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Properties

Label 16.8.13066428858...0625.1
Degree $16$
Signature $[8, 4]$
Discriminant $3^{8}\cdot 5^{12}\cdot 13^{8}$
Root discriminant $20.88$
Ramified primes $3, 5, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_4 \times D_4$ (as 16T19)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-29, 91, -82, 59, -160, 110, 209, -332, 136, 13, -46, 62, -37, 6, 4, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 4*x^14 + 6*x^13 - 37*x^12 + 62*x^11 - 46*x^10 + 13*x^9 + 136*x^8 - 332*x^7 + 209*x^6 + 110*x^5 - 160*x^4 + 59*x^3 - 82*x^2 + 91*x - 29)
 
gp: K = bnfinit(x^16 - 4*x^15 + 4*x^14 + 6*x^13 - 37*x^12 + 62*x^11 - 46*x^10 + 13*x^9 + 136*x^8 - 332*x^7 + 209*x^6 + 110*x^5 - 160*x^4 + 59*x^3 - 82*x^2 + 91*x - 29, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 4 x^{14} + 6 x^{13} - 37 x^{12} + 62 x^{11} - 46 x^{10} + 13 x^{9} + 136 x^{8} - 332 x^{7} + 209 x^{6} + 110 x^{5} - 160 x^{4} + 59 x^{3} - 82 x^{2} + 91 x - 29 \)

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:  \(1306642885859619140625=3^{8}\cdot 5^{12}\cdot 13^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.88$
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:  $3, 5, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{114} a^{14} - \frac{9}{38} a^{13} + \frac{3}{38} a^{11} + \frac{41}{114} a^{10} - \frac{4}{57} a^{9} + \frac{49}{114} a^{8} - \frac{47}{114} a^{7} + \frac{2}{57} a^{6} - \frac{5}{114} a^{5} + \frac{47}{114} a^{4} - \frac{2}{19} a^{3} + \frac{13}{38} a^{2} - \frac{7}{114} a - \frac{7}{57}$, $\frac{1}{30160145406} a^{15} + \frac{3746329}{10053381802} a^{14} + \frac{2276802861}{10053381802} a^{13} - \frac{21391071}{10053381802} a^{12} - \frac{7526181739}{30160145406} a^{11} - \frac{3775003073}{30160145406} a^{10} + \frac{3934970569}{30160145406} a^{9} + \frac{4790389423}{30160145406} a^{8} + \frac{5601164161}{30160145406} a^{7} - \frac{1845950321}{30160145406} a^{6} - \frac{2399561353}{30160145406} a^{5} - \frac{846340257}{10053381802} a^{4} - \frac{2104772605}{10053381802} a^{3} + \frac{3097781369}{30160145406} a^{2} + \frac{12732988375}{30160145406} a + \frac{82862447}{173334169}$

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:  \( 45306.2606629 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\times D_4$ (as 16T19):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_4 \times D_4$
Character table for $C_4 \times D_4$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{65}) \), \(\Q(\zeta_{15})^+\), 4.4.190125.1, 4.2.507.1, 4.2.12675.1, \(\Q(\sqrt{5}, \sqrt{13})\), 8.4.160655625.1, 8.8.36147515625.1, 8.4.4016390625.1

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 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$3$3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5Data not computed
$13$13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$