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Properties

Label 16.0.91778367105...8144.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 17^{6}\cdot 97^{4}$
Root discriminant $36.32$
Ramified primes $2, 17, 97$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $C_2^3.C_2^4.C_2$ (as 16T657)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![932249, -1918932, 1265966, -132104, -113236, -28944, 35626, -988, 869, -3616, 950, 380, -228, 44, 0, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 44*x^13 - 228*x^12 + 380*x^11 + 950*x^10 - 3616*x^9 + 869*x^8 - 988*x^7 + 35626*x^6 - 28944*x^5 - 113236*x^4 - 132104*x^3 + 1265966*x^2 - 1918932*x + 932249)
 
gp: K = bnfinit(x^16 - 4*x^15 + 44*x^13 - 228*x^12 + 380*x^11 + 950*x^10 - 3616*x^9 + 869*x^8 - 988*x^7 + 35626*x^6 - 28944*x^5 - 113236*x^4 - 132104*x^3 + 1265966*x^2 - 1918932*x + 932249, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 44 x^{13} - 228 x^{12} + 380 x^{11} + 950 x^{10} - 3616 x^{9} + 869 x^{8} - 988 x^{7} + 35626 x^{6} - 28944 x^{5} - 113236 x^{4} - 132104 x^{3} + 1265966 x^{2} - 1918932 x + 932249 \)

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:  \(9177836710508849627398144=2^{32}\cdot 17^{6}\cdot 97^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.32$
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:  $2, 17, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{42} a^{12} - \frac{1}{7} a^{11} - \frac{2}{21} a^{10} - \frac{4}{21} a^{9} + \frac{1}{42} a^{8} + \frac{1}{3} a^{7} + \frac{5}{21} a^{6} + \frac{2}{7} a^{5} + \frac{1}{14} a^{4} + \frac{2}{7} a^{3} - \frac{3}{7} a^{2} + \frac{1}{7} a + \frac{4}{21}$, $\frac{1}{42} a^{13} + \frac{1}{21} a^{11} + \frac{5}{21} a^{10} - \frac{5}{42} a^{9} - \frac{1}{42} a^{8} + \frac{5}{21} a^{7} - \frac{2}{7} a^{6} - \frac{3}{14} a^{5} + \frac{3}{14} a^{4} + \frac{2}{7} a^{3} - \frac{3}{7} a^{2} + \frac{1}{21} a - \frac{5}{14}$, $\frac{1}{294} a^{14} - \frac{1}{294} a^{13} - \frac{1}{294} a^{11} + \frac{5}{42} a^{10} - \frac{11}{147} a^{9} - \frac{9}{49} a^{8} - \frac{29}{294} a^{7} - \frac{101}{294} a^{6} - \frac{22}{49} a^{5} + \frac{3}{49} a^{4} + \frac{3}{98} a^{3} + \frac{1}{21} a^{2} + \frac{13}{294} a + \frac{10}{147}$, $\frac{1}{318635642633886601430844046974} a^{15} - \frac{227785400474371412478636781}{318635642633886601430844046974} a^{14} - \frac{826687740316320492087848289}{106211880877962200476948015658} a^{13} - \frac{1190937299394852968437436441}{318635642633886601430844046974} a^{12} - \frac{2730509918030996757629548195}{24510434048760507802372618998} a^{11} + \frac{7346007180547996898233228927}{53105940438981100238474007829} a^{10} - \frac{8275949434606482703950824779}{45519377519126657347263435282} a^{9} + \frac{5213027177624800931706210217}{106211880877962200476948015658} a^{8} - \frac{69943726062266039657339647063}{318635642633886601430844046974} a^{7} - \frac{26962259659274956048006037405}{159317821316943300715422023487} a^{6} - \frac{52304095480443457772720143375}{106211880877962200476948015658} a^{5} - \frac{1338870154567052922405088177}{8170144682920169267457539666} a^{4} + \frac{3204220362606036227681503936}{159317821316943300715422023487} a^{3} + \frac{136153403445391188962746455415}{318635642633886601430844046974} a^{2} - \frac{30078957397244891373901610477}{159317821316943300715422023487} a - \frac{16096234646222322663818343787}{159317821316943300715422023487}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( -\frac{2037875026721379917659270}{683767473463275968735716839} a^{15} + \frac{3232074342545189168940047}{455844982308850645823811226} a^{14} + \frac{16096822027035810364225315}{1367534946926551937471433678} a^{13} - \frac{25551793945325031495365871}{227922491154425322911905613} a^{12} + \frac{52236080356627154823421493}{105194995917427072113187206} a^{11} - \frac{218740393848844658906717402}{683767473463275968735716839} a^{10} - \frac{660179078106392407198047673}{195362135275221705353061954} a^{9} + \frac{1199783908318196417286170033}{227922491154425322911905613} a^{8} + \frac{2818441012038594488274672395}{455844982308850645823811226} a^{7} + \frac{8891815117940856007814062588}{683767473463275968735716839} a^{6} - \frac{39017557966050852352567673713}{455844982308850645823811226} a^{5} - \frac{960524196593047411834558451}{17532499319571178685531201} a^{4} + \frac{343937540350445688732576118673}{1367534946926551937471433678} a^{3} + \frac{370578227356131112754635278819}{455844982308850645823811226} a^{2} - \frac{1677452171195855144773650974932}{683767473463275968735716839} a + \frac{1142898265461454923155914898636}{683767473463275968735716839} \) (order $8$)
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:  \( 1056761.71526 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_2^4.C_2$ (as 16T657):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 34 conjugacy class representatives for $C_2^3.C_2^4.C_2$
Character table for $C_2^3.C_2^4.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-2}) \), 4.0.1088.2, 4.4.4352.1, \(\Q(\zeta_{8})\), 8.0.18939904.2

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$2$2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
$17$17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.4.2.2$x^{4} - 17 x^{2} + 867$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
$97$$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
97.2.1.2$x^{2} + 485$$2$$1$$1$$C_2$$[\ ]_{2}$
97.2.1.2$x^{2} + 485$$2$$1$$1$$C_2$$[\ ]_{2}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.2.2$x^{4} - 97 x^{2} + 47045$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$