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Properties

Label 16.0.37614514447...6416.3
Degree $16$
Signature $[0, 8]$
Discriminant $2^{52}\cdot 17^{4}$
Root discriminant $19.32$
Ramified primes $2, 17$
Class number $2$
Class group $[2]$
Galois group $C_2^4.C_2^3$ (as 16T268)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![17, 104, 24, -104, 980, -1728, 2900, -3248, 3292, -2664, 1784, -1008, 448, -160, 44, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 44*x^14 - 160*x^13 + 448*x^12 - 1008*x^11 + 1784*x^10 - 2664*x^9 + 3292*x^8 - 3248*x^7 + 2900*x^6 - 1728*x^5 + 980*x^4 - 104*x^3 + 24*x^2 + 104*x + 17)
 
gp: K = bnfinit(x^16 - 8*x^15 + 44*x^14 - 160*x^13 + 448*x^12 - 1008*x^11 + 1784*x^10 - 2664*x^9 + 3292*x^8 - 3248*x^7 + 2900*x^6 - 1728*x^5 + 980*x^4 - 104*x^3 + 24*x^2 + 104*x + 17, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 44 x^{14} - 160 x^{13} + 448 x^{12} - 1008 x^{11} + 1784 x^{10} - 2664 x^{9} + 3292 x^{8} - 3248 x^{7} + 2900 x^{6} - 1728 x^{5} + 980 x^{4} - 104 x^{3} + 24 x^{2} + 104 x + 17 \)

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:  \(376145144477611196416=2^{52}\cdot 17^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.32$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{9} a^{14} - \frac{1}{9} a^{13} + \frac{1}{9} a^{10} + \frac{4}{9} a^{9} - \frac{1}{9} a^{8} + \frac{4}{9} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{2}{9} a^{4} + \frac{2}{9} a^{3} + \frac{2}{9} a^{2} + \frac{1}{9} a - \frac{4}{9}$, $\frac{1}{299030074570311147} a^{15} - \frac{101669968707019}{5863334795496297} a^{14} + \frac{31250944833728243}{299030074570311147} a^{13} + \frac{1459507005910939}{99676691523437049} a^{12} - \frac{37830396332630813}{299030074570311147} a^{11} + \frac{35730790345843055}{299030074570311147} a^{10} - \frac{1478471111324675}{33225563841145683} a^{9} + \frac{621044078043905}{11075187947048561} a^{8} - \frac{68677956848921447}{299030074570311147} a^{7} - \frac{34055791484142973}{99676691523437049} a^{6} + \frac{42056089822305185}{299030074570311147} a^{5} - \frac{22732203313057022}{299030074570311147} a^{4} - \frac{60136089918437375}{299030074570311147} a^{3} + \frac{22623845329420190}{99676691523437049} a^{2} + \frac{742780966213373}{4746509120163669} a - \frac{917508088421132}{17590004386488891}$

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:  \( \frac{117194780502358054}{299030074570311147} a^{15} - \frac{18958676574021808}{5863334795496297} a^{14} + \frac{5394803909709439511}{299030074570311147} a^{13} - \frac{6690146118815543243}{99676691523437049} a^{12} + \frac{57355873056633224467}{299030074570311147} a^{11} - \frac{131861804633181262753}{299030074570311147} a^{10} + \frac{26703612562501145641}{33225563841145683} a^{9} - \frac{13646544084802844836}{11075187947048561} a^{8} + \frac{471211517762060295982}{299030074570311147} a^{7} - \frac{162903496114682813008}{99676691523437049} a^{6} + \frac{450363222315673658414}{299030074570311147} a^{5} - \frac{304227019061016304007}{299030074570311147} a^{4} + \frac{181809924173109946213}{299030074570311147} a^{3} - \frac{17483173029368495962}{99676691523437049} a^{2} + \frac{217462647578349782}{4746509120163669} a + \frac{530110373909856598}{17590004386488891} \) (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:  \( 30825.9919521 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3$ (as 16T268):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 29 conjugacy class representatives for $C_2^4.C_2^3$
Character table for $C_2^4.C_2^3$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-2}) \), \(\Q(\zeta_{16})^+\), 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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$$0$Trivial$[\ ]$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.1.2$x^{2} + 51$$2$$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}$