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Properties

Label 16.8.26997420280...5529.2
Degree $16$
Signature $[8, 4]$
Discriminant $53^{14}\cdot 89^{14}$
Root discriminant $1638.54$
Ramified primes $53, 89$
Class number $36$ (GRH)
Class group $[3, 12]$ (GRH)
Galois group $C_8.C_4$ (as 16T49)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1766411567, 2572687806, 17514522367, 20571597915, 9655116950, 1360249204, -391932779, -43358097, 83871983, 30799334, 2379681, -604947, -93130, 147, -37, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 37*x^14 + 147*x^13 - 93130*x^12 - 604947*x^11 + 2379681*x^10 + 30799334*x^9 + 83871983*x^8 - 43358097*x^7 - 391932779*x^6 + 1360249204*x^5 + 9655116950*x^4 + 20571597915*x^3 + 17514522367*x^2 + 2572687806*x - 1766411567)
 
gp: K = bnfinit(x^16 - 4*x^15 - 37*x^14 + 147*x^13 - 93130*x^12 - 604947*x^11 + 2379681*x^10 + 30799334*x^9 + 83871983*x^8 - 43358097*x^7 - 391932779*x^6 + 1360249204*x^5 + 9655116950*x^4 + 20571597915*x^3 + 17514522367*x^2 + 2572687806*x - 1766411567, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 37 x^{14} + 147 x^{13} - 93130 x^{12} - 604947 x^{11} + 2379681 x^{10} + 30799334 x^{9} + 83871983 x^{8} - 43358097 x^{7} - 391932779 x^{6} + 1360249204 x^{5} + 9655116950 x^{4} + 20571597915 x^{3} + 17514522367 x^{2} + 2572687806 x - 1766411567 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2699742028062005762663832012742665627137111010645529=53^{14}\cdot 89^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1638.54$
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:  $53, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{89} a^{8} - \frac{2}{89} a^{7} + \frac{24}{89} a^{6} - \frac{12}{89} a^{5} + \frac{26}{89} a^{4} - \frac{23}{89} a^{3} + \frac{14}{89} a^{2} - \frac{1}{89} a - \frac{25}{89}$, $\frac{1}{89} a^{9} + \frac{20}{89} a^{7} + \frac{36}{89} a^{6} + \frac{2}{89} a^{5} + \frac{29}{89} a^{4} - \frac{32}{89} a^{3} + \frac{27}{89} a^{2} - \frac{27}{89} a + \frac{39}{89}$, $\frac{1}{89} a^{10} - \frac{13}{89} a^{7} - \frac{33}{89} a^{6} + \frac{2}{89} a^{5} - \frac{18}{89} a^{4} + \frac{42}{89} a^{3} - \frac{40}{89} a^{2} - \frac{30}{89} a - \frac{34}{89}$, $\frac{1}{89} a^{11} + \frac{30}{89} a^{7} - \frac{42}{89} a^{6} + \frac{4}{89} a^{5} + \frac{24}{89} a^{4} + \frac{17}{89} a^{3} - \frac{26}{89} a^{2} + \frac{42}{89} a + \frac{31}{89}$, $\frac{1}{178} a^{12} - \frac{1}{178} a^{10} - \frac{1}{178} a^{8} - \frac{28}{89} a^{7} - \frac{42}{89} a^{6} - \frac{51}{178} a^{5} + \frac{15}{89} a^{4} - \frac{67}{178} a^{3} + \frac{2}{89} a^{2} + \frac{3}{178} a - \frac{81}{178}$, $\frac{1}{17266} a^{13} - \frac{19}{17266} a^{12} + \frac{61}{17266} a^{11} - \frac{61}{17266} a^{10} + \frac{19}{17266} a^{9} + \frac{33}{17266} a^{8} + \frac{379}{8633} a^{7} - \frac{6343}{17266} a^{6} + \frac{1355}{17266} a^{5} + \frac{597}{17266} a^{4} + \frac{5265}{17266} a^{3} + \frac{543}{17266} a^{2} + \frac{971}{8633} a + \frac{4143}{17266}$, $\frac{1}{2743722794} a^{14} + \frac{36195}{1371861397} a^{13} - \frac{1957682}{1371861397} a^{12} + \frac{5726052}{1371861397} a^{11} + \frac{993020}{1371861397} a^{10} + \frac{5432754}{1371861397} a^{9} + \frac{8010823}{2743722794} a^{8} + \frac{713120903}{2743722794} a^{7} - \frac{427062843}{1371861397} a^{6} + \frac{600187340}{1371861397} a^{5} - \frac{106150614}{1371861397} a^{4} + \frac{683368040}{1371861397} a^{3} - \frac{1213252805}{2743722794} a^{2} + \frac{474225209}{2743722794} a + \frac{1135494167}{2743722794}$, $\frac{1}{4027316604912136645729796420883320090687614934131761166} a^{15} - \frac{329832472491007741269130880973375338983259624}{2013658302456068322864898210441660045343807467065880583} a^{14} + \frac{406287518116006731791061125906793603858267655856}{22625374184899644077133687757771461183638286146807647} a^{13} + \frac{3450993721082275067986029046856281047132882176086855}{2013658302456068322864898210441660045343807467065880583} a^{12} + \frac{1978104307506392561148774473352918444696394299755058}{2013658302456068322864898210441660045343807467065880583} a^{11} - \frac{4579114168628773344388710162359638609605851101377400}{2013658302456068322864898210441660045343807467065880583} a^{10} - \frac{11431906177838617633285616418763047381718640929118793}{4027316604912136645729796420883320090687614934131761166} a^{9} - \frac{13940079568665434884503139175978570385780247134847605}{4027316604912136645729796420883320090687614934131761166} a^{8} + \frac{647611598845881047556663522742812976909580031352196799}{2013658302456068322864898210441660045343807467065880583} a^{7} + \frac{43145583829694617368765419340246392955288730481489326}{2013658302456068322864898210441660045343807467065880583} a^{6} - \frac{836197037073126385528683562985579374377525851278887639}{2013658302456068322864898210441660045343807467065880583} a^{5} - \frac{960229949639894801963199973613051255824983957145569814}{2013658302456068322864898210441660045343807467065880583} a^{4} - \frac{1259542898886590589170119443647640859748250550603632651}{4027316604912136645729796420883320090687614934131761166} a^{3} + \frac{1938011236023527287180037097030402366792348942369143707}{4027316604912136645729796420883320090687614934131761166} a^{2} + \frac{1403860431511103080883278501553297752694840957929499397}{4027316604912136645729796420883320090687614934131761166} a + \frac{163253429706134818914854119654545141626992592905542534}{2013658302456068322864898210441660045343807467065880583}$

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

Class group and class number

$C_{3}\times C_{12}$, which has order $36$ (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:  $11$
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:  \( 1147325462190000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_8.C_4$ (as 16T49):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_8.C_4$
Character table for $C_8.C_4$

Intermediate fields

\(\Q(\sqrt{4717}) \), \(\Q(\sqrt{53}) \), \(\Q(\sqrt{89}) \), \(\Q(\sqrt{53}, \sqrt{89})\), 8.8.11015272807216227454969.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$53$53.8.7.1$x^{8} - 53$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
53.8.7.1$x^{8} - 53$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$89$89.8.7.4$x^{8} - 64881$$8$$1$$7$$C_8$$[\ ]_{8}$
89.8.7.4$x^{8} - 64881$$8$$1$$7$$C_8$$[\ ]_{8}$