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Properties

Label 16.8.21524562403...5801.1
Degree $16$
Signature $[8, 4]$
Discriminant $7^{12}\cdot 41^{15}$
Root discriminant $139.90$
Ramified primes $7, 41$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{16} : C_2$ (as 16T22)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![10314571, -246301289, -9786670, 187459777, -20086013, -64805612, 3374410, 8895081, -51363, -528593, -23154, 12539, 2087, -28, -75, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 75*x^14 - 28*x^13 + 2087*x^12 + 12539*x^11 - 23154*x^10 - 528593*x^9 - 51363*x^8 + 8895081*x^7 + 3374410*x^6 - 64805612*x^5 - 20086013*x^4 + 187459777*x^3 - 9786670*x^2 - 246301289*x + 10314571)
 
gp: K = bnfinit(x^16 - 2*x^15 - 75*x^14 - 28*x^13 + 2087*x^12 + 12539*x^11 - 23154*x^10 - 528593*x^9 - 51363*x^8 + 8895081*x^7 + 3374410*x^6 - 64805612*x^5 - 20086013*x^4 + 187459777*x^3 - 9786670*x^2 - 246301289*x + 10314571, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 75 x^{14} - 28 x^{13} + 2087 x^{12} + 12539 x^{11} - 23154 x^{10} - 528593 x^{9} - 51363 x^{8} + 8895081 x^{7} + 3374410 x^{6} - 64805612 x^{5} - 20086013 x^{4} + 187459777 x^{3} - 9786670 x^{2} - 246301289 x + 10314571 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(21524562403589040109288671558095801=7^{12}\cdot 41^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $139.90$
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:  $7, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{5021837178142711871654324614589110726172743044245762382771127} a^{15} - \frac{2294568093222711063791058356847846637780145449744376790469292}{5021837178142711871654324614589110726172743044245762382771127} a^{14} + \frac{1165409136165317206137145921591235372919799296131559645639065}{5021837178142711871654324614589110726172743044245762382771127} a^{13} - \frac{303484936209502557010580010597710497715337800648323144571284}{5021837178142711871654324614589110726172743044245762382771127} a^{12} - \frac{1185687086403359419090614641689218203685981287241469262072815}{5021837178142711871654324614589110726172743044245762382771127} a^{11} + \frac{1987927152211137081775816059134064158512329776538010444221832}{5021837178142711871654324614589110726172743044245762382771127} a^{10} + \frac{203186707287371726346059245121616350656007070875000303384146}{5021837178142711871654324614589110726172743044245762382771127} a^{9} + \frac{396542765454808394393695261017856943292721315614972696773744}{5021837178142711871654324614589110726172743044245762382771127} a^{8} + \frac{241507872304326837233174494821181664961074694311934555904206}{5021837178142711871654324614589110726172743044245762382771127} a^{7} - \frac{360910816394853011424490409008566002615567146234938102773357}{5021837178142711871654324614589110726172743044245762382771127} a^{6} + \frac{1149766905170769383157993702891387615473747428326573300105807}{5021837178142711871654324614589110726172743044245762382771127} a^{5} + \frac{182042786441805355787742049179020498715727569970203681354712}{5021837178142711871654324614589110726172743044245762382771127} a^{4} + \frac{169148752709689022418575945871594504977392854910857723656025}{5021837178142711871654324614589110726172743044245762382771127} a^{3} + \frac{2049859238121182821342881823610833226836366480568386654142023}{5021837178142711871654324614589110726172743044245762382771127} a^{2} + \frac{1910246469035891956635655180986105743876943889879657981012577}{5021837178142711871654324614589110726172743044245762382771127} a - \frac{2129534193509310224159550244617055687187583942969444795198177}{5021837178142711871654324614589110726172743044245762382771127}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $11$
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:  \( 105232700409 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$OD_{32}$ (as 16T22):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_{16} : C_2$
Character table for $C_{16} : C_2$

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.68921.1, 8.8.467605011588281.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ R $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
7Data not computed
41Data not computed