↑

Properties

Label 16.8.94915668639...0625.1
Degree $16$
Signature $[8, 4]$
Discriminant $5^{14}\cdot 41^{15}$
Root discriminant $132.92$
Ramified primes $5, 41$
Class number $8$ (GRH)
Class group $[8]$ (GRH)
Galois group $C_2^3.C_8$ (as 16T104)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-55949, 224988, -99740, -309070, 288595, -119996, -47368, 115130, -83050, 42320, -14558, 3246, -450, -70, 30, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 30*x^14 - 70*x^13 - 450*x^12 + 3246*x^11 - 14558*x^10 + 42320*x^9 - 83050*x^8 + 115130*x^7 - 47368*x^6 - 119996*x^5 + 288595*x^4 - 309070*x^3 - 99740*x^2 + 224988*x - 55949)
 
gp: K = bnfinit(x^16 - 8*x^15 + 30*x^14 - 70*x^13 - 450*x^12 + 3246*x^11 - 14558*x^10 + 42320*x^9 - 83050*x^8 + 115130*x^7 - 47368*x^6 - 119996*x^5 + 288595*x^4 - 309070*x^3 - 99740*x^2 + 224988*x - 55949, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 30 x^{14} - 70 x^{13} - 450 x^{12} + 3246 x^{11} - 14558 x^{10} + 42320 x^{9} - 83050 x^{8} + 115130 x^{7} - 47368 x^{6} - 119996 x^{5} + 288595 x^{4} - 309070 x^{3} - 99740 x^{2} + 224988 x - 55949 \)

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:  \(9491566863962023381888433837890625=5^{14}\cdot 41^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $132.92$
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:  $5, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{40} a^{8} - \frac{1}{10} a^{7} + \frac{1}{20} a^{6} + \frac{1}{5} a^{5} - \frac{9}{20} a^{3} - \frac{3}{40} a^{2} + \frac{7}{20} a - \frac{9}{40}$, $\frac{1}{40} a^{9} + \frac{3}{20} a^{7} - \frac{1}{10} a^{6} - \frac{1}{5} a^{5} + \frac{1}{20} a^{4} + \frac{1}{8} a^{3} - \frac{9}{20} a^{2} + \frac{7}{40} a + \frac{1}{10}$, $\frac{1}{40} a^{10} - \frac{3}{20} a^{5} + \frac{1}{8} a^{4} + \frac{1}{4} a^{3} + \frac{1}{8} a^{2} - \frac{1}{2} a - \frac{3}{20}$, $\frac{1}{40} a^{11} - \frac{3}{20} a^{6} + \frac{1}{8} a^{5} - \frac{1}{4} a^{4} + \frac{1}{8} a^{3} - \frac{1}{2} a^{2} + \frac{7}{20} a - \frac{1}{2}$, $\frac{1}{320} a^{12} + \frac{1}{160} a^{11} - \frac{1}{160} a^{10} + \frac{1}{320} a^{9} - \frac{1}{320} a^{8} - \frac{9}{80} a^{7} - \frac{33}{320} a^{6} - \frac{1}{80} a^{5} + \frac{77}{320} a^{4} - \frac{27}{320} a^{3} - \frac{111}{320} a^{2} + \frac{141}{320} a + \frac{1}{64}$, $\frac{1}{320} a^{13} + \frac{1}{160} a^{11} - \frac{3}{320} a^{10} - \frac{3}{320} a^{9} - \frac{1}{160} a^{8} + \frac{71}{320} a^{7} + \frac{39}{160} a^{6} - \frac{51}{320} a^{5} + \frac{19}{320} a^{4} + \frac{127}{320} a^{3} - \frac{93}{320} a^{2} + \frac{123}{320} a + \frac{7}{32}$, $\frac{1}{229920449600} a^{14} - \frac{7}{229920449600} a^{13} + \frac{117368737}{114960224800} a^{12} - \frac{1408424753}{229920449600} a^{11} - \frac{3128489}{2673493600} a^{10} + \frac{64181113}{5346987200} a^{9} - \frac{1367399723}{229920449600} a^{8} - \frac{49573845219}{229920449600} a^{7} + \frac{35358678443}{229920449600} a^{6} - \frac{5142107667}{28740056200} a^{5} - \frac{22843513343}{114960224800} a^{4} - \frac{45635463859}{114960224800} a^{3} - \frac{28659949017}{114960224800} a^{2} - \frac{2481313189}{5346987200} a - \frac{35139042253}{114960224800}$, $\frac{1}{155656144379200} a^{15} + \frac{331}{155656144379200} a^{14} - \frac{3553629399}{3619910334400} a^{13} - \frac{143365591281}{155656144379200} a^{12} - \frac{273364877129}{77828072189600} a^{11} - \frac{13331763953}{1809955167200} a^{10} - \frac{176979058463}{77828072189600} a^{9} - \frac{1128229157083}{155656144379200} a^{8} + \frac{18372622022343}{77828072189600} a^{7} + \frac{6150067066877}{38914036094800} a^{6} + \frac{29481308269781}{155656144379200} a^{5} - \frac{27175364212801}{155656144379200} a^{4} + \frac{3553152418487}{155656144379200} a^{3} - \frac{21835304402377}{77828072189600} a^{2} - \frac{25628474930927}{155656144379200} a - \frac{896139308899}{77828072189600}$

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

Class group and class number

$C_{8}$, 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:  $11$
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:  \( 17870951998.4 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_8$ (as 16T104):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 22 conjugacy class representatives for $C_2^3.C_8$
Character table for $C_2^3.C_8$ is not computed

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.1723025.1, 8.8.3043035529390625.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ $16$ R $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ $16$ $16$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$5.8.7.4$x^{8} + 40$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
5.8.7.4$x^{8} + 40$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
41Data not computed