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Properties

Label 16.0.79443753352...7872.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{36}\cdot 3^{2}\cdot 383\cdot 761^{4}$
Root discriminant $41.57$
Ramified primes $2, 3, 383, 761$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1887

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![37289, 14710, 29185, 998, 15476, -1626, 7545, -2894, 4065, -1332, 1072, -448, 317, -102, 37, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 37*x^14 - 102*x^13 + 317*x^12 - 448*x^11 + 1072*x^10 - 1332*x^9 + 4065*x^8 - 2894*x^7 + 7545*x^6 - 1626*x^5 + 15476*x^4 + 998*x^3 + 29185*x^2 + 14710*x + 37289)
 
gp: K = bnfinit(x^16 - 6*x^15 + 37*x^14 - 102*x^13 + 317*x^12 - 448*x^11 + 1072*x^10 - 1332*x^9 + 4065*x^8 - 2894*x^7 + 7545*x^6 - 1626*x^5 + 15476*x^4 + 998*x^3 + 29185*x^2 + 14710*x + 37289, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 37 x^{14} - 102 x^{13} + 317 x^{12} - 448 x^{11} + 1072 x^{10} - 1332 x^{9} + 4065 x^{8} - 2894 x^{7} + 7545 x^{6} - 1626 x^{5} + 15476 x^{4} + 998 x^{3} + 29185 x^{2} + 14710 x + 37289 \)

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:  \(79443753352820378013007872=2^{36}\cdot 3^{2}\cdot 383\cdot 761^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.57$
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:  $2, 3, 383, 761$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{23} a^{14} + \frac{7}{23} a^{13} + \frac{3}{23} a^{12} + \frac{5}{23} a^{11} + \frac{7}{23} a^{10} + \frac{7}{23} a^{9} - \frac{11}{23} a^{8} - \frac{4}{23} a^{7} + \frac{6}{23} a^{6} + \frac{7}{23} a^{5} + \frac{9}{23} a^{4} + \frac{8}{23} a^{3} + \frac{11}{23} a^{2} + \frac{3}{23} a - \frac{4}{23}$, $\frac{1}{11626675549297340164319461340376821} a^{15} - \frac{183092667306250856133313249935829}{11626675549297340164319461340376821} a^{14} - \frac{1734999368913940659715933602451901}{11626675549297340164319461340376821} a^{13} + \frac{3213826373965896036692375264030617}{11626675549297340164319461340376821} a^{12} - \frac{691832871211262379407074161798439}{11626675549297340164319461340376821} a^{11} - \frac{5759514232538990394689802122523771}{11626675549297340164319461340376821} a^{10} - \frac{1580074750769934109849147384697302}{11626675549297340164319461340376821} a^{9} + \frac{1303582673986280835442898779658609}{11626675549297340164319461340376821} a^{8} + \frac{3755228742262013936507963251499208}{11626675549297340164319461340376821} a^{7} - \frac{830355132490620649913459051748828}{11626675549297340164319461340376821} a^{6} - \frac{1127439077694418938952660082615516}{11626675549297340164319461340376821} a^{5} - \frac{1948906976360226796176151285707899}{11626675549297340164319461340376821} a^{4} - \frac{3927106445871864308509693210049237}{11626675549297340164319461340376821} a^{3} - \frac{1135834545684018879095323387261474}{11626675549297340164319461340376821} a^{2} - \frac{4152837137284852169155905330558559}{11626675549297340164319461340376821} a - \frac{4059317118779176629139830330403717}{11626675549297340164319461340376821}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 1271827.27658 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1887:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 147456
The 148 conjugacy class representatives for t16n1887 are not computed
Character table for t16n1887 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 8.4.2372079616.2

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 R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }$

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$2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.12.30.108$x^{12} + 10 x^{10} - 3 x^{8} - 28 x^{6} - 21 x^{4} - 14 x^{2} - 17$$4$$3$$30$12T134$[2, 2, 2, 3, 7/2, 7/2, 7/2]^{3}$
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.12.0.1$x^{12} - x^{4} - x^{3} - x^{2} + x - 1$$1$$12$$0$$C_{12}$$[\ ]^{12}$
383Data not computed
761Data not computed