↑

Properties

Label 16.0.40219289990...7617.5
Degree $16$
Signature $[0, 8]$
Discriminant $17^{13}\cdot 67^{8}$
Root discriminant $81.80$
Ramified primes $17, 67$
Class number $16$ (GRH)
Class group $[4, 4]$ (GRH)
Galois group $C_4.D_4:C_4$ (as 16T289)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![444313, -457257, -788730, 624720, 755793, -230649, -281102, 68003, 70051, -4348, -3698, 2439, 702, 15, 15, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 15*x^14 + 15*x^13 + 702*x^12 + 2439*x^11 - 3698*x^10 - 4348*x^9 + 70051*x^8 + 68003*x^7 - 281102*x^6 - 230649*x^5 + 755793*x^4 + 624720*x^3 - 788730*x^2 - 457257*x + 444313)
 
gp: K = bnfinit(x^16 - 7*x^15 + 15*x^14 + 15*x^13 + 702*x^12 + 2439*x^11 - 3698*x^10 - 4348*x^9 + 70051*x^8 + 68003*x^7 - 281102*x^6 - 230649*x^5 + 755793*x^4 + 624720*x^3 - 788730*x^2 - 457257*x + 444313, 1)
 

Normalized defining polynomial

\( x^{16} - 7 x^{15} + 15 x^{14} + 15 x^{13} + 702 x^{12} + 2439 x^{11} - 3698 x^{10} - 4348 x^{9} + 70051 x^{8} + 68003 x^{7} - 281102 x^{6} - 230649 x^{5} + 755793 x^{4} + 624720 x^{3} - 788730 x^{2} - 457257 x + 444313 \)

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:  \(4021928999000637617913142677617=17^{13}\cdot 67^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $81.80$
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:  $17, 67$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{67} a^{14} - \frac{33}{67} a^{13} - \frac{33}{67} a^{12} + \frac{18}{67} a^{11} - \frac{18}{67} a^{10} - \frac{1}{67} a^{9} - \frac{27}{67} a^{8} + \frac{7}{67} a^{7} - \frac{5}{67} a^{6} + \frac{17}{67} a^{5} + \frac{31}{67} a^{4} - \frac{29}{67} a^{3} - \frac{30}{67} a^{2} - \frac{2}{67} a + \frac{24}{67}$, $\frac{1}{795847859839655459453807697852371141358160669} a^{15} - \frac{4264141433552657696306689068872776219586394}{795847859839655459453807697852371141358160669} a^{14} - \frac{105113150060320693363899203001426462376328262}{795847859839655459453807697852371141358160669} a^{13} + \frac{269856608567917006744667887897997208707302584}{795847859839655459453807697852371141358160669} a^{12} - \frac{213801245235331728778633557320957297996982008}{795847859839655459453807697852371141358160669} a^{11} + \frac{52709577008511038007219268681092733204004084}{795847859839655459453807697852371141358160669} a^{10} - \frac{63680040969175804918157190761234518556216350}{795847859839655459453807697852371141358160669} a^{9} - \frac{50187570870799094328960347435817194946499}{11878326266263514320206085042572703602360607} a^{8} + \frac{204188270386434525008506496760922764616974968}{795847859839655459453807697852371141358160669} a^{7} - \frac{10553815724987198464028162219990348180201090}{34602080862593715628426421645755267015572203} a^{6} + \frac{192111100598790469169476723231451025671341557}{795847859839655459453807697852371141358160669} a^{5} - \frac{189685489394710825134776784344527485311445748}{795847859839655459453807697852371141358160669} a^{4} + \frac{195490856238855024447378583037210311459213120}{795847859839655459453807697852371141358160669} a^{3} + \frac{7341788493842866046606399342480996293478357}{34602080862593715628426421645755267015572203} a^{2} - \frac{13355887501047723022559242042121937677609650}{41886729465245024181779352518545849545166351} a - \frac{43684905295992342487410169953806652647537352}{795847859839655459453807697852371141358160669}$

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

Class group and class number

$C_{4}\times C_{4}$, 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:  \( 33718131.2216 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4.D_4:C_4$ (as 16T289):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 44 conjugacy class representatives for $C_4.D_4:C_4$
Character table for $C_4.D_4:C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-67}) \), 4.0.76313.1, 8.0.28611710209697.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ $16$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$17$17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$
17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$
$67$67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$