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Properties

Label 16.0.28668679679...8249.1
Degree $16$
Signature $[0, 8]$
Discriminant $67^{12}\cdot 89^{9}$
Root discriminant $292.47$
Ramified primes $67, 89$
Class number $2$ (GRH)
Class group $[2]$ (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![26985991018919, 15396026241748, 4836624820184, 939964804413, 16288721013, -34254214176, -5102562617, 185231148, 90584933, 2446813, -823046, -18609, 10072, 523, -97, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 - 97*x^14 + 523*x^13 + 10072*x^12 - 18609*x^11 - 823046*x^10 + 2446813*x^9 + 90584933*x^8 + 185231148*x^7 - 5102562617*x^6 - 34254214176*x^5 + 16288721013*x^4 + 939964804413*x^3 + 4836624820184*x^2 + 15396026241748*x + 26985991018919)
 
gp: K = bnfinit(x^16 - 5*x^15 - 97*x^14 + 523*x^13 + 10072*x^12 - 18609*x^11 - 823046*x^10 + 2446813*x^9 + 90584933*x^8 + 185231148*x^7 - 5102562617*x^6 - 34254214176*x^5 + 16288721013*x^4 + 939964804413*x^3 + 4836624820184*x^2 + 15396026241748*x + 26985991018919, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} - 97 x^{14} + 523 x^{13} + 10072 x^{12} - 18609 x^{11} - 823046 x^{10} + 2446813 x^{9} + 90584933 x^{8} + 185231148 x^{7} - 5102562617 x^{6} - 34254214176 x^{5} + 16288721013 x^{4} + 939964804413 x^{3} + 4836624820184 x^{2} + 15396026241748 x + 26985991018919 \)

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:  \(2866867967976420459213670979269882298249=67^{12}\cdot 89^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $292.47$
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:  $67, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{134} a^{12} - \frac{5}{134} a^{11} - \frac{15}{67} a^{10} + \frac{27}{67} a^{9} + \frac{11}{67} a^{8} + \frac{17}{134} a^{7} + \frac{49}{134} a^{6} + \frac{20}{67} a^{5} + \frac{31}{67} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{134} a^{13} + \frac{6}{67} a^{11} - \frac{29}{134} a^{10} + \frac{12}{67} a^{9} - \frac{7}{134} a^{8} - \frac{25}{67} a^{6} + \frac{61}{134} a^{5} + \frac{21}{67} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{268} a^{14} + \frac{31}{268} a^{11} + \frac{49}{268} a^{10} + \frac{15}{268} a^{9} - \frac{65}{134} a^{8} + \frac{7}{134} a^{7} + \frac{9}{268} a^{6} - \frac{103}{268} a^{5} - \frac{7}{268} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{115736699512537931698871786448812115160628740405584644370553403546864224599072546934316} a^{15} - \frac{105463042451851234729727134882458124934860044845993328162610348092024981242915482217}{115736699512537931698871786448812115160628740405584644370553403546864224599072546934316} a^{14} - \frac{40456599253634965524687613666996173503284727507181149252601178276984020393627062171}{57868349756268965849435893224406057580314370202792322185276701773432112299536273467158} a^{13} - \frac{25195702058124954111220995888955565722487796317869839760402883788747510247909090511}{115736699512537931698871786448812115160628740405584644370553403546864224599072546934316} a^{12} + \frac{400381377719494385896110813869797175324630590224068077026624704461416435808378445765}{57868349756268965849435893224406057580314370202792322185276701773432112299536273467158} a^{11} + \frac{5572950556135331622548865976543405737602846548797453269043526986879771168361955905200}{28934174878134482924717946612203028790157185101396161092638350886716056149768136733579} a^{10} + \frac{13191152764914826880768113686904516566470281963267112386565114896831271387804040275311}{115736699512537931698871786448812115160628740405584644370553403546864224599072546934316} a^{9} - \frac{17106878664507963553969828003560368357512736568411538988439699125788570823822328681495}{57868349756268965849435893224406057580314370202792322185276701773432112299536273467158} a^{8} - \frac{14919679468920577827325771536652749044604215558742683646410655648864759254015839069155}{115736699512537931698871786448812115160628740405584644370553403546864224599072546934316} a^{7} - \frac{457977048182312840291863401387810764099869420134028496001458742355464225709489995866}{1258007603397151431509475939661001251745964569625920047506015255944176354337745075373} a^{6} + \frac{23871642389559066511370497928530998510468992719508159107484230598406255030964898168521}{57868349756268965849435893224406057580314370202792322185276701773432112299536273467158} a^{5} + \frac{4615916064624014352358782413770117198103628627814600864779261267046100268719617658953}{115736699512537931698871786448812115160628740405584644370553403546864224599072546934316} a^{4} + \frac{26394214800131247145686500831758689802195794313848640504969019231977491300358488660}{431853356390066909324148456898552668509808732856659120785647028159941136563703533337} a^{3} + \frac{809416855713388891763080612186242224295764989178313055958931023613636400324751709231}{1727413425560267637296593827594210674039234931426636483142588112639764546254814133348} a^{2} - \frac{387863498009951663225283998116898115473365041016646356592141695180268746991862115097}{863706712780133818648296913797105337019617465713318241571294056319882273127407066674} a - \frac{2990281678539995625602724833634935066971912268159796585263212327042012442574570553}{101612554444721625723329048682012392590543231260390381361328712508221443897342007844}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 10215687727400 \) (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

\(\Q(\sqrt{-67}) \), 4.0.399521.1, 8.0.14205915620249.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ $16$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ $16$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ $16$ $16$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$67$67.8.6.2$x^{8} + 1541 x^{4} + 646416$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
67.8.6.2$x^{8} + 1541 x^{4} + 646416$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
$89$89.4.2.2$x^{4} - 89 x^{2} + 47526$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
89.4.0.1$x^{4} - x + 27$$1$$4$$0$$C_4$$[\ ]^{4}$
89.8.7.4$x^{8} - 64881$$8$$1$$7$$C_8$$[\ ]_{8}$