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Properties

Label 16.8.25557302165...0625.3
Degree $16$
Signature $[8, 4]$
Discriminant $5^{10}\cdot 29^{6}\cdot 89^{6}\cdot 97^{4}$
Root discriminant $163.29$
Ramified primes $5, 29, 89, 97$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T790

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-129803237261, 184196436709, 252708127180, -69425863392, 16404250745, -15749698812, 2616789777, -473699896, 122313650, 5075165, 840025, 121886, -19523, -319, -99, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 - 99*x^14 - 319*x^13 - 19523*x^12 + 121886*x^11 + 840025*x^10 + 5075165*x^9 + 122313650*x^8 - 473699896*x^7 + 2616789777*x^6 - 15749698812*x^5 + 16404250745*x^4 - 69425863392*x^3 + 252708127180*x^2 + 184196436709*x - 129803237261)
 
gp: K = bnfinit(x^16 - 5*x^15 - 99*x^14 - 319*x^13 - 19523*x^12 + 121886*x^11 + 840025*x^10 + 5075165*x^9 + 122313650*x^8 - 473699896*x^7 + 2616789777*x^6 - 15749698812*x^5 + 16404250745*x^4 - 69425863392*x^3 + 252708127180*x^2 + 184196436709*x - 129803237261, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} - 99 x^{14} - 319 x^{13} - 19523 x^{12} + 121886 x^{11} + 840025 x^{10} + 5075165 x^{9} + 122313650 x^{8} - 473699896 x^{7} + 2616789777 x^{6} - 15749698812 x^{5} + 16404250745 x^{4} - 69425863392 x^{3} + 252708127180 x^{2} + 184196436709 x - 129803237261 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(255573021657714643907277003525390625=5^{10}\cdot 29^{6}\cdot 89^{6}\cdot 97^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $163.29$
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:  $5, 29, 89, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4}$, $\frac{1}{41585545308504665490515626454849291592624163904159893115674597826347734090888693335527023644} a^{15} + \frac{1251747289846270943284377984722758902479033815960440737036819949540229837259300431652064584}{10396386327126166372628906613712322898156040976039973278918649456586933522722173333881755911} a^{14} + \frac{1970672778606363084552931623828147634806439956303780840322015067167259617143671847709314118}{10396386327126166372628906613712322898156040976039973278918649456586933522722173333881755911} a^{13} + \frac{7468678656223831099351395404368853549132111413281535756358514700609400485485392916915740051}{41585545308504665490515626454849291592624163904159893115674597826347734090888693335527023644} a^{12} + \frac{2410932899360631327156921902123665111455321461618473715220228126731270447136383233913429130}{10396386327126166372628906613712322898156040976039973278918649456586933522722173333881755911} a^{11} + \frac{48823715259857594472026057567520780877997414755960309478871113411165486401567660468581683}{41585545308504665490515626454849291592624163904159893115674597826347734090888693335527023644} a^{10} + \frac{1286517645517359423146550192813002551075195803631389540630570011101210191451057827504075579}{20792772654252332745257813227424645796312081952079946557837298913173867045444346667763511822} a^{9} - \frac{6801279616406735551601385340809580474992103456833858634011641129890026863791853169703923577}{20792772654252332745257813227424645796312081952079946557837298913173867045444346667763511822} a^{8} + \frac{9668130077711242265280077543563846011255424575432186767093787881112953491003830995507069575}{20792772654252332745257813227424645796312081952079946557837298913173867045444346667763511822} a^{7} - \frac{7121973128218347675844211138466389544922623186000669650322365189299949975332978175131146989}{20792772654252332745257813227424645796312081952079946557837298913173867045444346667763511822} a^{6} - \frac{3424940789096431018789658736425129060832682234760738136172956778624913781023388067803058631}{41585545308504665490515626454849291592624163904159893115674597826347734090888693335527023644} a^{5} + \frac{15503555213592759430952398232984671154832104463069161795109422447462991809813813300531892213}{41585545308504665490515626454849291592624163904159893115674597826347734090888693335527023644} a^{4} - \frac{20293645026404741924879328061690052335899756723259732040186023832978131013716212711898574949}{41585545308504665490515626454849291592624163904159893115674597826347734090888693335527023644} a^{3} - \frac{538690293854421829271685558281424514935580671279370257682396495805544748274104812797698298}{10396386327126166372628906613712322898156040976039973278918649456586933522722173333881755911} a^{2} + \frac{13828877038324187566781278081541444742512264370178761754782138042755468965091873511993366619}{41585545308504665490515626454849291592624163904159893115674597826347734090888693335527023644} a - \frac{1667503683327707115267959618806451253866646641180595433886389959728129199201945569452196657}{10396386327126166372628906613712322898156040976039973278918649456586933522722173333881755911}$

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:  $11$
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:  \( 305658893134 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T790:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 62 conjugacy class representatives for t16n790 are not computed
Character table for t16n790 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.725.1, 4.4.2225.1, 4.4.64525.1, 8.8.4163475625.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.3.4$x^{4} + 232$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.3.3$x^{4} + 58$$4$$1$$3$$C_4$$[\ ]_{4}$
$89$89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
89.4.2.2$x^{4} - 89 x^{2} + 47526$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
89.4.0.1$x^{4} - x + 27$$1$$4$$0$$C_4$$[\ ]^{4}$
$97$97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.8.4.1$x^{8} + 432814 x^{4} - 912673 x^{2} + 46831989649$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$