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Properties

Label 16.0.37038789304...0625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 79^{8}$
Root discriminant $29.72$
Ramified primes $5, 79$
Class number $10$ (GRH)
Class group $[10]$ (GRH)
Galois group $QD_{16}$ (as 16T12)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -7, 69, 69, 684, 1715, 2252, 1943, 1263, 338, 187, -90, 99, -31, 4, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 4*x^14 - 31*x^13 + 99*x^12 - 90*x^11 + 187*x^10 + 338*x^9 + 1263*x^8 + 1943*x^7 + 2252*x^6 + 1715*x^5 + 684*x^4 + 69*x^3 + 69*x^2 - 7*x + 1)
 
gp: K = bnfinit(x^16 - 2*x^15 + 4*x^14 - 31*x^13 + 99*x^12 - 90*x^11 + 187*x^10 + 338*x^9 + 1263*x^8 + 1943*x^7 + 2252*x^6 + 1715*x^5 + 684*x^4 + 69*x^3 + 69*x^2 - 7*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 4 x^{14} - 31 x^{13} + 99 x^{12} - 90 x^{11} + 187 x^{10} + 338 x^{9} + 1263 x^{8} + 1943 x^{7} + 2252 x^{6} + 1715 x^{5} + 684 x^{4} + 69 x^{3} + 69 x^{2} - 7 x + 1 \)

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:  \(370387893043593994140625=5^{12}\cdot 79^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.72$
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:  $5, 79$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8} + \frac{1}{5} a^{6} + \frac{1}{5} a^{4} + \frac{1}{5} a^{2} + \frac{1}{5}$, $\frac{1}{5} a^{9} + \frac{1}{5} a^{7} + \frac{1}{5} a^{5} + \frac{1}{5} a^{3} + \frac{1}{5} a$, $\frac{1}{10} a^{10} - \frac{1}{2} a^{5} - \frac{1}{10}$, $\frac{1}{50} a^{11} - \frac{1}{25} a^{10} + \frac{2}{25} a^{9} + \frac{2}{25} a^{8} - \frac{8}{25} a^{7} + \frac{9}{50} a^{6} - \frac{8}{25} a^{5} + \frac{12}{25} a^{4} + \frac{12}{25} a^{3} + \frac{12}{25} a^{2} - \frac{17}{50} a - \frac{2}{25}$, $\frac{1}{50} a^{12} + \frac{1}{25} a^{9} + \frac{1}{25} a^{8} + \frac{17}{50} a^{7} + \frac{6}{25} a^{6} - \frac{9}{25} a^{5} - \frac{9}{25} a^{4} + \frac{6}{25} a^{3} - \frac{9}{50} a^{2} + \frac{1}{25} a + \frac{1}{25}$, $\frac{1}{50} a^{13} + \frac{1}{25} a^{10} + \frac{1}{25} a^{9} - \frac{3}{50} a^{8} + \frac{6}{25} a^{7} + \frac{6}{25} a^{6} - \frac{9}{25} a^{5} - \frac{4}{25} a^{4} - \frac{9}{50} a^{3} - \frac{9}{25} a^{2} + \frac{1}{25} a - \frac{2}{5}$, $\frac{1}{44750} a^{14} + \frac{1}{44750} a^{13} - \frac{439}{44750} a^{12} - \frac{2}{22375} a^{11} - \frac{1019}{44750} a^{10} + \frac{427}{44750} a^{9} + \frac{3967}{44750} a^{8} + \frac{2587}{44750} a^{7} + \frac{2341}{22375} a^{6} + \frac{9577}{44750} a^{5} + \frac{12051}{44750} a^{4} + \frac{15091}{44750} a^{3} - \frac{1749}{44750} a^{2} - \frac{2682}{22375} a + \frac{5071}{44750}$, $\frac{1}{1157015322250} a^{15} + \frac{2004528}{578507661125} a^{14} + \frac{9002686771}{1157015322250} a^{13} + \frac{6191238861}{1157015322250} a^{12} + \frac{1830657431}{1157015322250} a^{11} - \frac{20543327023}{1157015322250} a^{10} + \frac{46057622931}{578507661125} a^{9} - \frac{19236968693}{1157015322250} a^{8} + \frac{153497143177}{1157015322250} a^{7} - \frac{21361325103}{1157015322250} a^{6} - \frac{155651226879}{1157015322250} a^{5} + \frac{32042188803}{578507661125} a^{4} - \frac{131134847289}{1157015322250} a^{3} - \frac{33960784809}{1157015322250} a^{2} + \frac{457966303641}{1157015322250} a + \frac{19006767657}{115701532225}$

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

Class group and class number

$C_{10}$, which has order $10$ (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:  \( 24561.9499263 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$SD_{16}$ (as 16T12):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 7 conjugacy class representatives for $QD_{16}$
Character table for $QD_{16}$

Intermediate fields

\(\Q(\sqrt{-395}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-79}) \), \(\Q(\sqrt{5}, \sqrt{-79})\), 4.2.1975.1 x2, 4.0.31205.1 x2, 8.0.24343800625.1, 8.2.7703734375.1 x4

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

Sibling fields

Degree 8 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$5$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
$79$79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$