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Properties

Label 16.8.24948698345...0000.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{32}\cdot 5^{10}\cdot 29^{6}$
Root discriminant $38.66$
Ramified primes $2, 5, 29$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^3.C_2^4.C_2$ (as 16T456)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3631, 580, 20184, -200, -23865, 3704, 878, 32, 3990, 832, -308, 8, 64, -12, -18, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 18*x^14 - 12*x^13 + 64*x^12 + 8*x^11 - 308*x^10 + 832*x^9 + 3990*x^8 + 32*x^7 + 878*x^6 + 3704*x^5 - 23865*x^4 - 200*x^3 + 20184*x^2 + 580*x - 3631)
 
gp: K = bnfinit(x^16 - 18*x^14 - 12*x^13 + 64*x^12 + 8*x^11 - 308*x^10 + 832*x^9 + 3990*x^8 + 32*x^7 + 878*x^6 + 3704*x^5 - 23865*x^4 - 200*x^3 + 20184*x^2 + 580*x - 3631, 1)
 

Normalized defining polynomial

\( x^{16} - 18 x^{14} - 12 x^{13} + 64 x^{12} + 8 x^{11} - 308 x^{10} + 832 x^{9} + 3990 x^{8} + 32 x^{7} + 878 x^{6} + 3704 x^{5} - 23865 x^{4} - 200 x^{3} + 20184 x^{2} + 580 x - 3631 \)

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:  \(24948698345635840000000000=2^{32}\cdot 5^{10}\cdot 29^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.66$
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:  $2, 5, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{15} a^{14} - \frac{4}{15} a^{13} - \frac{1}{15} a^{12} + \frac{1}{5} a^{11} + \frac{2}{5} a^{10} + \frac{2}{15} a^{9} + \frac{1}{3} a^{8} + \frac{4}{15} a^{7} + \frac{4}{15} a^{6} + \frac{1}{3} a^{5} + \frac{7}{15} a^{4} + \frac{2}{5} a^{3} - \frac{2}{15} a^{2} - \frac{2}{5} a + \frac{1}{15}$, $\frac{1}{29069201420752640949097713898004955} a^{15} - \frac{304035744529363156131226589619679}{9689733806917546983032571299334985} a^{14} - \frac{13621215734896713776560924680990314}{29069201420752640949097713898004955} a^{13} + \frac{13312431228203084025792807128392871}{29069201420752640949097713898004955} a^{12} - \frac{260903542079349064275920984678951}{9689733806917546983032571299334985} a^{11} + \frac{7017039393125904580207220348593904}{29069201420752640949097713898004955} a^{10} - \frac{2868510343102830820909120170801026}{29069201420752640949097713898004955} a^{9} - \frac{4081880726048229887673927891540837}{9689733806917546983032571299334985} a^{8} + \frac{11846657733164708286860356671533417}{29069201420752640949097713898004955} a^{7} + \frac{2546927470939048492594958756254401}{9689733806917546983032571299334985} a^{6} - \frac{3705606017520183311969168091275911}{9689733806917546983032571299334985} a^{5} - \frac{882866250502950665203907518312270}{5813840284150528189819542779600991} a^{4} + \frac{1845554985428988204607525758743897}{5813840284150528189819542779600991} a^{3} + \frac{1399324091722147626865435305314633}{5813840284150528189819542779600991} a^{2} - \frac{5894935519381192930812821454418811}{29069201420752640949097713898004955} a + \frac{9940858135791003513945287341664872}{29069201420752640949097713898004955}$

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:  \( 5036751.87521 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_2^4.C_2$ (as 16T456):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 46 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

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), 4.4.46400.1, \(\Q(\sqrt{2}, \sqrt{5})\), 4.4.725.1, 8.8.2152960000.1

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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
$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$$[\ ]^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$