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Properties

Label 16.8.98376126526...6864.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{16}\cdot 3^{14}\cdot 11^{12}$
Root discriminant $31.59$
Ramified primes $2, 3, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $SD_{16}:C_2$ (as 16T50)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2351, 3322, 4391, -9798, 3215, 5144, -6833, 2946, 844, -1014, -41, 280, -145, 48, -4, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 4*x^14 + 48*x^13 - 145*x^12 + 280*x^11 - 41*x^10 - 1014*x^9 + 844*x^8 + 2946*x^7 - 6833*x^6 + 5144*x^5 + 3215*x^4 - 9798*x^3 + 4391*x^2 + 3322*x - 2351)
 
gp: K = bnfinit(x^16 - 4*x^15 - 4*x^14 + 48*x^13 - 145*x^12 + 280*x^11 - 41*x^10 - 1014*x^9 + 844*x^8 + 2946*x^7 - 6833*x^6 + 5144*x^5 + 3215*x^4 - 9798*x^3 + 4391*x^2 + 3322*x - 2351, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 4 x^{14} + 48 x^{13} - 145 x^{12} + 280 x^{11} - 41 x^{10} - 1014 x^{9} + 844 x^{8} + 2946 x^{7} - 6833 x^{6} + 5144 x^{5} + 3215 x^{4} - 9798 x^{3} + 4391 x^{2} + 3322 x - 2351 \)

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:  \(983761265267629401636864=2^{16}\cdot 3^{14}\cdot 11^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.59$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{22} a^{12} + \frac{4}{11} a^{11} + \frac{5}{22} a^{10} + \frac{2}{11} a^{9} + \frac{9}{22} a^{8} - \frac{1}{11} a^{7} - \frac{7}{22} a^{6} + \frac{5}{11} a^{5} - \frac{5}{11} a^{4} - \frac{1}{11} a^{3} - \frac{3}{11} a^{2} - \frac{5}{11} a + \frac{9}{22}$, $\frac{1}{22} a^{13} + \frac{7}{22} a^{11} + \frac{4}{11} a^{10} - \frac{1}{22} a^{9} - \frac{4}{11} a^{8} + \frac{9}{22} a^{7} - \frac{1}{11} a^{5} - \frac{5}{11} a^{4} + \frac{5}{11} a^{3} - \frac{3}{11} a^{2} + \frac{1}{22} a - \frac{3}{11}$, $\frac{1}{3586} a^{14} - \frac{16}{1793} a^{13} - \frac{45}{3586} a^{12} + \frac{487}{1793} a^{11} - \frac{67}{326} a^{10} - \frac{708}{1793} a^{9} + \frac{1403}{3586} a^{8} - \frac{114}{1793} a^{7} + \frac{346}{1793} a^{6} - \frac{827}{1793} a^{5} + \frac{755}{1793} a^{4} - \frac{804}{1793} a^{3} - \frac{1343}{3586} a^{2} + \frac{10}{1793} a - \frac{677}{1793}$, $\frac{1}{783837283308818904286} a^{15} + \frac{46446949471538857}{391918641654409452143} a^{14} - \frac{245610478911487279}{71257934846256264026} a^{13} - \frac{17351216047808457073}{783837283308818904286} a^{12} + \frac{32384129747353390039}{71257934846256264026} a^{11} - \frac{326542074485075268475}{783837283308818904286} a^{10} - \frac{318028955464271804105}{783837283308818904286} a^{9} - \frac{279093725616589354235}{783837283308818904286} a^{8} - \frac{137777963527838927155}{391918641654409452143} a^{7} + \frac{244475680486718177645}{783837283308818904286} a^{6} + \frac{17512420894975223860}{35628967423128132013} a^{5} + \frac{8849941114669015005}{35628967423128132013} a^{4} - \frac{67984584179111462381}{783837283308818904286} a^{3} + \frac{101680209355973655039}{391918641654409452143} a^{2} - \frac{93009279087426189546}{391918641654409452143} a - \frac{104405009323598180815}{783837283308818904286}$

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:  \( 751195.186818 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$SD_{16}:C_2$ (as 16T50):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $SD_{16}:C_2$
Character table for $SD_{16}:C_2$

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{33}) \), \(\Q(\sqrt{11}) \), 4.4.4752.1 x2, 4.4.13068.1 x2, \(\Q(\sqrt{3}, \sqrt{11})\), 8.8.2732361984.1

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 sibling: 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 ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
3Data not computed
$11$11.8.6.1$x^{8} + 143 x^{4} + 5929$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
11.8.6.1$x^{8} + 143 x^{4} + 5929$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$