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Properties

Label 16.0.40219289990...7617.7
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 16T260)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1087469, -231515, 791589, -223361, 332988, -111143, 62463, -34360, 13197, -4682, 2591, -744, 241, -79, 18, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 18*x^14 - 79*x^13 + 241*x^12 - 744*x^11 + 2591*x^10 - 4682*x^9 + 13197*x^8 - 34360*x^7 + 62463*x^6 - 111143*x^5 + 332988*x^4 - 223361*x^3 + 791589*x^2 - 231515*x + 1087469)
 
gp: K = bnfinit(x^16 - 6*x^15 + 18*x^14 - 79*x^13 + 241*x^12 - 744*x^11 + 2591*x^10 - 4682*x^9 + 13197*x^8 - 34360*x^7 + 62463*x^6 - 111143*x^5 + 332988*x^4 - 223361*x^3 + 791589*x^2 - 231515*x + 1087469, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 18 x^{14} - 79 x^{13} + 241 x^{12} - 744 x^{11} + 2591 x^{10} - 4682 x^{9} + 13197 x^{8} - 34360 x^{7} + 62463 x^{6} - 111143 x^{5} + 332988 x^{4} - 223361 x^{3} + 791589 x^{2} - 231515 x + 1087469 \)

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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{1273} a^{14} - \frac{495}{1273} a^{13} + \frac{505}{1273} a^{12} - \frac{611}{1273} a^{11} - \frac{385}{1273} a^{10} + \frac{453}{1273} a^{9} - \frac{397}{1273} a^{8} + \frac{369}{1273} a^{7} + \frac{560}{1273} a^{6} + \frac{553}{1273} a^{5} + \frac{629}{1273} a^{4} + \frac{336}{1273} a^{3} - \frac{344}{1273} a^{2} + \frac{499}{1273} a - \frac{587}{1273}$, $\frac{1}{4548959238471825221993884580522023978343} a^{15} - \frac{18978571189117542173481988826752342}{4548959238471825221993884580522023978343} a^{14} + \frac{858796159656622497616842822917622652620}{4548959238471825221993884580522023978343} a^{13} + \frac{1156115755330767211110427519627653595714}{4548959238471825221993884580522023978343} a^{12} + \frac{204015285943956711058424229008401846576}{4548959238471825221993884580522023978343} a^{11} - \frac{404748500321339447774471490265665828423}{4548959238471825221993884580522023978343} a^{10} - \frac{391091344859014168802724667744984212295}{4548959238471825221993884580522023978343} a^{9} + \frac{298903341753223730756620331863586722419}{4548959238471825221993884580522023978343} a^{8} - \frac{470892976743961669685415871980207501859}{4548959238471825221993884580522023978343} a^{7} + \frac{483428070129748518498378817446963969271}{4548959238471825221993884580522023978343} a^{6} + \frac{2241977819712126474603561734783590553580}{4548959238471825221993884580522023978343} a^{5} + \frac{19637046851586148370607997854198689}{3573416526686429868023475711329162591} a^{4} + \frac{1885652556409758945805114997557586667500}{4548959238471825221993884580522023978343} a^{3} + \frac{1768828498259892201766263209468768366168}{4548959238471825221993884580522023978343} a^{2} + \frac{258281440780674929825804230885371815007}{4548959238471825221993884580522023978343} a - \frac{1175181436606303746506682275596715468875}{4548959238471825221993884580522023978343}$

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:  \( 35047489.4828 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4.D_4:C_4$ (as 16T260):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 32 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 sibling: 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 ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ $16$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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}$