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Properties

Label 16.0.17606641095...625.12
Degree $16$
Signature $[0, 8]$
Discriminant $5^{6}\cdot 101^{12}$
Root discriminant $58.26$
Ramified primes $5, 101$
Class number $50$ (GRH)
Class group $[5, 10]$ (GRH)
Galois group $(C_2\times C_4).D_4$ (as 16T121)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![464375, -814425, 1504070, -1714545, 1790895, -1403525, 988724, -548960, 276033, -109674, 40043, -11120, 2915, -518, 94, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 94*x^14 - 518*x^13 + 2915*x^12 - 11120*x^11 + 40043*x^10 - 109674*x^9 + 276033*x^8 - 548960*x^7 + 988724*x^6 - 1403525*x^5 + 1790895*x^4 - 1714545*x^3 + 1504070*x^2 - 814425*x + 464375)
 
gp: K = bnfinit(x^16 - 8*x^15 + 94*x^14 - 518*x^13 + 2915*x^12 - 11120*x^11 + 40043*x^10 - 109674*x^9 + 276033*x^8 - 548960*x^7 + 988724*x^6 - 1403525*x^5 + 1790895*x^4 - 1714545*x^3 + 1504070*x^2 - 814425*x + 464375, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 94 x^{14} - 518 x^{13} + 2915 x^{12} - 11120 x^{11} + 40043 x^{10} - 109674 x^{9} + 276033 x^{8} - 548960 x^{7} + 988724 x^{6} - 1403525 x^{5} + 1790895 x^{4} - 1714545 x^{3} + 1504070 x^{2} - 814425 x + 464375 \)

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:  \(17606641095812026885331265625=5^{6}\cdot 101^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $58.26$
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:  $5, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5} a^{10} - \frac{2}{5} a^{8} - \frac{2}{5} a^{7} - \frac{2}{5} a^{6} + \frac{2}{5} a^{5} - \frac{2}{5} a^{4}$, $\frac{1}{5} a^{11} - \frac{2}{5} a^{9} - \frac{2}{5} a^{8} - \frac{2}{5} a^{7} + \frac{2}{5} a^{6} - \frac{2}{5} a^{5}$, $\frac{1}{5} a^{12} - \frac{2}{5} a^{9} - \frac{1}{5} a^{8} - \frac{2}{5} a^{7} - \frac{1}{5} a^{6} - \frac{1}{5} a^{5} + \frac{1}{5} a^{4}$, $\frac{1}{5} a^{13} - \frac{1}{5} a^{9} - \frac{1}{5} a^{8} + \frac{1}{5} a^{4}$, $\frac{1}{231461921275} a^{14} - \frac{7}{231461921275} a^{13} - \frac{19926984743}{231461921275} a^{12} - \frac{19315244216}{231461921275} a^{11} - \frac{19747378516}{231461921275} a^{10} - \frac{25107199931}{231461921275} a^{9} + \frac{8424307639}{17804763175} a^{8} - \frac{83257025437}{231461921275} a^{7} + \frac{24465309326}{231461921275} a^{6} - \frac{8284059219}{231461921275} a^{5} - \frac{111604381}{9258476851} a^{4} - \frac{12865415744}{46292384255} a^{3} + \frac{2749000229}{9258476851} a^{2} + \frac{8009753191}{46292384255} a - \frac{3361890088}{9258476851}$, $\frac{1}{32400271201995775} a^{15} + \frac{69983}{32400271201995775} a^{14} - \frac{1348146741748783}{32400271201995775} a^{13} + \frac{888199952128539}{32400271201995775} a^{12} - \frac{3121975586814791}{32400271201995775} a^{11} + \frac{1061630835286679}{32400271201995775} a^{10} - \frac{9244658806112103}{32400271201995775} a^{9} - \frac{2482534263761527}{32400271201995775} a^{8} - \frac{9829852042019134}{32400271201995775} a^{7} + \frac{559960549119852}{2492328553999675} a^{6} + \frac{1020785710319462}{6480054240399155} a^{5} - \frac{1746211589672031}{6480054240399155} a^{4} - \frac{285208086750537}{1296010848079831} a^{3} - \frac{1042532183118269}{6480054240399155} a^{2} - \frac{83310766093378}{1296010848079831} a - \frac{186293569286777}{1296010848079831}$

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

Class group and class number

$C_{5}\times C_{10}$, which has order $50$ (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:  \( 3064881.8401 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times C_4).D_4$ (as 16T121):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 28 conjugacy class representatives for $(C_2\times C_4).D_4$
Character table for $(C_2\times C_4).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{101}) \), 4.4.51005.1, 4.0.1030301.1, 4.0.5151505.1, 8.4.132690018825125.1, 8.4.13007550125.1, 8.0.26538003765025.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
$101$101.4.3.2$x^{4} - 404$$4$$1$$3$$C_4$$[\ ]_{4}$
101.4.3.2$x^{4} - 404$$4$$1$$3$$C_4$$[\ ]_{4}$
101.4.3.2$x^{4} - 404$$4$$1$$3$$C_4$$[\ ]_{4}$
101.4.3.2$x^{4} - 404$$4$$1$$3$$C_4$$[\ ]_{4}$