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Properties

Label 16.0.65044924825...3184.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 17^{6}\cdot 89^{4}$
Root discriminant $35.55$
Ramified primes $2, 17, 89$
Class number $2$ (GRH)
Class group $[2]$ (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![1187794, -494628, 1267074, -709592, 716343, -383132, 224334, -102396, 39301, -15124, 4186, -1448, 377, -108, 30, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 30*x^14 - 108*x^13 + 377*x^12 - 1448*x^11 + 4186*x^10 - 15124*x^9 + 39301*x^8 - 102396*x^7 + 224334*x^6 - 383132*x^5 + 716343*x^4 - 709592*x^3 + 1267074*x^2 - 494628*x + 1187794)
 
gp: K = bnfinit(x^16 - 4*x^15 + 30*x^14 - 108*x^13 + 377*x^12 - 1448*x^11 + 4186*x^10 - 15124*x^9 + 39301*x^8 - 102396*x^7 + 224334*x^6 - 383132*x^5 + 716343*x^4 - 709592*x^3 + 1267074*x^2 - 494628*x + 1187794, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 30 x^{14} - 108 x^{13} + 377 x^{12} - 1448 x^{11} + 4186 x^{10} - 15124 x^{9} + 39301 x^{8} - 102396 x^{7} + 224334 x^{6} - 383132 x^{5} + 716343 x^{4} - 709592 x^{3} + 1267074 x^{2} - 494628 x + 1187794 \)

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:  \(6504492482542510154973184=2^{32}\cdot 17^{6}\cdot 89^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $35.55$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7}$, $\frac{1}{4} a^{12} - \frac{1}{2} a^{6} + \frac{1}{4} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{13} - \frac{1}{2} a^{7} + \frac{1}{4} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{16} a^{14} + \frac{1}{16} a^{12} - \frac{1}{4} a^{10} - \frac{1}{8} a^{8} - \frac{1}{4} a^{7} - \frac{1}{16} a^{6} - \frac{1}{4} a^{5} - \frac{5}{16} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a - \frac{3}{8}$, $\frac{1}{96120817696921336456638353831027437424} a^{15} - \frac{1459537218954352521761889449461598391}{48060408848460668228319176915513718712} a^{14} + \frac{3241962270580915732724676854980471169}{96120817696921336456638353831027437424} a^{13} - \frac{2223203636775710410630905925864890705}{48060408848460668228319176915513718712} a^{12} + \frac{5477400192624281432415808937130672991}{24030204424230334114159588457756859356} a^{11} + \frac{1348162344319231010635584507551072005}{12015102212115167057079794228878429678} a^{10} - \frac{10857765403457419297134056658190653525}{48060408848460668228319176915513718712} a^{9} - \frac{335571896288109664989226614342233471}{12015102212115167057079794228878429678} a^{8} - \frac{44386442534349188692113088661319438665}{96120817696921336456638353831027437424} a^{7} + \frac{17816715726813868995310065689474283809}{48060408848460668228319176915513718712} a^{6} + \frac{18140450674295597610352463091070510635}{96120817696921336456638353831027437424} a^{5} - \frac{1359258251950327991901909076952690253}{48060408848460668228319176915513718712} a^{4} - \frac{2335477345964464096959412463321022994}{6007551106057583528539897114439214839} a^{3} + \frac{9799290716859769651288826136694212183}{24030204424230334114159588457756859356} a^{2} - \frac{8322646702980342808747948870342984295}{48060408848460668228319176915513718712} a + \frac{3196623362190269074764598166541637223}{24030204424230334114159588457756859356}$

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:  \( -\frac{30968005237190175991392570913}{7334107866391067942670406976272504} a^{15} + \frac{395237163614447670164987700581}{14668215732782135885340813952545008} a^{14} - \frac{1094909129584366176186091917249}{7334107866391067942670406976272504} a^{13} + \frac{10270672993983243252495822480669}{14668215732782135885340813952545008} a^{12} - \frac{2004201624610258734176279757014}{916763483298883492833800872034063} a^{11} + \frac{30187678734831336309178892388853}{3667053933195533971335203488136252} a^{10} - \frac{97404113014398117481323029414265}{3667053933195533971335203488136252} a^{9} + \frac{605839803314331414782319474113983}{7334107866391067942670406976272504} a^{8} - \frac{1879138703164575184679759889636309}{7334107866391067942670406976272504} a^{7} + \frac{8776995300882305610446574740570859}{14668215732782135885340813952545008} a^{6} - \frac{10067166299846804537848241215012561}{7334107866391067942670406976272504} a^{5} + \frac{35198921054656811085561773009759863}{14668215732782135885340813952545008} a^{4} - \frac{12915431958618117274175059945726795}{3667053933195533971335203488136252} a^{3} + \frac{4611789019255211503754475462356788}{916763483298883492833800872034063} a^{2} - \frac{2973070873639051884163163075534827}{916763483298883492833800872034063} a + \frac{41461469303993354008905227103860865}{7334107866391067942670406976272504} \) (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:  \( 4458007.68326 \) (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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), 4.0.1088.2, 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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$2$2.4.8.2$x^{4} + 6 x^{2} + 1$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.8.2$x^{4} + 6 x^{2} + 1$$4$$1$$8$$C_2^2$$[2, 3]$
2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
$17$17.4.3.3$x^{4} + 51$$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$$[\ ]^{4}$
$89$$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
89.2.1.1$x^{2} - 89$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.1$x^{2} - 89$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.1$x^{2} - 89$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.1$x^{2} - 89$$2$$1$$1$$C_2$$[\ ]_{2}$