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Properties

Label 16.0.10407187972...0944.3
Degree $16$
Signature $[0, 8]$
Discriminant $2^{36}\cdot 17^{6}\cdot 89^{4}$
Root discriminant $42.27$
Ramified primes $2, 17, 89$
Class number $8$ (GRH)
Class group $[2, 2, 2]$ (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![77313634, -135856912, 143123840, -100180644, 57244536, -26323352, 10775414, -3647616, 1145051, -289244, 73000, -13200, 2798, -324, 66, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 66*x^14 - 324*x^13 + 2798*x^12 - 13200*x^11 + 73000*x^10 - 289244*x^9 + 1145051*x^8 - 3647616*x^7 + 10775414*x^6 - 26323352*x^5 + 57244536*x^4 - 100180644*x^3 + 143123840*x^2 - 135856912*x + 77313634)
 
gp: K = bnfinit(x^16 - 4*x^15 + 66*x^14 - 324*x^13 + 2798*x^12 - 13200*x^11 + 73000*x^10 - 289244*x^9 + 1145051*x^8 - 3647616*x^7 + 10775414*x^6 - 26323352*x^5 + 57244536*x^4 - 100180644*x^3 + 143123840*x^2 - 135856912*x + 77313634, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 66 x^{14} - 324 x^{13} + 2798 x^{12} - 13200 x^{11} + 73000 x^{10} - 289244 x^{9} + 1145051 x^{8} - 3647616 x^{7} + 10775414 x^{6} - 26323352 x^{5} + 57244536 x^{4} - 100180644 x^{3} + 143123840 x^{2} - 135856912 x + 77313634 \)

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:  \(104071879720680162479570944=2^{36}\cdot 17^{6}\cdot 89^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.27$
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:  $2, 17, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{8} a^{12} - \frac{1}{4} a^{11} + \frac{1}{4} a^{10} + \frac{1}{4} a^{9} - \frac{3}{8} a^{8} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{8} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} + \frac{1}{8} a^{9} + \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{56} a^{14} + \frac{1}{28} a^{13} + \frac{1}{28} a^{12} + \frac{13}{28} a^{11} - \frac{19}{56} a^{10} + \frac{3}{14} a^{9} + \frac{11}{28} a^{8} + \frac{3}{28} a^{6} + \frac{1}{7} a^{5} - \frac{3}{7} a^{4} + \frac{3}{14} a^{3} + \frac{13}{28} a^{2} + \frac{3}{7} a + \frac{1}{7}$, $\frac{1}{3963781762575012139114130245091896049652261962164408} a^{15} + \frac{1156427859958457772781535328907304739687134006451}{990945440643753034778532561272974012413065490541102} a^{14} - \frac{56769532060005630327507719645898935038665568858}{5444755168372269421860069017983373694577282915061} a^{13} - \frac{550500786695798451460789006468955853479106473159}{495472720321876517389266280636487006206532745270551} a^{12} - \frac{1291822728410976897975858861834821896415318826790951}{3963781762575012139114130245091896049652261962164408} a^{11} + \frac{319273148412473385416582871814258510312794647921251}{1981890881287506069557065122545948024826130981082204} a^{10} - \frac{27128648047007832865311603140982198156750448066863}{76226572357211771906040966251767231724081960810854} a^{9} + \frac{628962073336235795923915146562442813757804858003585}{1981890881287506069557065122545948024826130981082204} a^{8} - \frac{500042776704187618654397951994322584320096373251223}{1981890881287506069557065122545948024826130981082204} a^{7} + \frac{205865566276470719414631085951128412131100076605407}{495472720321876517389266280636487006206532745270551} a^{6} - \frac{490882880351559318650577613368481270151888340663091}{990945440643753034778532561272974012413065490541102} a^{5} - \frac{138914774676714779978928175600838139062773270840502}{495472720321876517389266280636487006206532745270551} a^{4} + \frac{14557957815299631979712893313704748345106022292679}{283127268755358009936723588935135432118018711583172} a^{3} - \frac{21652793760995776558611584400018994446856128186813}{76226572357211771906040966251767231724081960810854} a^{2} + \frac{80347688433943234071394072111755955301407780224111}{990945440643753034778532561272974012413065490541102} a - \frac{77438846045668573009552193288607041809052583818025}{990945440643753034778532561272974012413065490541102}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}$, which has order $8$ (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:  \( -\frac{8639361832890622549794948411923083292641}{33042804312932019599314184388765295222970031112} a^{15} + \frac{3530195762768803561368107669389344743496}{4130350539116502449914273048595661902871253889} a^{14} - \frac{10297481646816738605812821621696486267871}{635438544479461915371426622860871061980192906} a^{13} + \frac{2346479449368480448006446764959724712534917}{33042804312932019599314184388765295222970031112} a^{12} - \frac{21479258917137992490347547956264727995335771}{33042804312932019599314184388765295222970031112} a^{11} + \frac{11696872691272819487411558995552150897057337}{4130350539116502449914273048595661902871253889} a^{10} - \frac{20021025026801084144833271687223938880776775}{1270877088958923830742853245721742123960385812} a^{9} + \frac{1926150766056279154084671737381151642201998487}{33042804312932019599314184388765295222970031112} a^{8} - \frac{3699330683782849041094805322428911628331024981}{16521402156466009799657092194382647611485015556} a^{7} + \frac{10998794453181949256076639660529328671242924271}{16521402156466009799657092194382647611485015556} a^{6} - \frac{15206731624524713491694021791557821616523542779}{8260701078233004899828546097191323805742507778} a^{5} + \frac{67864561914429019708407962499783774609734564779}{16521402156466009799657092194382647611485015556} a^{4} - \frac{131709538821963912752830068913967506060531691929}{16521402156466009799657092194382647611485015556} a^{3} + \frac{7627960926370600673413303052039928523866999467}{635438544479461915371426622860871061980192906} a^{2} - \frac{54129732762252390792156258560852610000263899296}{4130350539116502449914273048595661902871253889} a + \frac{111196726194959047068725403972284504758179471451}{16521402156466009799657092194382647611485015556} \) (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:  \( 3247748.4223 \) (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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), 4.4.4352.1, 4.0.1088.2, \(\Q(\zeta_{8})\), 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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$2.8.18.53$x^{8} + 2 x^{6} + 4 x^{3} + 2$$8$$1$$18$$D_4\times C_2$$[2, 2, 3]^{2}$
2.8.18.53$x^{8} + 2 x^{6} + 4 x^{3} + 2$$8$$1$$18$$D_4\times C_2$$[2, 2, 3]^{2}$
$17$17.4.3.3$x^{4} + 51$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.3.3$x^{4} + 51$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
89Data not computed