↑

Properties

Label 16.0.11740491199...2081.2
Degree $16$
Signature $[0, 8]$
Discriminant $17^{12}\cdot 67^{4}$
Root discriminant $23.95$
Ramified primes $17, 67$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\wr C_4$ (as 16T158)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![404, -692, 955, 1109, -1605, -270, 971, 160, -105, -137, 140, -56, 15, -8, 4, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 4*x^14 - 8*x^13 + 15*x^12 - 56*x^11 + 140*x^10 - 137*x^9 - 105*x^8 + 160*x^7 + 971*x^6 - 270*x^5 - 1605*x^4 + 1109*x^3 + 955*x^2 - 692*x + 404)
 
gp: K = bnfinit(x^16 - 2*x^15 + 4*x^14 - 8*x^13 + 15*x^12 - 56*x^11 + 140*x^10 - 137*x^9 - 105*x^8 + 160*x^7 + 971*x^6 - 270*x^5 - 1605*x^4 + 1109*x^3 + 955*x^2 - 692*x + 404, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 4 x^{14} - 8 x^{13} + 15 x^{12} - 56 x^{11} + 140 x^{10} - 137 x^{9} - 105 x^{8} + 160 x^{7} + 971 x^{6} - 270 x^{5} - 1605 x^{4} + 1109 x^{3} + 955 x^{2} - 692 x + 404 \)

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:  \(11740491199707618712081=17^{12}\cdot 67^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.95$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{26} a^{14} + \frac{1}{26} a^{13} + \frac{2}{13} a^{12} + \frac{1}{26} a^{11} + \frac{3}{13} a^{10} + \frac{11}{26} a^{9} - \frac{1}{26} a^{8} + \frac{9}{26} a^{7} + \frac{3}{26} a^{6} + \frac{6}{13} a^{5} + \frac{5}{13} a^{4} + \frac{5}{13} a^{3} + \frac{7}{26} a^{2} + \frac{4}{13} a - \frac{2}{13}$, $\frac{1}{36200577396303545725674772} a^{15} + \frac{10497439100833211773045}{1392329899857828681756722} a^{14} + \frac{2784441940080073299887115}{18100288698151772862837386} a^{13} - \frac{598655720551164486570390}{9050144349075886431418693} a^{12} + \frac{14480010838280112078381745}{36200577396303545725674772} a^{11} - \frac{1276138595483892364863487}{18100288698151772862837386} a^{10} + \frac{7912135768213897905242643}{18100288698151772862837386} a^{9} - \frac{1124657781881368824641867}{36200577396303545725674772} a^{8} + \frac{15084651533360267504364167}{36200577396303545725674772} a^{7} - \frac{784194015044588590423866}{9050144349075886431418693} a^{6} + \frac{7453483433171892357272367}{36200577396303545725674772} a^{5} - \frac{236344136338468205027696}{696164949928914340878361} a^{4} + \frac{5273483612496720347630251}{36200577396303545725674772} a^{3} + \frac{6378808671443872694125889}{36200577396303545725674772} a^{2} + \frac{11845661353853991441667605}{36200577396303545725674772} a + \frac{791195071600086197019445}{18100288698151772862837386}$

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:  $7$
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:  \( 93737.0374562 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\wr C_4$ (as 16T158):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 13 conjugacy class representatives for $C_2\wr C_4$
Character table for $C_2\wr C_4$

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, 4.2.19363.1, 4.2.329171.1, 8.2.1617217123.1 x2, 8.0.6373738073.1 x2, 8.4.108353547241.1

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

Sibling fields

Degree 8 siblings: data not computed
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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{8}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$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}$
17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$
$67$67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.4.2.1$x^{4} + 1541 x^{2} + 646416$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
67.4.2.1$x^{4} + 1541 x^{2} + 646416$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$