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Properties

Label 16.16.2039628312...0625.1
Degree $16$
Signature $[16, 0]$
Discriminant $5^{14}\cdot 109^{14}$
Root discriminant $247.94$
Ramified primes $5, 109$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $OD_{16}.C_2$ (as 16T40)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![625, -6875, -2000, 309625, -1557975, 3089875, -2098230, -984145, 1550036, 50243, -337032, -28129, 13545, 901, -202, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 - 202*x^14 + 901*x^13 + 13545*x^12 - 28129*x^11 - 337032*x^10 + 50243*x^9 + 1550036*x^8 - 984145*x^7 - 2098230*x^6 + 3089875*x^5 - 1557975*x^4 + 309625*x^3 - 2000*x^2 - 6875*x + 625)
 
gp: K = bnfinit(x^16 - 7*x^15 - 202*x^14 + 901*x^13 + 13545*x^12 - 28129*x^11 - 337032*x^10 + 50243*x^9 + 1550036*x^8 - 984145*x^7 - 2098230*x^6 + 3089875*x^5 - 1557975*x^4 + 309625*x^3 - 2000*x^2 - 6875*x + 625, 1)
 

Normalized defining polynomial

\( x^{16} - 7 x^{15} - 202 x^{14} + 901 x^{13} + 13545 x^{12} - 28129 x^{11} - 337032 x^{10} + 50243 x^{9} + 1550036 x^{8} - 984145 x^{7} - 2098230 x^{6} + 3089875 x^{5} - 1557975 x^{4} + 309625 x^{3} - 2000 x^{2} - 6875 x + 625 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(203962831252212947906041415777587890625=5^{14}\cdot 109^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $247.94$
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:  $5, 109$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{20} a^{10} + \frac{3}{20} a^{9} - \frac{7}{20} a^{8} + \frac{3}{10} a^{7} - \frac{1}{4} a^{6} - \frac{1}{5} a^{5} - \frac{7}{20} a^{4} - \frac{1}{10} a^{3} - \frac{9}{20} a^{2} - \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{40} a^{11} + \frac{1}{10} a^{9} + \frac{7}{40} a^{8} - \frac{3}{40} a^{7} - \frac{9}{40} a^{6} + \frac{1}{8} a^{5} - \frac{1}{40} a^{4} + \frac{17}{40} a^{3} + \frac{1}{20} a^{2} - \frac{1}{2} a + \frac{1}{8}$, $\frac{1}{400} a^{12} + \frac{3}{400} a^{11} + \frac{1}{50} a^{10} + \frac{71}{400} a^{9} - \frac{13}{40} a^{8} + \frac{63}{200} a^{7} + \frac{39}{200} a^{6} - \frac{61}{200} a^{5} - \frac{47}{200} a^{4} + \frac{1}{80} a^{3} - \frac{1}{8} a^{2} + \frac{5}{16} a + \frac{3}{16}$, $\frac{1}{800} a^{13} - \frac{1}{800} a^{11} + \frac{7}{800} a^{10} - \frac{63}{800} a^{9} + \frac{99}{200} a^{8} - \frac{7}{40} a^{7} + \frac{61}{200} a^{6} + \frac{1}{25} a^{5} - \frac{233}{800} a^{4} - \frac{77}{160} a^{3} - \frac{33}{160} a^{2} + \frac{3}{8} a - \frac{1}{32}$, $\frac{1}{232000} a^{14} + \frac{133}{232000} a^{13} + \frac{7}{8000} a^{12} + \frac{1063}{116000} a^{11} + \frac{37}{11600} a^{10} - \frac{27419}{232000} a^{9} - \frac{337}{14500} a^{8} - \frac{4819}{29000} a^{7} + \frac{26609}{58000} a^{6} - \frac{769}{9280} a^{5} - \frac{3427}{23200} a^{4} + \frac{1157}{4640} a^{3} - \frac{2717}{9280} a^{2} + \frac{459}{1856} a - \frac{37}{1856}$, $\frac{1}{7099839869065517840144000} a^{15} - \frac{5065777411217156273}{3549919934532758920072000} a^{14} + \frac{500974288983487021637}{887479983633189730018000} a^{13} - \frac{7793807504220175720431}{7099839869065517840144000} a^{12} - \frac{22693398852722957154487}{3549919934532758920072000} a^{11} - \frac{153312569622715957624439}{7099839869065517840144000} a^{10} - \frac{1394061146209965543825411}{7099839869065517840144000} a^{9} + \frac{47793746025438919556027}{887479983633189730018000} a^{8} - \frac{685831970724323526814069}{1774959967266379460036000} a^{7} - \frac{1509837137026084649733429}{7099839869065517840144000} a^{6} - \frac{92253408709232940714691}{1419967973813103568028800} a^{5} - \frac{1535007205743574809413}{12241103222526754896800} a^{4} + \frac{11699508162713623961257}{56798718952524142721152} a^{3} + \frac{8469327051040373318673}{141996797381310356802880} a^{2} + \frac{10312990029278664127855}{28399359476262071360576} a + \frac{24077497301802371423347}{56798718952524142721152}$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  $15$
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:  \( 34323318670400 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$OD_{16}.C_2$ (as 16T40):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $OD_{16}.C_2$
Character table for $OD_{16}.C_2$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{545}) \), \(\Q(\sqrt{109}) \), 4.4.161878625.1, 4.4.161878625.2, \(\Q(\sqrt{5}, \sqrt{109})\), 8.8.26204689231890625.1

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

Sibling fields

Galois closure: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$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}$
$109$109.8.7.1$x^{8} - 109$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
109.8.7.1$x^{8} - 109$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$