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Properties

Label 16.0.13604889600...0000.3
Degree $16$
Signature $[0, 8]$
Discriminant $2^{20}\cdot 3^{12}\cdot 5^{12}$
Root discriminant $18.13$
Ramified primes $2, 3, 5$
Class number $2$ (GRH)
Class group $[2]$ (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![16, 48, 28, -8, 103, -12, -101, 96, 28, -232, 361, -320, 218, -104, 37, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 37*x^14 - 104*x^13 + 218*x^12 - 320*x^11 + 361*x^10 - 232*x^9 + 28*x^8 + 96*x^7 - 101*x^6 - 12*x^5 + 103*x^4 - 8*x^3 + 28*x^2 + 48*x + 16)
 
gp: K = bnfinit(x^16 - 8*x^15 + 37*x^14 - 104*x^13 + 218*x^12 - 320*x^11 + 361*x^10 - 232*x^9 + 28*x^8 + 96*x^7 - 101*x^6 - 12*x^5 + 103*x^4 - 8*x^3 + 28*x^2 + 48*x + 16, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 37 x^{14} - 104 x^{13} + 218 x^{12} - 320 x^{11} + 361 x^{10} - 232 x^{9} + 28 x^{8} + 96 x^{7} - 101 x^{6} - 12 x^{5} + 103 x^{4} - 8 x^{3} + 28 x^{2} + 48 x + 16 \)

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:  \(136048896000000000000=2^{20}\cdot 3^{12}\cdot 5^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.13$
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:  $2, 3, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{13} - \frac{1}{2} a$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{12} - \frac{1}{4} a^{8} - \frac{1}{2} a^{6} + \frac{1}{4} a^{4} + \frac{1}{4} a^{2}$, $\frac{1}{1731530204898568} a^{15} + \frac{15495333020546}{216441275612321} a^{14} + \frac{232607080807017}{1731530204898568} a^{13} + \frac{83490839933585}{432882551224642} a^{12} + \frac{118981937561257}{865765102449284} a^{11} + \frac{1856975323686}{11391646084859} a^{10} - \frac{18245813310393}{91133168678872} a^{9} - \frac{91173690541161}{432882551224642} a^{8} + \frac{133730528837019}{432882551224642} a^{7} + \frac{51581748616681}{216441275612321} a^{6} + \frac{335589061147655}{1731530204898568} a^{5} + \frac{54505618134007}{216441275612321} a^{4} + \frac{21414980360253}{91133168678872} a^{3} + \frac{90876054932836}{216441275612321} a^{2} + \frac{98865192422497}{432882551224642} a + \frac{101210009039574}{216441275612321}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( -\frac{208358580}{6543162599} a^{15} + \frac{7389382705}{26172650396} a^{14} - \frac{9118976275}{6543162599} a^{13} + \frac{112050507145}{26172650396} a^{12} - \frac{62554249555}{6543162599} a^{11} + \frac{202269713339}{13086325198} a^{10} - \frac{123427520750}{6543162599} a^{9} + \frac{403574124525}{26172650396} a^{8} - \frac{38879557495}{6543162599} a^{7} - \frac{14143802310}{6543162599} a^{6} + \frac{23452735268}{6543162599} a^{5} - \frac{3900436725}{26172650396} a^{4} - \frac{27619516490}{6543162599} a^{3} + \frac{70268964955}{26172650396} a^{2} - \frac{4235352685}{6543162599} a - \frac{1651515427}{6543162599} \) (order $6$)
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:  \( 6942.44507571 \) (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

\(\Q(\sqrt{-15}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-3}) \), 4.2.3600.1, 4.2.400.1, 4.0.13500.2, 4.0.13500.1, \(\Q(\sqrt{-3}, \sqrt{5})\), 8.0.182250000.1, 8.0.12960000.2, 8.0.2916000000.5

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$2.4.4.3$x^{4} + 2 x^{2} + 4 x + 4$$2$$2$$4$$D_{4}$$[2, 2]^{2}$
2.4.4.3$x^{4} + 2 x^{2} + 4 x + 4$$2$$2$$4$$D_{4}$$[2, 2]^{2}$
2.8.12.13$x^{8} + 12 x^{4} + 16$$4$$2$$12$$D_4$$[2, 2]^{2}$
3Data not computed
$5$5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$