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Properties

Label 16.0.58195704056...8329.1
Degree $16$
Signature $[0, 8]$
Discriminant $37^{14}\cdot 71^{8}$
Root discriminant $198.52$
Ramified primes $37, 71$
Class number $42$ (GRH)
Class group $[42]$ (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![165649, 0, 1308165747, 0, 421389152, 0, 56172808, 0, 1970546, 0, -29785, 0, -555, 0, 74, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 74*x^14 - 555*x^12 - 29785*x^10 + 1970546*x^8 + 56172808*x^6 + 421389152*x^4 + 1308165747*x^2 + 165649)
 
gp: K = bnfinit(x^16 + 74*x^14 - 555*x^12 - 29785*x^10 + 1970546*x^8 + 56172808*x^6 + 421389152*x^4 + 1308165747*x^2 + 165649, 1)
 

Normalized defining polynomial

\( x^{16} + 74 x^{14} - 555 x^{12} - 29785 x^{10} + 1970546 x^{8} + 56172808 x^{6} + 421389152 x^{4} + 1308165747 x^{2} + 165649 \)

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:  \(5819570405692025964063507186598868329=37^{14}\cdot 71^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $198.52$
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:  $37, 71$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{74} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{74} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{74} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{74} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{1554} a^{12} - \frac{4}{777} a^{10} - \frac{1}{259} a^{8} - \frac{2}{7} a^{6} + \frac{5}{42} a^{4} - \frac{1}{2} a^{3} + \frac{1}{21} a^{2} - \frac{19}{42}$, $\frac{1}{17094} a^{13} - \frac{25}{8547} a^{11} + \frac{19}{5698} a^{9} - \frac{25}{154} a^{7} - \frac{1}{2} a^{6} + \frac{89}{462} a^{5} - \frac{1}{2} a^{4} + \frac{85}{231} a^{3} - \frac{1}{2} a^{2} + \frac{43}{231} a$, $\frac{1}{21925658500183705210347558} a^{14} - \frac{1869369429673463112947}{7308552833394568403449186} a^{12} + \frac{100666127576570574898079}{21925658500183705210347558} a^{10} - \frac{37663132252979859562267}{7308552833394568403449186} a^{8} - \frac{1}{2} a^{7} - \frac{92950024737935205933289}{592585364869829870549934} a^{6} - \frac{1}{2} a^{5} + \frac{2090302783982130367725}{197528454956609956849978} a^{4} + \frac{17539388419347763753418}{98764227478304978424989} a^{2} + \frac{15375018692291917297865}{53871396806348170049994}$, $\frac{1}{811249364506797092782859646} a^{15} + \frac{333757404858441459761}{21925658500183705210347558} a^{13} + \frac{65302303789200245643548}{10962829250091852605173779} a^{11} + \frac{40234948934960854734799}{7308552833394568403449186} a^{9} - \frac{267460235519107431195802}{996620840917441145924889} a^{7} + \frac{235553515100312627455465}{592585364869829870549934} a^{5} - \frac{1}{2} a^{4} - \frac{291021817790050065024413}{592585364869829870549934} a^{3} - \frac{1}{2} a^{2} + \frac{99229723608025798224334}{296292682434914935274967} a$

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

Class group and class number

$C_{42}$, which has order $42$ (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:  \( 11603181354.1 \) (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

\(\Q(\sqrt{37}) \), \(\Q(\sqrt{-2627}) \), \(\Q(\sqrt{-71}) \), 4.4.255341773.1, 4.0.50653.1, \(\Q(\sqrt{37}, \sqrt{-71})\), 8.4.2412378578434990573.1 x2, 8.0.478551592627453.1 x2, 8.0.65199421038783529.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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}$
$71$71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.4.2.1$x^{4} + 1491 x^{2} + 609961$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
71.4.2.1$x^{4} + 1491 x^{2} + 609961$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$