Properties

Label 16.8.373...553.1
Degree $16$
Signature $[8, 4]$
Discriminant $3.730\times 10^{23}$
Root discriminant $29.73$
Ramified primes $17, 19$
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 - x^15 + 18*x^14 - 35*x^13 + 52*x^12 - 239*x^11 + 154*x^10 + 356*x^9 + 171*x^8 - 511*x^7 - 2056*x^6 + 3518*x^5 - 33*x^4 - 2432*x^3 + 851*x^2 + 186*x - 67)
 
gp: K = bnfinit(x^16 - x^15 + 18*x^14 - 35*x^13 + 52*x^12 - 239*x^11 + 154*x^10 + 356*x^9 + 171*x^8 - 511*x^7 - 2056*x^6 + 3518*x^5 - 33*x^4 - 2432*x^3 + 851*x^2 + 186*x - 67, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-67, 186, 851, -2432, -33, 3518, -2056, -511, 171, 356, 154, -239, 52, -35, 18, -1, 1]);
 

\( x^{16} - x^{15} + 18 x^{14} - 35 x^{13} + 52 x^{12} - 239 x^{11} + 154 x^{10} + 356 x^{9} + 171 x^{8} - 511 x^{7} - 2056 x^{6} + 3518 x^{5} - 33 x^{4} - 2432 x^{3} + 851 x^{2} + 186 x - 67 \)

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

Invariants

Degree:  $16$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[8, 4]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(373033834495810703959553\)\(\medspace = 17^{15}\cdot 19^{4}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $29.73$
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:  $17, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{137} a^{14} - \frac{59}{137} a^{13} + \frac{33}{137} a^{12} + \frac{3}{137} a^{11} + \frac{61}{137} a^{10} - \frac{24}{137} a^{9} + \frac{41}{137} a^{8} + \frac{12}{137} a^{7} - \frac{61}{137} a^{6} - \frac{45}{137} a^{5} + \frac{4}{137} a^{4} + \frac{10}{137} a^{3} + \frac{7}{137} a^{2} - \frac{55}{137} a + \frac{57}{137}$, $\frac{1}{29345375458215303446803} a^{15} + \frac{62148182013505022436}{29345375458215303446803} a^{14} - \frac{7729347102623594877862}{29345375458215303446803} a^{13} - \frac{6727023575548304481719}{29345375458215303446803} a^{12} - \frac{8198318362972618905168}{29345375458215303446803} a^{11} - \frac{8790303164322315576657}{29345375458215303446803} a^{10} + \frac{1344153994178670948225}{29345375458215303446803} a^{9} + \frac{2823906535042360480067}{29345375458215303446803} a^{8} + \frac{11100012059662492923682}{29345375458215303446803} a^{7} - \frac{2290608535404597005230}{29345375458215303446803} a^{6} - \frac{2607657434615127125109}{29345375458215303446803} a^{5} - \frac{83696155522023871701}{29345375458215303446803} a^{4} - \frac{2801953230466366886358}{29345375458215303446803} a^{3} + \frac{308850175838207418883}{29345375458215303446803} a^{2} + \frac{14428841887259340947523}{29345375458215303446803} a + \frac{5295317482415839586512}{29345375458215303446803}$

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:  $11$
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:  \( 454664.771694 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{8}\cdot(2\pi)^{4}\cdot 454664.771694 \cdot 1}{2\sqrt{373033834495810703959553}}\approx 0.148506901315$ (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 R $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$19$19.8.4.2$x^{8} - 13718 x^{2} + 1303210$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
19.8.0.1$x^{8} - x + 2$$1$$8$$0$$C_8$$[\ ]^{8}$