Properties

Label 16.8.139...033.1
Degree $16$
Signature $[8, 4]$
Discriminant $1.397\times 10^{25}$
Root discriminant $37.29$
Ramified primes $17, 47$
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 - 8*x^14 + 126*x^13 - 514*x^12 + 811*x^11 + 1001*x^10 - 9174*x^9 + 20365*x^8 - 13699*x^7 - 20001*x^6 + 53713*x^5 - 48412*x^4 + 11089*x^3 + 17631*x^2 - 12722*x + 409)
 
gp: K = bnfinit(x^16 - 3*x^15 - 8*x^14 + 126*x^13 - 514*x^12 + 811*x^11 + 1001*x^10 - 9174*x^9 + 20365*x^8 - 13699*x^7 - 20001*x^6 + 53713*x^5 - 48412*x^4 + 11089*x^3 + 17631*x^2 - 12722*x + 409, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![409, -12722, 17631, 11089, -48412, 53713, -20001, -13699, 20365, -9174, 1001, 811, -514, 126, -8, -3, 1]);
 

\( x^{16} - 3 x^{15} - 8 x^{14} + 126 x^{13} - 514 x^{12} + 811 x^{11} + 1001 x^{10} - 9174 x^{9} + 20365 x^{8} - 13699 x^{7} - 20001 x^{6} + 53713 x^{5} - 48412 x^{4} + 11089 x^{3} + 17631 x^{2} - 12722 x + 409 \)

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:  \(13967711378414469438602033\)\(\medspace = 17^{15}\cdot 47^{4}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $37.29$
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:  $17, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{101} a^{14} - \frac{39}{101} a^{13} + \frac{38}{101} a^{12} + \frac{8}{101} a^{11} + \frac{13}{101} a^{10} - \frac{17}{101} a^{9} + \frac{18}{101} a^{8} + \frac{33}{101} a^{7} - \frac{15}{101} a^{6} + \frac{1}{101} a^{5} + \frac{30}{101} a^{4} - \frac{33}{101} a^{3} + \frac{7}{101} a^{2} + \frac{45}{101}$, $\frac{1}{337525567380184211763964079681087567} a^{15} - \frac{647180613728654044075109235047375}{337525567380184211763964079681087567} a^{14} - \frac{68843981923014449516391190541084255}{337525567380184211763964079681087567} a^{13} + \frac{77529193373236901273365485299680250}{337525567380184211763964079681087567} a^{12} + \frac{40063111951315037195811837586153115}{337525567380184211763964079681087567} a^{11} - \frac{110052661721686328284567769060431775}{337525567380184211763964079681087567} a^{10} - \frac{73248132303868723933723318055509052}{337525567380184211763964079681087567} a^{9} - \frac{77178774742281179616519021709759854}{337525567380184211763964079681087567} a^{8} - \frac{126801683294809692247879743931179123}{337525567380184211763964079681087567} a^{7} + \frac{62596030482770424531873813632408765}{337525567380184211763964079681087567} a^{6} + \frac{121711436454707792606185484806653988}{337525567380184211763964079681087567} a^{5} + \frac{90175351844972649799328758792479914}{337525567380184211763964079681087567} a^{4} - \frac{144539859372673989314162005189743701}{337525567380184211763964079681087567} a^{3} - \frac{165324393653652701810095077461377507}{337525567380184211763964079681087567} a^{2} - \frac{31254735111469271224681093749885120}{337525567380184211763964079681087567} a + \frac{70301888154481670662106569804734059}{337525567380184211763964079681087567}$

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:  \( 3120523.02485 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{8}\cdot(2\pi)^{4}\cdot 3120523.02485 \cdot 1}{2\sqrt{13967711378414469438602033}}\approx 0.166569011429$ (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 ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$47$47.4.2.2$x^{4} - 47 x^{2} + 28717$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
47.4.0.1$x^{4} - x + 39$$1$$4$$0$$C_4$$[\ ]^{4}$
47.4.2.2$x^{4} - 47 x^{2} + 28717$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
47.4.0.1$x^{4} - x + 39$$1$$4$$0$$C_4$$[\ ]^{4}$