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Properties

Label 16.8.34793159156...9536.2
Degree $16$
Signature $[8, 4]$
Discriminant $2^{38}\cdot 3^{10}\cdot 11^{8}$
Root discriminant $34.19$
Ramified primes $2, 3, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\wr C_2^2$ (as 16T127)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-72, -864, -2880, -2544, 4084, 7560, 1884, -2784, -2379, -1212, -652, -288, -75, -12, -6, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^14 - 12*x^13 - 75*x^12 - 288*x^11 - 652*x^10 - 1212*x^9 - 2379*x^8 - 2784*x^7 + 1884*x^6 + 7560*x^5 + 4084*x^4 - 2544*x^3 - 2880*x^2 - 864*x - 72)
 
gp: K = bnfinit(x^16 - 6*x^14 - 12*x^13 - 75*x^12 - 288*x^11 - 652*x^10 - 1212*x^9 - 2379*x^8 - 2784*x^7 + 1884*x^6 + 7560*x^5 + 4084*x^4 - 2544*x^3 - 2880*x^2 - 864*x - 72, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{14} - 12 x^{13} - 75 x^{12} - 288 x^{11} - 652 x^{10} - 1212 x^{9} - 2379 x^{8} - 2784 x^{7} + 1884 x^{6} + 7560 x^{5} + 4084 x^{4} - 2544 x^{3} - 2880 x^{2} - 864 x - 72 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3479315915610802970689536=2^{38}\cdot 3^{10}\cdot 11^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.19$
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:  $2, 3, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{8} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{6} a^{11} - \frac{1}{3} a^{9} - \frac{1}{2} a^{7} - \frac{1}{3} a^{5} + \frac{1}{6} a^{3}$, $\frac{1}{66} a^{12} - \frac{5}{66} a^{11} - \frac{1}{33} a^{10} - \frac{13}{33} a^{9} - \frac{3}{22} a^{8} + \frac{5}{22} a^{7} + \frac{14}{33} a^{6} - \frac{16}{33} a^{5} - \frac{29}{66} a^{4} + \frac{19}{66} a^{3} + \frac{3}{11} a^{2} + \frac{3}{11} a - \frac{4}{11}$, $\frac{1}{66} a^{13} - \frac{5}{66} a^{11} + \frac{4}{33} a^{10} + \frac{5}{22} a^{9} + \frac{7}{33} a^{8} - \frac{29}{66} a^{7} - \frac{4}{11} a^{6} + \frac{31}{66} a^{5} - \frac{8}{33} a^{4} + \frac{1}{22} a^{3} + \frac{10}{33} a^{2} + \frac{2}{11}$, $\frac{1}{660} a^{14} - \frac{1}{330} a^{13} + \frac{1}{165} a^{11} + \frac{7}{60} a^{10} - \frac{39}{110} a^{9} + \frac{25}{66} a^{8} - \frac{16}{33} a^{7} + \frac{7}{220} a^{6} - \frac{163}{330} a^{5} - \frac{1}{30} a^{4} + \frac{21}{55} a^{3} - \frac{27}{55} a^{2} + \frac{14}{55} a - \frac{12}{55}$, $\frac{1}{26835431761182660} a^{15} + \frac{106830731607}{447257196019711} a^{14} - \frac{72609709203457}{13417715880591330} a^{13} - \frac{489536265553}{638938851456730} a^{12} - \frac{133759971180431}{5367086352236532} a^{11} + \frac{206328281781905}{1341771588059133} a^{10} - \frac{39962107528659}{2236285980098555} a^{9} + \frac{116062676942197}{2683543176118266} a^{8} + \frac{1606535940577201}{26835431761182660} a^{7} - \frac{69791626268082}{203298725463505} a^{6} - \frac{3247306337174236}{6708857940295665} a^{5} - \frac{4250063662405741}{13417715880591330} a^{4} - \frac{10456822305458}{21996255541953} a^{3} + \frac{271932003936077}{1341771588059133} a^{2} + \frac{633966531425441}{2236285980098555} a - \frac{191671118361554}{2236285980098555}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $11$
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:  \( 5932651.45643 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\wr C_2^2$ (as 16T127):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 16 conjugacy class representatives for $C_2\wr C_2^2$
Character table for $C_2\wr C_2^2$

Intermediate fields

\(\Q(\sqrt{22}) \), \(\Q(\sqrt{66}) \), \(\Q(\sqrt{33}) \), \(\Q(\sqrt{11}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\sqrt{3}, \sqrt{22})\), \(\Q(\sqrt{3}, \sqrt{11})\), \(\Q(\sqrt{6}, \sqrt{22})\), \(\Q(\sqrt{6}, \sqrt{11})\), \(\Q(\sqrt{2}, \sqrt{11})\), \(\Q(\sqrt{2}, \sqrt{33})\), 8.8.77720518656.1, 8.4.932646223872.3, 8.4.233161555968.3

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

Sibling fields

Degree 8 siblings: data not computed
Degree 16 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$2.8.22.83$x^{8} + 4 x^{7} + 2 x^{4} + 4 x^{2} + 14$$8$$1$$22$$D_4$$[2, 3, 7/2]$
2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
$11$11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$