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Properties

Label 16.0.86038720425...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 3^{8}\cdot 5^{6}\cdot 29^{8}$
Root discriminant $48.24$
Ramified primes $2, 3, 5, 29$
Class number $6$ (GRH)
Class group $[6]$ (GRH)
Galois group $C_2^2:D_4$ (as 16T34)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![313, -1646, 4363, -8278, 13448, -17876, 19291, -20452, 21523, -17224, 9353, -3360, 804, -134, 25, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 25*x^14 - 134*x^13 + 804*x^12 - 3360*x^11 + 9353*x^10 - 17224*x^9 + 21523*x^8 - 20452*x^7 + 19291*x^6 - 17876*x^5 + 13448*x^4 - 8278*x^3 + 4363*x^2 - 1646*x + 313)
 
gp: K = bnfinit(x^16 - 6*x^15 + 25*x^14 - 134*x^13 + 804*x^12 - 3360*x^11 + 9353*x^10 - 17224*x^9 + 21523*x^8 - 20452*x^7 + 19291*x^6 - 17876*x^5 + 13448*x^4 - 8278*x^3 + 4363*x^2 - 1646*x + 313, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 25 x^{14} - 134 x^{13} + 804 x^{12} - 3360 x^{11} + 9353 x^{10} - 17224 x^{9} + 21523 x^{8} - 20452 x^{7} + 19291 x^{6} - 17876 x^{5} + 13448 x^{4} - 8278 x^{3} + 4363 x^{2} - 1646 x + 313 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(860387204251548647424000000=2^{24}\cdot 3^{8}\cdot 5^{6}\cdot 29^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $48.24$
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:  $2, 3, 5, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{23} a^{13} + \frac{11}{23} a^{12} - \frac{7}{23} a^{11} - \frac{2}{23} a^{10} + \frac{7}{23} a^{9} - \frac{10}{23} a^{8} + \frac{7}{23} a^{7} + \frac{2}{23} a^{6} + \frac{2}{23} a^{5} - \frac{5}{23} a^{4} + \frac{7}{23} a^{3} - \frac{3}{23} a^{2} - \frac{10}{23} a - \frac{4}{23}$, $\frac{1}{529} a^{14} - \frac{10}{529} a^{13} + \frac{61}{529} a^{12} + \frac{191}{529} a^{11} - \frac{204}{529} a^{10} - \frac{134}{529} a^{9} + \frac{125}{529} a^{8} - \frac{260}{529} a^{7} - \frac{155}{529} a^{6} + \frac{252}{529} a^{5} - \frac{141}{529} a^{4} + \frac{195}{529} a^{3} + \frac{7}{529} a^{2} - \frac{93}{529} a - \frac{54}{529}$, $\frac{1}{4333793305862664901391161} a^{15} + \frac{19293401650403819754}{25950858118938113181983} a^{14} - \frac{30562845140496439421878}{4333793305862664901391161} a^{13} + \frac{574160485354352755198484}{4333793305862664901391161} a^{12} + \frac{542054912675858796828026}{4333793305862664901391161} a^{11} + \frac{1352916050995305228810257}{4333793305862664901391161} a^{10} + \frac{2014299521600344811381725}{4333793305862664901391161} a^{9} - \frac{1742853349682400624987701}{4333793305862664901391161} a^{8} + \frac{72498405672339386118968}{4333793305862664901391161} a^{7} + \frac{1966243836416396780562105}{4333793305862664901391161} a^{6} + \frac{1105580540481097999391858}{4333793305862664901391161} a^{5} - \frac{1206148287746999321681245}{4333793305862664901391161} a^{4} - \frac{569735956342711220415499}{4333793305862664901391161} a^{3} - \frac{1093015705433907398201282}{4333793305862664901391161} a^{2} + \frac{1364396229668464390940215}{4333793305862664901391161} a - \frac{1303233909538205637866912}{4333793305862664901391161}$

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

Class group and class number

$C_{6}$, which has order $6$ (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:  $7$
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:  \( 1602254.01058 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:D_4$ (as 16T34):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_2^2:D_4$
Character table for $C_2^2:D_4$

Intermediate fields

\(\Q(\sqrt{174}) \), \(\Q(\sqrt{29}) \), \(\Q(\sqrt{6}) \), 4.0.269120.1, 4.0.37845.1, 4.4.83520.1, \(\Q(\sqrt{6}, \sqrt{29})\), 4.4.83520.2, 8.8.5866471526400.3, 8.0.5866471526400.4, 8.0.5866471526400.8

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

Sibling fields

Galois closure: data not computed
Degree 16 siblings: 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 R R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$2$2.8.12.2$x^{8} + 2 x^{6} + 8 x^{4} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.2$x^{8} + 2 x^{6} + 8 x^{4} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$29$29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$