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Properties

Label 16.0.24722989956...4209.3
Degree $16$
Signature $[0, 8]$
Discriminant $17^{14}\cdot 59^{8}$
Root discriminant $91.64$
Ramified primes $17, 59$
Class number $1252$ (GRH)
Class group $[2, 626]$ (GRH)
Galois group $C_2^2 : C_8$ (as 16T24)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4936153, 3924023, 5459082, 2574663, 836728, -50668, -50075, -64314, 18564, 5577, 796, -1060, 411, -108, 30, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 30*x^14 - 108*x^13 + 411*x^12 - 1060*x^11 + 796*x^10 + 5577*x^9 + 18564*x^8 - 64314*x^7 - 50075*x^6 - 50668*x^5 + 836728*x^4 + 2574663*x^3 + 5459082*x^2 + 3924023*x + 4936153)
 
gp: K = bnfinit(x^16 - 6*x^15 + 30*x^14 - 108*x^13 + 411*x^12 - 1060*x^11 + 796*x^10 + 5577*x^9 + 18564*x^8 - 64314*x^7 - 50075*x^6 - 50668*x^5 + 836728*x^4 + 2574663*x^3 + 5459082*x^2 + 3924023*x + 4936153, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 30 x^{14} - 108 x^{13} + 411 x^{12} - 1060 x^{11} + 796 x^{10} + 5577 x^{9} + 18564 x^{8} - 64314 x^{7} - 50075 x^{6} - 50668 x^{5} + 836728 x^{4} + 2574663 x^{3} + 5459082 x^{2} + 3924023 x + 4936153 \)

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:  \(24722989956581301387479701814209=17^{14}\cdot 59^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $91.64$
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:  $17, 59$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{26} a^{14} - \frac{2}{13} a^{13} - \frac{1}{13} a^{12} - \frac{3}{26} a^{11} - \frac{1}{13} a^{10} - \frac{2}{13} a^{9} - \frac{9}{26} a^{8} + \frac{4}{13} a^{6} - \frac{1}{2} a^{5} + \frac{2}{13} a^{4} - \frac{6}{13} a^{3} - \frac{7}{26} a^{2} - \frac{6}{13} a + \frac{3}{13}$, $\frac{1}{134883544356867694290703551727695838910793158546} a^{15} + \frac{238803146175798762839794773638983165176985659}{134883544356867694290703551727695838910793158546} a^{14} - \frac{9255426350370241252799964715206762991424950808}{67441772178433847145351775863847919455396579273} a^{13} - \frac{13055652136887722676938572015695483921844699612}{67441772178433847145351775863847919455396579273} a^{12} + \frac{4464950813577111334712607009396008733865233785}{10375657258220591868515657825207372223907166042} a^{11} + \frac{28634797134644514397771821024861247971737665800}{67441772178433847145351775863847919455396579273} a^{10} - \frac{7218539245007196031248958801983764199529908060}{67441772178433847145351775863847919455396579273} a^{9} + \frac{24967212508780205429148078589353666749110763201}{134883544356867694290703551727695838910793158546} a^{8} + \frac{7857267902556709424000192677391463152407043023}{67441772178433847145351775863847919455396579273} a^{7} + \frac{4809494180723679491760526750457681908621087334}{67441772178433847145351775863847919455396579273} a^{6} + \frac{54515836300095634062955917835937448529366485243}{134883544356867694290703551727695838910793158546} a^{5} + \frac{32241027960306865841487351875236805345562285940}{67441772178433847145351775863847919455396579273} a^{4} - \frac{373589619973311972691344147333407213447229538}{757772721105998282532042425436493477026927857} a^{3} - \frac{66280565334047400869222837663875638662183826847}{134883544356867694290703551727695838910793158546} a^{2} - \frac{27930261890873025704981211784291676824662809007}{67441772178433847145351775863847919455396579273} a + \frac{34032761470618324229001202252329197699445032763}{134883544356867694290703551727695838910793158546}$

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

Class group and class number

$C_{2}\times C_{626}$, which has order $1252$ (assuming GRH)

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:  \( -1 \) (order $2$)
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1288346.89197 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:C_8$ (as 16T24):

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 $C_2^2 : C_8$
Character table for $C_2^2 : C_8$

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, 4.2.289867.1, 4.2.17051.1, 8.4.1428388920713.1, 8.0.4972221833001953.2, 8.4.84022877689.1

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 sibling: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{8}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ R

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
17Data not computed
$59$59.4.2.2$x^{4} - 59 x^{2} + 6962$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
59.4.2.2$x^{4} - 59 x^{2} + 6962$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
59.8.4.1$x^{8} + 97468 x^{4} - 205379 x^{2} + 2375002756$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$