Properties

Label 16.16.179...113.1
Degree $16$
Signature $[16, 0]$
Discriminant $1.796\times 10^{26}$
Root discriminant $43.74$
Ramified primes $17, 89$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_{16} : C_2$ (as 16T22)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - 59*x^14 + 177*x^13 + 1203*x^12 - 3371*x^11 - 11052*x^10 + 25557*x^9 + 53804*x^8 - 84215*x^7 - 137046*x^6 + 109014*x^5 + 148448*x^4 - 32074*x^3 - 32876*x^2 - 4698*x - 67)
 
gp: K = bnfinit(x^16 - 3*x^15 - 59*x^14 + 177*x^13 + 1203*x^12 - 3371*x^11 - 11052*x^10 + 25557*x^9 + 53804*x^8 - 84215*x^7 - 137046*x^6 + 109014*x^5 + 148448*x^4 - 32074*x^3 - 32876*x^2 - 4698*x - 67, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-67, -4698, -32876, -32074, 148448, 109014, -137046, -84215, 53804, 25557, -11052, -3371, 1203, 177, -59, -3, 1]);
 

\( x^{16} - 3 x^{15} - 59 x^{14} + 177 x^{13} + 1203 x^{12} - 3371 x^{11} - 11052 x^{10} + 25557 x^{9} + 53804 x^{8} - 84215 x^{7} - 137046 x^{6} + 109014 x^{5} + 148448 x^{4} - 32074 x^{3} - 32876 x^{2} - 4698 x - 67 \)

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

Invariants

Degree:  $16$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[16, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(179594836941784276350012113\)\(\medspace = 17^{15}\cdot 89^{4}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $43.74$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $17, 89$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $8$
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{67} a^{13} - \frac{5}{67} a^{12} + \frac{33}{67} a^{11} + \frac{14}{67} a^{10} - \frac{29}{67} a^{9} - \frac{10}{67} a^{8} + \frac{17}{67} a^{7} + \frac{15}{67} a^{6} - \frac{21}{67} a^{5} + \frac{31}{67} a^{4} - \frac{1}{67} a^{3} - \frac{4}{67} a^{2} + \frac{24}{67} a$, $\frac{1}{67} a^{14} + \frac{8}{67} a^{12} - \frac{22}{67} a^{11} - \frac{26}{67} a^{10} - \frac{21}{67} a^{9} - \frac{33}{67} a^{8} + \frac{33}{67} a^{7} - \frac{13}{67} a^{6} - \frac{7}{67} a^{5} + \frac{20}{67} a^{4} - \frac{9}{67} a^{3} + \frac{4}{67} a^{2} - \frac{14}{67} a$, $\frac{1}{16529681543978371585987202239} a^{15} - \frac{105584950941387831643068685}{16529681543978371585987202239} a^{14} + \frac{54920849472778948993354107}{16529681543978371585987202239} a^{13} + \frac{1863555253709387484093466137}{16529681543978371585987202239} a^{12} + \frac{3780521972495617398828012336}{16529681543978371585987202239} a^{11} - \frac{6676124600565621848527478965}{16529681543978371585987202239} a^{10} - \frac{8058283812787967485022177865}{16529681543978371585987202239} a^{9} + \frac{3540217780520032372332905042}{16529681543978371585987202239} a^{8} - \frac{1177885760423882371412638343}{16529681543978371585987202239} a^{7} - \frac{6153682512348614214897494907}{16529681543978371585987202239} a^{6} - \frac{1450045851492115511059076687}{16529681543978371585987202239} a^{5} + \frac{6381162574421113005849211686}{16529681543978371585987202239} a^{4} + \frac{3744039332865653523908140270}{16529681543978371585987202239} a^{3} + \frac{1264974038102167884094621249}{16529681543978371585987202239} a^{2} - \frac{923823844458563618277222993}{16529681543978371585987202239} a - \frac{107733437102736885846397888}{246711664835498083372943317}$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $15$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 73340696.5653 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{16}\cdot(2\pi)^{0}\cdot 73340696.5653 \cdot 1}{2\sqrt{179594836941784276350012113}}\approx 0.179327973915$ (assuming GRH)

Galois group

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

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
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

\(\Q(\sqrt{17}) \), 4.4.4913.1, \(\Q(\zeta_{17})^+\)

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

Sibling fields

Galois closure: Deg 32

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
17Data not computed
$89$89.4.0.1$x^{4} - x + 27$$1$$4$$0$$C_4$$[\ ]^{4}$
89.4.2.2$x^{4} - 89 x^{2} + 47526$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
89.4.2.2$x^{4} - 89 x^{2} + 47526$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
89.4.0.1$x^{4} - x + 27$$1$$4$$0$$C_4$$[\ ]^{4}$