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Properties

Label 16.4.15712755557...5489.9
Degree $16$
Signature $[4, 6]$
Discriminant $23^{10}\cdot 41^{14}$
Root discriminant $182.92$
Ramified primes $23, 41$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 16T817

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![249941449, 716455696, 759573920, 382690212, 296743745, -4230432, 19968220, -4981822, 296072, -143382, -45460, 6480, -3078, 388, -14, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 14*x^14 + 388*x^13 - 3078*x^12 + 6480*x^11 - 45460*x^10 - 143382*x^9 + 296072*x^8 - 4981822*x^7 + 19968220*x^6 - 4230432*x^5 + 296743745*x^4 + 382690212*x^3 + 759573920*x^2 + 716455696*x + 249941449)
 
gp: K = bnfinit(x^16 - 2*x^15 - 14*x^14 + 388*x^13 - 3078*x^12 + 6480*x^11 - 45460*x^10 - 143382*x^9 + 296072*x^8 - 4981822*x^7 + 19968220*x^6 - 4230432*x^5 + 296743745*x^4 + 382690212*x^3 + 759573920*x^2 + 716455696*x + 249941449, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 14 x^{14} + 388 x^{13} - 3078 x^{12} + 6480 x^{11} - 45460 x^{10} - 143382 x^{9} + 296072 x^{8} - 4981822 x^{7} + 19968220 x^{6} - 4230432 x^{5} + 296743745 x^{4} + 382690212 x^{3} + 759573920 x^{2} + 716455696 x + 249941449 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1571275555715210001755383712793895489=23^{10}\cdot 41^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $182.92$
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:  $23, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a$, $\frac{1}{236} a^{10} + \frac{1}{236} a^{9} + \frac{1}{236} a^{8} + \frac{5}{118} a^{7} + \frac{27}{118} a^{6} + \frac{11}{118} a^{5} + \frac{3}{236} a^{4} + \frac{65}{236} a^{3} - \frac{16}{59} a^{2} + \frac{33}{236} a + \frac{33}{236}$, $\frac{1}{236} a^{11} + \frac{9}{236} a^{8} + \frac{11}{59} a^{7} - \frac{8}{59} a^{6} - \frac{19}{236} a^{5} - \frac{14}{59} a^{4} + \frac{107}{236} a^{3} + \frac{97}{236} a^{2} - \frac{1}{2} a + \frac{85}{236}$, $\frac{1}{472} a^{12} + \frac{9}{472} a^{9} - \frac{15}{472} a^{8} - \frac{4}{59} a^{7} + \frac{99}{472} a^{6} + \frac{31}{236} a^{5} + \frac{107}{472} a^{4} - \frac{21}{472} a^{3} + \frac{3}{8} a^{2} - \frac{151}{472} a + \frac{1}{8}$, $\frac{1}{472} a^{13} - \frac{1}{472} a^{10} - \frac{25}{472} a^{9} - \frac{21}{236} a^{8} - \frac{1}{472} a^{7} - \frac{3}{236} a^{6} - \frac{113}{472} a^{5} - \frac{51}{472} a^{4} - \frac{1}{472} a^{3} + \frac{17}{472} a^{2} + \frac{201}{472} a + \frac{71}{236}$, $\frac{1}{472} a^{14} - \frac{1}{472} a^{11} - \frac{1}{472} a^{10} - \frac{9}{236} a^{9} + \frac{23}{472} a^{8} - \frac{1}{236} a^{7} + \frac{3}{472} a^{6} + \frac{5}{472} a^{5} + \frac{71}{472} a^{4} + \frac{161}{472} a^{3} - \frac{155}{472} a^{2} + \frac{113}{236} a + \frac{21}{118}$, $\frac{1}{1250413793451088884968722562726893522207100775174912536} a^{15} - \frac{629883535486839168353557766075256456582451939841507}{625206896725544442484361281363446761103550387587456268} a^{14} - \frac{104668192052196397858474907703788753350618619176695}{625206896725544442484361281363446761103550387587456268} a^{13} - \frac{469197912297710839545918854192869637989643313918963}{625206896725544442484361281363446761103550387587456268} a^{12} + \frac{1550235037205197791255195927163518150905406025240753}{1250413793451088884968722562726893522207100775174912536} a^{11} + \frac{200225122474086304029620807494722591644042468989084}{156301724181386110621090320340861690275887596896864067} a^{10} - \frac{10734617683180833825166460641500556367406788904076704}{156301724181386110621090320340861690275887596896864067} a^{9} + \frac{65264002393139984019898945423289239964174935609100719}{1250413793451088884968722562726893522207100775174912536} a^{8} + \frac{149990794438610088575781109500899213091908237751386649}{1250413793451088884968722562726893522207100775174912536} a^{7} + \frac{12507132187473354745157114939441317804518815060816609}{312603448362772221242180640681723380551775193793728134} a^{6} - \frac{203100703092535984229502542912920482538539987073832679}{1250413793451088884968722562726893522207100775174912536} a^{5} + \frac{37218864869307431094602296471345975636705425305148267}{156301724181386110621090320340861690275887596896864067} a^{4} - \frac{8199761463877557877626022835239357373568024407997975}{625206896725544442484361281363446761103550387587456268} a^{3} - \frac{60506413993204196524450560793573097870206411716014801}{1250413793451088884968722562726893522207100775174912536} a^{2} + \frac{69503721092868264310047082881826546019936443022662813}{1250413793451088884968722562726893522207100775174912536} a - \frac{590974544610860326136145123326397666198987926673703239}{1250413793451088884968722562726893522207100775174912536}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  $9$
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:  \( 167735292148 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T817:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 32 conjugacy class representatives for t16n817
Character table for t16n817 is not computed

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.68921.1, 8.8.54500230757132921.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{12}$

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
$23$23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.4.3.2$x^{4} - 23$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
23.4.3.2$x^{4} - 23$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
$41$41.8.7.3$x^{8} - 53136$$8$$1$$7$$C_8$$[\ ]_{8}$
41.8.7.3$x^{8} - 53136$$8$$1$$7$$C_8$$[\ ]_{8}$