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Properties

Label 16.0.24073289246...0624.5
Degree $16$
Signature $[0, 8]$
Discriminant $2^{58}\cdot 17^{4}$
Root discriminant $25.05$
Ramified primes $2, 17$
Class number $2$
Class group $[2]$
Galois group $C_2^4:C_2^2.C_2$ (as 16T317)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![401, -1496, 2104, -576, -2416, 2080, 1368, -2376, 1112, 1048, -704, 96, 88, -80, 32, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 32*x^14 - 80*x^13 + 88*x^12 + 96*x^11 - 704*x^10 + 1048*x^9 + 1112*x^8 - 2376*x^7 + 1368*x^6 + 2080*x^5 - 2416*x^4 - 576*x^3 + 2104*x^2 - 1496*x + 401)
 
gp: K = bnfinit(x^16 - 8*x^15 + 32*x^14 - 80*x^13 + 88*x^12 + 96*x^11 - 704*x^10 + 1048*x^9 + 1112*x^8 - 2376*x^7 + 1368*x^6 + 2080*x^5 - 2416*x^4 - 576*x^3 + 2104*x^2 - 1496*x + 401, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 32 x^{14} - 80 x^{13} + 88 x^{12} + 96 x^{11} - 704 x^{10} + 1048 x^{9} + 1112 x^{8} - 2376 x^{7} + 1368 x^{6} + 2080 x^{5} - 2416 x^{4} - 576 x^{3} + 2104 x^{2} - 1496 x + 401 \)

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:  \(24073289246567116570624=2^{58}\cdot 17^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.05$
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:  $2, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{7709} a^{14} - \frac{1903}{7709} a^{13} + \frac{175}{593} a^{12} + \frac{3320}{7709} a^{11} - \frac{196}{7709} a^{10} - \frac{1019}{7709} a^{9} + \frac{1}{593} a^{8} + \frac{3018}{7709} a^{7} - \frac{121}{593} a^{6} + \frac{3310}{7709} a^{5} - \frac{1065}{7709} a^{4} - \frac{2161}{7709} a^{3} + \frac{1965}{7709} a^{2} + \frac{391}{7709} a - \frac{3010}{7709}$, $\frac{1}{58025941937878111515877} a^{15} - \frac{1095759637586670588}{58025941937878111515877} a^{14} - \frac{10059927611087686031054}{58025941937878111515877} a^{13} + \frac{24336780841796006197346}{58025941937878111515877} a^{12} - \frac{25787909645239565164338}{58025941937878111515877} a^{11} + \frac{421351406190082043197}{58025941937878111515877} a^{10} - \frac{12880417443763210793690}{58025941937878111515877} a^{9} + \frac{25278339455383309749599}{58025941937878111515877} a^{8} + \frac{1580074545700116497199}{58025941937878111515877} a^{7} - \frac{2486729951813243032291}{58025941937878111515877} a^{6} - \frac{20858132167771338737651}{58025941937878111515877} a^{5} + \frac{4550812805305611819549}{58025941937878111515877} a^{4} - \frac{9408254199392077062348}{58025941937878111515877} a^{3} + \frac{508011217072537387554}{4463533995221393193529} a^{2} - \frac{11961929535775269800270}{58025941937878111515877} a - \frac{28583930376248726795982}{58025941937878111515877}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( \frac{121085215078204695040}{58025941937878111515877} a^{15} - \frac{880898241839033019155}{58025941937878111515877} a^{14} + \frac{3331987590598618656740}{58025941937878111515877} a^{13} - \frac{7774547153036831359087}{58025941937878111515877} a^{12} + \frac{6649790934374865378929}{58025941937878111515877} a^{11} + \frac{13828577361613685653131}{58025941937878111515877} a^{10} - \frac{76421848906322911079131}{58025941937878111515877} a^{9} + \frac{84616194255518468787624}{58025941937878111515877} a^{8} + \frac{152913992934521617940712}{58025941937878111515877} a^{7} - \frac{189936702505497827693087}{58025941937878111515877} a^{6} + \frac{129486408720277301491036}{58025941937878111515877} a^{5} + \frac{303023994230515753657951}{58025941937878111515877} a^{4} - \frac{121743682315803219600229}{58025941937878111515877} a^{3} - \frac{32184499915039369447426}{58025941937878111515877} a^{2} + \frac{208743124082762238695793}{58025941937878111515877} a - \frac{95804163468564611830734}{58025941937878111515877} \) (order $16$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 209331.862536 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4:C_2^2.C_2$ (as 16T317):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 23 conjugacy class representatives for $C_2^4:C_2^2.C_2$
Character table for $C_2^4:C_2^2.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), \(\Q(\zeta_{16})^+\), 4.0.2048.2, \(\Q(\zeta_{8})\), \(\Q(\zeta_{16})\)

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 R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
$17$$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$