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Properties

Label 16.8.76476099559...0000.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{54}\cdot 3^{20}\cdot 5^{16}\cdot 7^{20}$
Root discriminant $2331.96$
Ramified primes $2, 3, 5, 7$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $A_{16}$ (as 16T1953)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![837176934680100, 0, 41342070848400, -66238050412800, -170131978800, 0, -857661210000, 0, -1225230300, 0, 39488904, 0, 21168, 0, -648, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 648*x^14 + 21168*x^12 + 39488904*x^10 - 1225230300*x^8 - 857661210000*x^6 - 170131978800*x^4 - 66238050412800*x^3 + 41342070848400*x^2 + 837176934680100)
 
gp: K = bnfinit(x^16 - 648*x^14 + 21168*x^12 + 39488904*x^10 - 1225230300*x^8 - 857661210000*x^6 - 170131978800*x^4 - 66238050412800*x^3 + 41342070848400*x^2 + 837176934680100, 1)
 

Normalized defining polynomial

\( x^{16} - 648 x^{14} + 21168 x^{12} + 39488904 x^{10} - 1225230300 x^{8} - 857661210000 x^{6} - 170131978800 x^{4} - 66238050412800 x^{3} + 41342070848400 x^{2} + 837176934680100 \)

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:  \(764760995596249771707719623311297085440000000000000000=2^{54}\cdot 3^{20}\cdot 5^{16}\cdot 7^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $2331.96$
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:  $2, 3, 5, 7$
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 \(a\))

$1$, $a$, $\frac{1}{3} a^{2}$, $\frac{1}{3} a^{3}$, $\frac{1}{9} a^{4}$, $\frac{1}{27} a^{5} + \frac{1}{9} a^{3}$, $\frac{1}{189} a^{6} + \frac{1}{63} a^{4}$, $\frac{1}{567} a^{7} + \frac{1}{189} a^{5}$, $\frac{1}{68040} a^{8} + \frac{1}{2835} a^{7} - \frac{13}{5670} a^{6} + \frac{8}{945} a^{5} + \frac{11}{378} a^{4} - \frac{1}{9} a^{3} - \frac{1}{6} a^{2} + \frac{1}{4}$, $\frac{1}{476280} a^{9} + \frac{1}{4410} a^{7} + \frac{23}{1890} a^{5} - \frac{1}{6} a^{3} - \frac{1}{4} a$, $\frac{1}{1428840} a^{10} + \frac{1}{476280} a^{8} + \frac{4}{2835} a^{6} - \frac{4}{189} a^{4} + \frac{1}{12} a^{2} - \frac{1}{4}$, $\frac{1}{4286520} a^{11} - \frac{31}{39690} a^{7} - \frac{83}{5670} a^{5} - \frac{1}{36} a^{3} - \frac{1}{3} a$, $\frac{1}{12859560} a^{12} + \frac{1}{238140} a^{8} - \frac{2}{2835} a^{7} - \frac{1}{3402} a^{6} - \frac{16}{945} a^{5} - \frac{11}{756} a^{4} - \frac{1}{9} a^{3} - \frac{1}{9} a^{2} - \frac{1}{2}$, $\frac{1}{810152280} a^{13} - \frac{1}{10001880} a^{11} + \frac{1}{1071630} a^{9} + \frac{251}{535815} a^{7} - \frac{89}{11340} a^{5} + \frac{53}{324} a^{3}$, $\frac{1}{218741115600} a^{14} + \frac{1}{2025380700} a^{13} - \frac{1}{54010152} a^{12} - \frac{1}{25004700} a^{11} - \frac{187}{578680200} a^{10} - \frac{1}{2143260} a^{9} - \frac{391}{57868020} a^{8} - \frac{11}{21870} a^{7} + \frac{1061}{612360} a^{6} - \frac{43}{5670} a^{5} + \frac{2495}{61236} a^{4} - \frac{11}{162} a^{3} - \frac{5}{36} a^{2} - \frac{17}{54} a$, $\frac{1}{10611709972547558736263275767686309394247091235150000} a^{15} + \frac{214213318165049554145384830984398617663}{98256573819884803113548849700799161057843437362500} a^{14} - \frac{3672145682234395634088634144643047943773}{9357768935227124106052271400076110576937470225000} a^{13} + \frac{30809697887282768694418054637544402140653}{2426088242466291434902440733353065705131936725000} a^{12} - \frac{985209160654463307519558493698101462739361}{196513147639769606227097699401598322115686874725000} a^{11} + \frac{158155072350369185102519503494796377559429}{1039752103914124900672474600008456730770830025000} a^{10} + \frac{501906888756962322264556197308276592110347}{561466136113627446363136284004566634616248213500} a^{9} - \frac{291165744623374564196360659035953981218037}{41590084156564996026898984000338269230833201000} a^{8} + \frac{590681169583127996478980626908807219892447}{1663603366262599841075959360013530769233328040} a^{7} + \frac{1815154266097305985828159589978456218912}{1100266776628703598595211216940165852667545} a^{6} - \frac{1003236394778132863688904910877630674150123}{118828811875899988648282811429537912088094860} a^{5} + \frac{24089916385241596260170622068861486053493}{880213421302962878876168973552132682134036} a^{4} + \frac{178944960243728496304591742413617145090847}{1746455200997942219992398757047882305821500} a^{3} - \frac{102057614635655623328951960232284036591813}{1047873120598765331995439254228729383492900} a^{2} + \frac{19540461883978757494919715944445799946719}{48512644472165061666455521029107841828375} a - \frac{628407463752035469089405124805940215473}{7187058440320749876511929041349309900500}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 35342415376800000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$A_{16}$ (as 16T1953):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 10461394944000
The 123 conjugacy class representatives for $A_{16}$ are not computed
Character table for $A_{16}$ is not computed

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R R R R ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.14.0.1}{14} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ $15{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.14.0.1}{14} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.14.0.1}{14} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.9.0.1}{9} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.14.0.1}{14} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.7.0.1}{7} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.9.0.1}{9} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.11.0.1}{11} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
2Data not computed
$3$3.3.5.1$x^{3} + 3$$3$$1$$5$$S_3$$[5/2]_{2}$
3.3.5.2$x^{3} + 21$$3$$1$$5$$S_3$$[5/2]_{2}$
3.4.3.2$x^{4} - 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.6.7.4$x^{6} + 3 x^{2} + 3$$6$$1$$7$$S_3$$[3/2]_{2}$
$5$5.3.2.1$x^{3} - 5$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
5.3.2.1$x^{3} - 5$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
5.5.6.4$x^{5} + 20 x^{2} + 5$$5$$1$$6$$F_5$$[3/2]_{2}^{2}$
5.5.6.4$x^{5} + 20 x^{2} + 5$$5$$1$$6$$F_5$$[3/2]_{2}^{2}$
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.7.10.6$x^{7} + 42 x^{4} + 7$$7$$1$$10$$F_7$$[5/3]_{3}^{2}$
7.7.10.6$x^{7} + 42 x^{4} + 7$$7$$1$$10$$F_7$$[5/3]_{3}^{2}$