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Properties

Label 16.0.20297795652...8249.1
Degree $16$
Signature $[0, 8]$
Discriminant $37^{14}\cdot 83^{8}$
Root discriminant $214.64$
Ramified primes $37, 83$
Class number $114$ (GRH)
Class group $[114]$ (GRH)
Galois group $C_2^3.C_4$ (as 16T36)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4879681, 0, 7455375, 0, 3647932, 0, 535839, 0, -42617, 0, -12672, 0, 58, 0, 60, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 60*x^14 + 58*x^12 - 12672*x^10 - 42617*x^8 + 535839*x^6 + 3647932*x^4 + 7455375*x^2 + 4879681)
 
gp: K = bnfinit(x^16 + 60*x^14 + 58*x^12 - 12672*x^10 - 42617*x^8 + 535839*x^6 + 3647932*x^4 + 7455375*x^2 + 4879681, 1)
 

Normalized defining polynomial

\( x^{16} + 60 x^{14} + 58 x^{12} - 12672 x^{10} - 42617 x^{8} + 535839 x^{6} + 3647932 x^{4} + 7455375 x^{2} + 4879681 \)

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:  \(20297795652530455918821504664845058249=37^{14}\cdot 83^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $214.64$
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:  $37, 83$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7} a^{6} - \frac{1}{7}$, $\frac{1}{7} a^{7} - \frac{1}{7} a$, $\frac{1}{14} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} + \frac{3}{7} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{14} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} + \frac{3}{7} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{14} a^{10} - \frac{1}{14} a^{6} - \frac{1}{2} a^{5} + \frac{3}{7} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{3}{7}$, $\frac{1}{14} a^{11} - \frac{1}{14} a^{7} - \frac{1}{14} a^{6} + \frac{3}{7} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{3}{7} a - \frac{3}{7}$, $\frac{1}{686} a^{12} + \frac{1}{49} a^{10} - \frac{1}{98} a^{8} - \frac{1}{14} a^{7} - \frac{15}{343} a^{6} - \frac{1}{2} a^{5} + \frac{5}{98} a^{4} + \frac{11}{49} a^{2} - \frac{3}{7} a + \frac{88}{343}$, $\frac{1}{32242} a^{13} + \frac{163}{4606} a^{11} + \frac{31}{2303} a^{9} - \frac{1157}{32242} a^{7} - \frac{1107}{2303} a^{5} - \frac{1}{2} a^{4} - \frac{437}{2303} a^{3} - \frac{1}{2} a^{2} + \frac{4596}{16121} a - \frac{1}{2}$, $\frac{1}{9557213219086246} a^{14} - \frac{140455996938}{682658087077589} a^{12} - \frac{20259170988249}{682658087077589} a^{10} - \frac{156195601556919}{4778606609543123} a^{8} - \frac{1}{14} a^{7} + \frac{23768544515909}{682658087077589} a^{6} + \frac{351081091268741}{1365316174155178} a^{4} - \frac{1}{2} a^{3} - \frac{799860132935307}{4778606609543123} a^{2} + \frac{1}{14} a - \frac{106125684929}{309034896821}$, $\frac{1}{449189021297053562} a^{15} - \frac{140455996938}{32084930092646683} a^{13} - \frac{138040925844725}{64169860185293366} a^{11} - \frac{16013527205898385}{449189021297053562} a^{9} - \frac{3560798514092581}{64169860185293366} a^{7} - \frac{1}{14} a^{6} + \frac{6368224621266785}{32084930092646683} a^{5} - \frac{52681874750832071}{224594510648526781} a^{3} - \frac{5006536191662}{14524640150587} a + \frac{1}{14}$

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

Class group and class number

$C_{114}$, which has order $114$ (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:  \( 51875715919.7 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_4$ (as 16T36):

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

\(\Q(\sqrt{37}) \), \(\Q(\sqrt{-83}) \), \(\Q(\sqrt{-3071}) \), 4.4.348948517.1, 4.0.50653.1, \(\Q(\sqrt{37}, \sqrt{-83})\), 8.4.4505307498110473693.1 x2, 8.0.653985701569237.1 x2, 8.0.121765067516499289.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$37$37.8.7.2$x^{8} - 148$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
37.8.7.2$x^{8} - 148$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$83$83.2.1.2$x^{2} + 249$$2$$1$$1$$C_2$$[\ ]_{2}$
83.2.1.2$x^{2} + 249$$2$$1$$1$$C_2$$[\ ]_{2}$
83.2.1.2$x^{2} + 249$$2$$1$$1$$C_2$$[\ ]_{2}$
83.2.1.2$x^{2} + 249$$2$$1$$1$$C_2$$[\ ]_{2}$
83.4.2.1$x^{4} + 249 x^{2} + 27556$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
83.4.2.1$x^{4} + 249 x^{2} + 27556$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$