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Properties

Label 16.4.51952739971...6889.1
Degree $16$
Signature $[4, 6]$
Discriminant $7^{8}\cdot 37^{14}$
Root discriminant $62.33$
Ramified primes $7, 37$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^3.C_4$ (as 16T41)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-59643, 368010, -264567, -213083, 115074, 8343, -30968, 18398, 3020, -5670, 1629, -166, -33, 0, 20, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 20*x^14 - 33*x^12 - 166*x^11 + 1629*x^10 - 5670*x^9 + 3020*x^8 + 18398*x^7 - 30968*x^6 + 8343*x^5 + 115074*x^4 - 213083*x^3 - 264567*x^2 + 368010*x - 59643)
 
gp: K = bnfinit(x^16 - 8*x^15 + 20*x^14 - 33*x^12 - 166*x^11 + 1629*x^10 - 5670*x^9 + 3020*x^8 + 18398*x^7 - 30968*x^6 + 8343*x^5 + 115074*x^4 - 213083*x^3 - 264567*x^2 + 368010*x - 59643, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 20 x^{14} - 33 x^{12} - 166 x^{11} + 1629 x^{10} - 5670 x^{9} + 3020 x^{8} + 18398 x^{7} - 30968 x^{6} + 8343 x^{5} + 115074 x^{4} - 213083 x^{3} - 264567 x^{2} + 368010 x - 59643 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(51952739971213319841055446889=7^{8}\cdot 37^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $62.33$
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:  $7, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{5}$, $\frac{1}{1066722485934169557} a^{14} - \frac{7}{1066722485934169557} a^{13} + \frac{2999464197364744}{1066722485934169557} a^{12} - \frac{17996785184188373}{1066722485934169557} a^{11} - \frac{64974481750028567}{1066722485934169557} a^{10} + \frac{14918753069682915}{118524720659352173} a^{9} - \frac{118615164572590871}{355574161978056519} a^{8} + \frac{139934890694047075}{355574161978056519} a^{7} - \frac{393984551346168862}{1066722485934169557} a^{6} - \frac{165502686749403446}{1066722485934169557} a^{5} + \frac{343069682381347994}{1066722485934169557} a^{4} - \frac{288049309851853852}{1066722485934169557} a^{3} + \frac{122629917934756577}{1066722485934169557} a^{2} + \frac{87860264792219648}{355574161978056519} a + \frac{1101099710332127}{2521802567220259}$, $\frac{1}{17907070371376904353359} a^{15} + \frac{8386}{17907070371376904353359} a^{14} - \frac{108802694101087988821}{17907070371376904353359} a^{13} + \frac{244131171465884174770}{5969023457125634784453} a^{12} + \frac{347042724193260155017}{5969023457125634784453} a^{11} + \frac{40789662389474526716}{17907070371376904353359} a^{10} - \frac{724625788783133343872}{5969023457125634784453} a^{9} + \frac{634414475692685203220}{1989674485708544928151} a^{8} - \frac{5229431368855796082622}{17907070371376904353359} a^{7} - \frac{6803594151027352470058}{17907070371376904353359} a^{6} - \frac{1780208132543202001703}{17907070371376904353359} a^{5} - \frac{2538440007234519413992}{5969023457125634784453} a^{4} + \frac{551148117690446676608}{1989674485708544928151} a^{3} + \frac{8161247448814668211288}{17907070371376904353359} a^{2} + \frac{511065716280234332968}{5969023457125634784453} a - \frac{10113304834597945914}{42333499695926487833}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $9$
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:  \( 870646429.869 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_4$ (as 16T41):

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 $C_2^3.C_4$
Character table for $C_2^3.C_4$

Intermediate fields

\(\Q(\sqrt{37}) \), 4.4.2481997.1, 4.2.354571.1, 4.2.9583.1, 8.4.227931436996333.1 x2, 8.4.6160309108009.1

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 8 siblings: data not computed
Degree 16 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$7$7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
37Data not computed