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Properties

Label 16.0.22245513193...9081.1
Degree $16$
Signature $[0, 8]$
Discriminant $61^{14}\cdot 83^{8}$
Root discriminant $332.43$
Ramified primes $61, 83$
Class number $54$ (GRH)
Class group $[3, 18]$ (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![41151773881, 0, 8629333501, 0, 426352493, 0, -11265729, 0, -2269931, 0, -109236, 0, 3183, 0, 194, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 194*x^14 + 3183*x^12 - 109236*x^10 - 2269931*x^8 - 11265729*x^6 + 426352493*x^4 + 8629333501*x^2 + 41151773881)
 
gp: K = bnfinit(x^16 + 194*x^14 + 3183*x^12 - 109236*x^10 - 2269931*x^8 - 11265729*x^6 + 426352493*x^4 + 8629333501*x^2 + 41151773881, 1)
 

Normalized defining polynomial

\( x^{16} + 194 x^{14} + 3183 x^{12} - 109236 x^{10} - 2269931 x^{8} - 11265729 x^{6} + 426352493 x^{4} + 8629333501 x^{2} + 41151773881 \)

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:  \(22245513193027862267173879060120840909081=61^{14}\cdot 83^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $332.43$
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:  $61, 83$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{10} a^{8} + \frac{3}{10} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} + \frac{3}{10} a^{2} - \frac{1}{2} a - \frac{1}{10}$, $\frac{1}{10} a^{9} + \frac{3}{10} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} + \frac{3}{10} a^{3} - \frac{1}{2} a^{2} - \frac{1}{10} a$, $\frac{1}{10} a^{10} - \frac{2}{5} a^{6} - \frac{1}{2} a^{5} - \frac{1}{5} a^{4} - \frac{1}{2} a + \frac{3}{10}$, $\frac{1}{10} a^{11} - \frac{2}{5} a^{7} - \frac{1}{2} a^{6} - \frac{1}{5} a^{5} - \frac{1}{2} a^{2} + \frac{3}{10} a$, $\frac{1}{650} a^{12} - \frac{16}{325} a^{10} + \frac{1}{25} a^{8} - \frac{1}{2} a^{7} - \frac{142}{325} a^{6} - \frac{103}{325} a^{4} - \frac{1}{2} a^{3} - \frac{177}{650} a^{2} + \frac{22}{325}$, $\frac{1}{650} a^{13} - \frac{16}{325} a^{11} + \frac{1}{25} a^{9} - \frac{142}{325} a^{7} - \frac{1}{2} a^{6} - \frac{103}{325} a^{5} + \frac{74}{325} a^{3} - \frac{1}{2} a^{2} - \frac{281}{650} a - \frac{1}{2}$, $\frac{1}{722079426190502241832234598750} a^{14} + \frac{347169231078600075956337141}{722079426190502241832234598750} a^{12} + \frac{4208873784956052564147165267}{144415885238100448366446919750} a^{10} - \frac{4066558374557804884649062808}{361039713095251120916117299375} a^{8} + \frac{33599490189669563713714724796}{361039713095251120916117299375} a^{6} - \frac{1}{2} a^{5} + \frac{18656610490454268545886873027}{72207942619050224183223459875} a^{4} - \frac{15775561459499961695403836221}{361039713095251120916117299375} a^{2} - \frac{200197734610111487885953943823}{722079426190502241832234598750}$, $\frac{1}{146480310317579094275845278467826250} a^{15} - \frac{9529608503961727594666367224317}{73240155158789547137922639233913125} a^{13} + \frac{698233684884642381376576267380677}{29296062063515818855169055693565250} a^{11} - \frac{3620394776769362317993015769659508}{73240155158789547137922639233913125} a^{9} + \frac{23839229239250467328242841690621721}{73240155158789547137922639233913125} a^{7} + \frac{359906597476396478685691400352667}{14648031031757909427584527846782625} a^{5} + \frac{10186460891719047204344482027780929}{73240155158789547137922639233913125} a^{3} - \frac{1}{2} a^{2} + \frac{23101785237821262851892733704073226}{73240155158789547137922639233913125} a - \frac{1}{2}$

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

Class group and class number

$C_{3}\times C_{18}$, which has order $54$ (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:  \( 432906818173 \) (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

\(\Q(\sqrt{61}) \), \(\Q(\sqrt{-83}) \), \(\Q(\sqrt{-5063}) \), 4.4.1563672109.1, 4.0.226981.1, \(\Q(\sqrt{61}, \sqrt{-83})\), 8.4.149149298332334980741.1 x2, 8.0.21650355397348669.1 x2, 8.0.2445070464464507881.2

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$61$61.8.7.2$x^{8} - 244$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
61.8.7.2$x^{8} - 244$$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}$