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Properties

Label 16.0.21458877493...2329.1
Degree $16$
Signature $[0, 8]$
Discriminant $37^{14}\cdot 47^{8}$
Root discriminant $161.52$
Ramified primes $37, 47$
Class number $250$ (GRH)
Class group $[5, 5, 10]$ (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![6901129, 0, 60835622, 0, 55276113, 0, 9339318, 0, 1059384, 0, 74037, 0, 777, 0, -74, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 74*x^14 + 777*x^12 + 74037*x^10 + 1059384*x^8 + 9339318*x^6 + 55276113*x^4 + 60835622*x^2 + 6901129)
 
gp: K = bnfinit(x^16 - 74*x^14 + 777*x^12 + 74037*x^10 + 1059384*x^8 + 9339318*x^6 + 55276113*x^4 + 60835622*x^2 + 6901129, 1)
 

Normalized defining polynomial

\( x^{16} - 74 x^{14} + 777 x^{12} + 74037 x^{10} + 1059384 x^{8} + 9339318 x^{6} + 55276113 x^{4} + 60835622 x^{2} + 6901129 \)

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:  \(214588774932298492495221177464492329=37^{14}\cdot 47^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $161.52$
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:  $37, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{74} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{74} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{74} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{5254} a^{11} + \frac{6}{2627} a^{9} - \frac{3}{142} a^{7} - \frac{28}{71} a^{5} + \frac{20}{71} a^{3} - \frac{1}{2} a^{2} + \frac{67}{142} a - \frac{1}{2}$, $\frac{1}{15762} a^{12} + \frac{2}{2627} a^{10} + \frac{31}{15762} a^{8} + \frac{43}{213} a^{6} + \frac{91}{213} a^{4} - \frac{1}{2} a^{3} - \frac{25}{142} a^{2} - \frac{1}{2} a - \frac{1}{3}$, $\frac{1}{15762} a^{13} + \frac{50}{7881} a^{9} - \frac{91}{426} a^{7} - \frac{1}{2} a^{6} + \frac{1}{213} a^{5} - \frac{1}{2} a^{4} + \frac{14}{71} a^{3} - \frac{1}{2} a^{2} + \frac{119}{426} a$, $\frac{1}{1564461766318079828202} a^{14} + \frac{22935016642206154}{782230883159039914101} a^{12} + \frac{3205937027584436377}{1564461766318079828202} a^{10} + \frac{3061040443141831447}{1564461766318079828202} a^{8} + \frac{1614498947390715565}{14094250147009728182} a^{6} + \frac{20644276910554568941}{42282750441029184546} a^{4} - \frac{1}{2} a^{3} + \frac{2545687703196649643}{42282750441029184546} a^{2} - \frac{4025203077250013}{8387770371162306}$, $\frac{1}{57885085353768953643474} a^{15} - \frac{6807994997077015}{1564461766318079828202} a^{13} - \frac{65320822610468026}{782230883159039914101} a^{11} - \frac{1379058161663224408}{782230883159039914101} a^{9} + \frac{83338017059882604144}{260743627719679971367} a^{7} + \frac{11567242086910291825}{42282750441029184546} a^{5} + \frac{8778255078246130783}{21141375220514592273} a^{3} + \frac{3562291092248380}{297765848176261863} a - \frac{1}{2}$

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

Class group and class number

$C_{5}\times C_{5}\times C_{10}$, which has order $250$ (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:  \( 339086333.465 \) (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

\(\Q(\sqrt{37}) \), \(\Q(\sqrt{-1739}) \), \(\Q(\sqrt{-47}) \), 4.4.111892477.1, 4.0.50653.1, \(\Q(\sqrt{37}, \sqrt{-47})\), 8.4.463237277140234573.1 x2, 8.0.209704516586797.1 x2, 8.0.12519926409195529.1

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 ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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}$
$47$47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$