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Properties

Label 16.0.12196812568...9233.1
Degree $16$
Signature $[0, 8]$
Discriminant $37^{2}\cdot 73^{15}$
Root discriminant $87.68$
Ramified primes $37, 73$
Class number $89$ (GRH)
Class group $[89]$ (GRH)
Galois group 16T841

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![375808, -841088, 1053996, -136764, -590233, 615758, 71748, -209562, 17710, 22860, -4264, -246, 132, -70, 30, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 30*x^14 - 70*x^13 + 132*x^12 - 246*x^11 - 4264*x^10 + 22860*x^9 + 17710*x^8 - 209562*x^7 + 71748*x^6 + 615758*x^5 - 590233*x^4 - 136764*x^3 + 1053996*x^2 - 841088*x + 375808)
 
gp: K = bnfinit(x^16 - 8*x^15 + 30*x^14 - 70*x^13 + 132*x^12 - 246*x^11 - 4264*x^10 + 22860*x^9 + 17710*x^8 - 209562*x^7 + 71748*x^6 + 615758*x^5 - 590233*x^4 - 136764*x^3 + 1053996*x^2 - 841088*x + 375808, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 30 x^{14} - 70 x^{13} + 132 x^{12} - 246 x^{11} - 4264 x^{10} + 22860 x^{9} + 17710 x^{8} - 209562 x^{7} + 71748 x^{6} + 615758 x^{5} - 590233 x^{4} - 136764 x^{3} + 1053996 x^{2} - 841088 x + 375808 \)

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:  \(12196812568090154832283098089233=37^{2}\cdot 73^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $87.68$
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:  $37, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{8} a^{8} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{3}{8} a^{2} - \frac{1}{4} a$, $\frac{1}{8} a^{9} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{3}{8} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{16} a^{10} - \frac{1}{16} a^{9} - \frac{1}{16} a^{8} + \frac{1}{8} a^{7} - \frac{1}{4} a^{6} + \frac{3}{16} a^{4} - \frac{3}{16} a^{3} + \frac{1}{16} a^{2} + \frac{1}{8} a$, $\frac{1}{16} a^{11} - \frac{1}{16} a^{8} - \frac{1}{8} a^{7} + \frac{3}{16} a^{5} - \frac{1}{4} a^{4} + \frac{5}{16} a^{2} - \frac{1}{8} a$, $\frac{1}{64} a^{12} - \frac{1}{32} a^{11} - \frac{1}{32} a^{10} + \frac{1}{64} a^{9} - \frac{1}{16} a^{8} - \frac{1}{4} a^{7} + \frac{11}{64} a^{6} - \frac{3}{32} a^{5} + \frac{7}{32} a^{4} - \frac{21}{64} a^{3} + \frac{3}{16} a^{2} + \frac{3}{16} a$, $\frac{1}{64} a^{13} - \frac{1}{32} a^{11} + \frac{1}{64} a^{10} + \frac{1}{32} a^{9} + \frac{11}{64} a^{7} - \frac{1}{4} a^{6} - \frac{1}{32} a^{5} + \frac{3}{64} a^{4} - \frac{1}{32} a^{3} + \frac{3}{16} a^{2} - \frac{1}{8} a$, $\frac{1}{22399474766822144} a^{14} - \frac{7}{22399474766822144} a^{13} + \frac{146075194732143}{22399474766822144} a^{12} + \frac{523516004533617}{22399474766822144} a^{11} - \frac{449309946974555}{22399474766822144} a^{10} - \frac{919051938271433}{22399474766822144} a^{9} - \frac{969962029458329}{22399474766822144} a^{8} + \frac{2553131240994531}{22399474766822144} a^{7} - \frac{4523422188962983}{22399474766822144} a^{6} + \frac{1455462028259351}{22399474766822144} a^{5} + \frac{5483972120247819}{22399474766822144} a^{4} - \frac{8730881479630191}{22399474766822144} a^{3} - \frac{69482210976051}{174995896615798} a^{2} - \frac{2018820191839395}{5599868691705536} a - \frac{174122605716647}{349991793231596}$, $\frac{1}{7772617744087283968} a^{15} + \frac{83}{3886308872043641984} a^{14} + \frac{1086494178377521}{1943154436021820992} a^{13} - \frac{726200587949129}{1943154436021820992} a^{12} + \frac{94758314057557225}{3886308872043641984} a^{11} + \frac{6793401082892129}{971577218010910496} a^{10} + \frac{180760912282330901}{3886308872043641984} a^{9} + \frac{79071058545349159}{3886308872043641984} a^{8} - \frac{36462986660015191}{485788609005455248} a^{7} - \frac{243398015734928627}{1943154436021820992} a^{6} + \frac{182538187662223555}{3886308872043641984} a^{5} + \frac{15613279941908449}{485788609005455248} a^{4} - \frac{895650843442253499}{7772617744087283968} a^{3} + \frac{191864094056570457}{1943154436021820992} a^{2} - \frac{738432794607363671}{1943154436021820992} a + \frac{55974770345992685}{121447152251363812}$

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

Class group and class number

$C_{89}$, which has order $89$ (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:  \( 4232340654.47 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T841:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 32 conjugacy class representatives for t16n841
Character table for t16n841 is not computed

Intermediate fields

\(\Q(\sqrt{73}) \), 4.4.389017.1, 8.0.11047398519097.1

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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ $16$ $16$ R ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ $16$ $16$ $16$ $16$

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
$37$$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
37.2.1.1$x^{2} - 37$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.1.1$x^{2} - 37$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
73Data not computed