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Properties

Label 16.0.11256645592...249.14
Degree $16$
Signature $[0, 8]$
Discriminant $13^{8}\cdot 53^{14}$
Root discriminant $116.34$
Ramified primes $13, 53$
Class number $16$ (GRH)
Class group $[2, 2, 2, 2]$ (GRH)
Galois group 16T1263

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![43741763, 13550329, 42347991, -3692254, 8653181, -1459696, -365599, -30297, 26459, -6845, 1940, -1975, 667, -29, 9, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 9*x^14 - 29*x^13 + 667*x^12 - 1975*x^11 + 1940*x^10 - 6845*x^9 + 26459*x^8 - 30297*x^7 - 365599*x^6 - 1459696*x^5 + 8653181*x^4 - 3692254*x^3 + 42347991*x^2 + 13550329*x + 43741763)
 
gp: K = bnfinit(x^16 - 8*x^15 + 9*x^14 - 29*x^13 + 667*x^12 - 1975*x^11 + 1940*x^10 - 6845*x^9 + 26459*x^8 - 30297*x^7 - 365599*x^6 - 1459696*x^5 + 8653181*x^4 - 3692254*x^3 + 42347991*x^2 + 13550329*x + 43741763, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 9 x^{14} - 29 x^{13} + 667 x^{12} - 1975 x^{11} + 1940 x^{10} - 6845 x^{9} + 26459 x^{8} - 30297 x^{7} - 365599 x^{6} - 1459696 x^{5} + 8653181 x^{4} - 3692254 x^{3} + 42347991 x^{2} + 13550329 x + 43741763 \)

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:  \(1125664559289128829386632937086249=13^{8}\cdot 53^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $116.34$
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 not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{13} a^{13} + \frac{1}{13} a^{12} + \frac{1}{13} a^{11} + \frac{2}{13} a^{10} + \frac{5}{13} a^{9} - \frac{1}{13} a^{8} - \frac{3}{13} a^{6} + \frac{3}{13} a^{5} + \frac{6}{13} a^{4} + \frac{3}{13} a^{3} - \frac{1}{13} a^{2} - \frac{4}{13} a$, $\frac{1}{13} a^{14} + \frac{1}{13} a^{11} + \frac{3}{13} a^{10} - \frac{6}{13} a^{9} + \frac{1}{13} a^{8} - \frac{3}{13} a^{7} + \frac{6}{13} a^{6} + \frac{3}{13} a^{5} - \frac{3}{13} a^{4} - \frac{4}{13} a^{3} - \frac{3}{13} a^{2} + \frac{4}{13} a$, $\frac{1}{31556300454149836014696752814836349641375372988555728183} a^{15} - \frac{1067898556893437644526583327917905799366173000516271095}{31556300454149836014696752814836349641375372988555728183} a^{14} + \frac{242981271699112282192571356710634560905563053025879205}{31556300454149836014696752814836349641375372988555728183} a^{13} + \frac{800541828981763691894822943211386278906183168515292553}{31556300454149836014696752814836349641375372988555728183} a^{12} - \frac{7409160771699432745741810857204554845280344313864814731}{31556300454149836014696752814836349641375372988555728183} a^{11} - \frac{5635310974261652215974195457218286898002841421080235447}{31556300454149836014696752814836349641375372988555728183} a^{10} - \frac{3526363179784200555076621712538504655377035841391267519}{31556300454149836014696752814836349641375372988555728183} a^{9} + \frac{9825959848290574627609213395087442700909208510159271555}{31556300454149836014696752814836349641375372988555728183} a^{8} - \frac{7821695744372094632870719040219667481167682200805458934}{31556300454149836014696752814836349641375372988555728183} a^{7} + \frac{172306837510765724881637002880036172517648316558828516}{31556300454149836014696752814836349641375372988555728183} a^{6} - \frac{6228835778940313364774670881565950737041009876813264138}{31556300454149836014696752814836349641375372988555728183} a^{5} - \frac{1213329095851875687135414731544799805055015153537383187}{31556300454149836014696752814836349641375372988555728183} a^{4} + \frac{4653435213559717115146358489480598271815483200981684121}{31556300454149836014696752814836349641375372988555728183} a^{3} - \frac{13970493656190758075784289385754696696118707951029854360}{31556300454149836014696752814836349641375372988555728183} a^{2} - \frac{996017921472499404677047287781928377087547952292227865}{2427407727242295078053596370372026895490413306811979091} a + \frac{16398567226396347801717648563206616192021611766751927}{186723671326330390619507413105540530422339485139383007}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}$, which has order $16$ (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:  \( 354589050.14 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1263:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 34 conjugacy class representatives for t16n1263
Character table for t16n1263 is not computed

Intermediate fields

\(\Q(\sqrt{53}) \), 4.0.148877.1, 8.0.288136694677.1

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

Sibling fields

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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$13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
$53$53.8.7.2$x^{8} - 212$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
53.8.7.2$x^{8} - 212$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$