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Properties

Label 16.16.4007349801...6161.1
Degree $16$
Signature $[16, 0]$
Discriminant $13^{8}\cdot 53^{12}$
Root discriminant $70.82$
Ramified primes $13, 53$
Class number $1$ (GRH)
Class group Trivial (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![201601, 597619, -267121, -2820970, -3462683, -537602, 1827770, 1261949, 62158, -190092, -45478, 8716, 3396, -126, -97, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 97*x^14 - 126*x^13 + 3396*x^12 + 8716*x^11 - 45478*x^10 - 190092*x^9 + 62158*x^8 + 1261949*x^7 + 1827770*x^6 - 537602*x^5 - 3462683*x^4 - 2820970*x^3 - 267121*x^2 + 597619*x + 201601)
 
gp: K = bnfinit(x^16 - 97*x^14 - 126*x^13 + 3396*x^12 + 8716*x^11 - 45478*x^10 - 190092*x^9 + 62158*x^8 + 1261949*x^7 + 1827770*x^6 - 537602*x^5 - 3462683*x^4 - 2820970*x^3 - 267121*x^2 + 597619*x + 201601, 1)
 

Normalized defining polynomial

\( x^{16} - 97 x^{14} - 126 x^{13} + 3396 x^{12} + 8716 x^{11} - 45478 x^{10} - 190092 x^{9} + 62158 x^{8} + 1261949 x^{7} + 1827770 x^{6} - 537602 x^{5} - 3462683 x^{4} - 2820970 x^{3} - 267121 x^{2} + 597619 x + 201601 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(400734980167009195224860426161=13^{8}\cdot 53^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $70.82$
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:  $13, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{7} a^{13} - \frac{2}{7} a^{12} + \frac{2}{7} a^{11} + \frac{1}{7} a^{10} + \frac{2}{7} a^{9} - \frac{1}{7} a^{8} + \frac{3}{7} a^{7} + \frac{2}{7} a^{6} + \frac{1}{7} a^{4} + \frac{3}{7} a^{3} + \frac{3}{7} a^{2} - \frac{2}{7} a + \frac{1}{7}$, $\frac{1}{8701} a^{14} - \frac{24}{791} a^{13} + \frac{3956}{8701} a^{12} - \frac{544}{8701} a^{11} + \frac{461}{8701} a^{10} - \frac{388}{1243} a^{9} - \frac{3991}{8701} a^{8} + \frac{186}{1243} a^{7} - \frac{2820}{8701} a^{6} - \frac{2008}{8701} a^{5} - \frac{619}{1243} a^{4} - \frac{3065}{8701} a^{3} - \frac{1803}{8701} a^{2} - \frac{40}{113} a - \frac{80}{8701}$, $\frac{1}{8229086706986454309228952945529} a^{15} + \frac{83737967294738260249226}{2618226760097503757311152703} a^{14} + \frac{520712379044100021261581267110}{8229086706986454309228952945529} a^{13} - \frac{385868593594197014996141420477}{1175583815283779187032707563647} a^{12} + \frac{2595069165677232630171809246232}{8229086706986454309228952945529} a^{11} + \frac{215524071850001113441535149684}{8229086706986454309228952945529} a^{10} + \frac{59691670884594059005946198841}{748098791544223119020813904139} a^{9} + \frac{62184896745733011573816844040}{748098791544223119020813904139} a^{8} + \frac{3515471572065574542418031448}{72823776168021719550698698633} a^{7} + \frac{1481952732044940038466438529764}{8229086706986454309228952945529} a^{6} - \frac{1220734389097988576331089112321}{8229086706986454309228952945529} a^{5} - \frac{2558306423203135012155603453464}{8229086706986454309228952945529} a^{4} - \frac{3606000211602793639572769203665}{8229086706986454309228952945529} a^{3} - \frac{2604936841331342177929781353932}{8229086706986454309228952945529} a^{2} + \frac{3522508651097804824802476554190}{8229086706986454309228952945529} a + \frac{892617049687828442509380274}{18327587320682526301178068921}$

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:  $15$
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:  \( 3296211055.51 \) (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

\(\Q(\sqrt{689}) \), \(\Q(\sqrt{53}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{13}, \sqrt{53})\), 4.4.36517.1 x2, 4.4.8957.1 x2, 8.8.225360027841.1, 8.8.48695101400413.1 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$13$13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$53$53.4.3.2$x^{4} - 212$$4$$1$$3$$C_4$$[\ ]_{4}$
53.4.3.2$x^{4} - 212$$4$$1$$3$$C_4$$[\ ]_{4}$
53.4.3.2$x^{4} - 212$$4$$1$$3$$C_4$$[\ ]_{4}$
53.4.3.2$x^{4} - 212$$4$$1$$3$$C_4$$[\ ]_{4}$