Properties

Label 16.8.978...193.1
Degree $16$
Signature $[8, 4]$
Discriminant $9.786\times 10^{24}$
Root discriminant $36.47$
Ramified primes $17, 43$
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 - 16*x^14 + 33*x^13 - 16*x^12 - 171*x^11 + 698*x^10 - 1208*x^9 + 1089*x^8 + 1529*x^7 - 8176*x^6 + 18614*x^5 - 29307*x^4 + 31602*x^3 - 22983*x^2 + 8312*x - 1087)
 
gp: K = bnfinit(x^16 - x^15 - 16*x^14 + 33*x^13 - 16*x^12 - 171*x^11 + 698*x^10 - 1208*x^9 + 1089*x^8 + 1529*x^7 - 8176*x^6 + 18614*x^5 - 29307*x^4 + 31602*x^3 - 22983*x^2 + 8312*x - 1087, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1087, 8312, -22983, 31602, -29307, 18614, -8176, 1529, 1089, -1208, 698, -171, -16, 33, -16, -1, 1]);
 

\( x^{16} - x^{15} - 16 x^{14} + 33 x^{13} - 16 x^{12} - 171 x^{11} + 698 x^{10} - 1208 x^{9} + 1089 x^{8} + 1529 x^{7} - 8176 x^{6} + 18614 x^{5} - 29307 x^{4} + 31602 x^{3} - 22983 x^{2} + 8312 x - 1087 \)

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:  \(9786054790924809742924193\)\(\medspace = 17^{15}\cdot 43^{4}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $36.47$
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:  $17, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{6767} a^{14} + \frac{2573}{6767} a^{13} + \frac{1933}{6767} a^{12} + \frac{2484}{6767} a^{11} + \frac{2130}{6767} a^{10} + \frac{3057}{6767} a^{9} + \frac{515}{6767} a^{8} - \frac{2610}{6767} a^{7} + \frac{1580}{6767} a^{6} - \frac{2845}{6767} a^{5} - \frac{2985}{6767} a^{4} - \frac{913}{6767} a^{3} + \frac{2761}{6767} a^{2} + \frac{2055}{6767} a - \frac{1135}{6767}$, $\frac{1}{1491148817736927361330597} a^{15} - \frac{45934711283690975826}{1491148817736927361330597} a^{14} - \frac{170723663225853974497145}{1491148817736927361330597} a^{13} + \frac{293207805458118522620981}{1491148817736927361330597} a^{12} - \frac{622008360555620105873019}{1491148817736927361330597} a^{11} - \frac{629313852632608949707675}{1491148817736927361330597} a^{10} + \frac{222522481845445367739686}{1491148817736927361330597} a^{9} - \frac{394673418766422007536799}{1491148817736927361330597} a^{8} + \frac{196372026311366921994798}{1491148817736927361330597} a^{7} + \frac{671209624024009892848630}{1491148817736927361330597} a^{6} + \frac{575529083703201205701483}{1491148817736927361330597} a^{5} - \frac{276577773330158820932465}{1491148817736927361330597} a^{4} + \frac{16669550600096525221522}{1491148817736927361330597} a^{3} + \frac{479116737288740367480768}{1491148817736927361330597} a^{2} + \frac{544145345512620619380797}{1491148817736927361330597} a - \frac{166786993642416100882123}{1491148817736927361330597}$

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:  \( 2917719.3712 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{8}\cdot(2\pi)^{4}\cdot 2917719.3712 \cdot 1}{2\sqrt{9786054790924809742924193}}\approx 0.18606690783$ (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 ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ R ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$43$43.8.4.2$x^{8} - 79507 x^{2} + 68376020$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
43.8.0.1$x^{8} - 3 x + 18$$1$$8$$0$$C_8$$[\ ]^{8}$