cdot 5^{12}\cdot 61^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.69$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{6} a^{11} - \frac{1}{6} a^{10} + \frac{1}{6} a^{9} - \frac{1}{6} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{2} a^{3} - \frac{1}{6} a^{2} - \frac{1}{3} a$, $\frac{1}{30} a^{12} + \frac{1}{30} a^{11} - \frac{7}{30} a^{10} - \frac{1}{6} a^{9} - \frac{2}{15} a^{8} + \frac{1}{3} a^{7} + \frac{1}{15} a^{6} + \frac{2}{5} a^{5} + \frac{11}{30} a^{4} + \frac{1}{6} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{30} a^{13} + \frac{1}{15} a^{11} + \frac{7}{30} a^{10} - \frac{2}{15} a^{9} + \frac{2}{15} a^{8} - \frac{13}{30} a^{7} - \frac{1}{2} a^{6} + \frac{3}{10} a^{5} + \frac{2}{15} a^{4} - \frac{1}{2} a^{3} + \frac{1}{6} a^{2} + \frac{1}{6} a - \frac{1}{2}$, $\frac{1}{18450} a^{14} - \frac{139}{9225} a^{13} + \frac{104}{9225} a^{12} - \frac{614}{9225} a^{11} - \frac{107}{18450} a^{10} + \frac{31}{450} a^{9} + \frac{1193}{9225} a^{8} - \frac{3431}{18450} a^{7} - \frac{2429}{18450} a^{6} - \frac{3583}{9225} a^{5} - \frac{6631}{18450} a^{4} + \frac{59}{123} a^{3} - \frac{559}{1845} a^{2} + \frac{223}{1230} a - \frac{92}{205}$, $\frac{1}{3805580449350} a^{15} - \frac{1079983}{76111608987} a^{14} - \frac{10233301558}{1902790224675} a^{13} + \frac{57609138461}{3805580449350} a^{12} + \frac{23167414922}{1902790224675} a^{11} + \frac{51912157304}{380558044935} a^{10} - \frac{53727221308}{1902790224675} a^{9} - \frac{419675290084}{1902790224675} a^{8} - \frac{545159984176}{1902790224675} a^{7} - \frac{1254325642973}{3805580449350} a^{6} + \frac{1901113460891}{3805580449350} a^{5} - \frac{177121044371}{1268526816450} a^{4} - \frac{266720238383}{761116089870} a^{3} + \frac{8884663763}{50741072658} a^{2} + \frac{10781266883}{28189484810} a - \frac{9075292379}{28189484810}$

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:  \( \frac{14474275553}{761116089870} a^{15} - \frac{219343330697}{3805580449350} a^{14} + \frac{259900509893}{1902790224675} a^{13} + \frac{40242851429}{3805580449350} a^{12} + \frac{469222733161}{3805580449350} a^{11} - \frac{4643840476231}{3805580449350} a^{10} + \frac{10157410750829}{1902790224675} a^{9} - \frac{29576288846207}{3805580449350} a^{8} + \frac{9326673711997}{3805580449350} a^{7} + \frac{54487819505773}{3805580449350} a^{6} - \frac{77744889706663}{3805580449350} a^{5} + \frac{1653192370457}{634263408225} a^{4} + \frac{1660303014034}{76111608987} a^{3} - \frac{3653524372193}{253705363290} a^{2} - \frac{294901600411}{42284227215} a + \frac{91058151138}{14094742405} \) (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:  \( 29399.1026781 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3$ (as 16T268):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 29 conjugacy class representatives for $C_2^4.C_2^3$
Character table for $C_2^4.C_2^3$ is not computed

Intermediate fields

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

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

Sibling fields

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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}$
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
61Data not computed