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Properties

Label 16.0.30467756702...9696.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{52}\cdot 3^{4}\cdot 17^{4}$
Root discriminant $25.42$
Ramified primes $2, 3, 17$
Class number $2$
Class group $[2]$
Galois group $C_2^4.C_2^3$ (as 16T203)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3106, -12288, 31624, -54032, 70592, -70344, 55728, -34576, 16982, -6416, 1876, -416, 92, -16, 8, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 8*x^14 - 16*x^13 + 92*x^12 - 416*x^11 + 1876*x^10 - 6416*x^9 + 16982*x^8 - 34576*x^7 + 55728*x^6 - 70344*x^5 + 70592*x^4 - 54032*x^3 + 31624*x^2 - 12288*x + 3106)
 
gp: K = bnfinit(x^16 + 8*x^14 - 16*x^13 + 92*x^12 - 416*x^11 + 1876*x^10 - 6416*x^9 + 16982*x^8 - 34576*x^7 + 55728*x^6 - 70344*x^5 + 70592*x^4 - 54032*x^3 + 31624*x^2 - 12288*x + 3106, 1)
 

Normalized defining polynomial

\( x^{16} + 8 x^{14} - 16 x^{13} + 92 x^{12} - 416 x^{11} + 1876 x^{10} - 6416 x^{9} + 16982 x^{8} - 34576 x^{7} + 55728 x^{6} - 70344 x^{5} + 70592 x^{4} - 54032 x^{3} + 31624 x^{2} - 12288 x + 3106 \)

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:  \(30467756702686506909696=2^{52}\cdot 3^{4}\cdot 17^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.42$
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:  $2, 3, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{17} a^{12} + \frac{5}{17} a^{11} - \frac{1}{17} a^{10} + \frac{3}{17} a^{9} - \frac{7}{17} a^{8} + \frac{4}{17} a^{7} - \frac{8}{17} a^{6} + \frac{1}{17} a^{5} + \frac{4}{17} a^{4} - \frac{1}{17} a^{3} + \frac{6}{17} a^{2} - \frac{4}{17} a - \frac{3}{17}$, $\frac{1}{17} a^{13} + \frac{8}{17} a^{11} + \frac{8}{17} a^{10} - \frac{5}{17} a^{9} + \frac{5}{17} a^{8} + \frac{6}{17} a^{7} + \frac{7}{17} a^{6} - \frac{1}{17} a^{5} - \frac{4}{17} a^{4} - \frac{6}{17} a^{3} - \frac{2}{17}$, $\frac{1}{85} a^{14} + \frac{1}{85} a^{13} - \frac{2}{85} a^{12} - \frac{2}{5} a^{11} + \frac{6}{17} a^{10} + \frac{38}{85} a^{9} - \frac{21}{85} a^{8} - \frac{2}{17} a^{7} + \frac{18}{85} a^{6} + \frac{2}{85} a^{5} - \frac{33}{85} a^{4} + \frac{21}{85} a^{3} - \frac{9}{85} a^{2} + \frac{38}{85} a + \frac{11}{85}$, $\frac{1}{3181751646084745390645} a^{15} - \frac{11355673803450537991}{3181751646084745390645} a^{14} - \frac{12602770340093609039}{3181751646084745390645} a^{13} - \frac{13880011606828291845}{636350329216949078129} a^{12} - \frac{288167682800546079437}{3181751646084745390645} a^{11} + \frac{332491511660071017348}{3181751646084745390645} a^{10} + \frac{1022492112912829114963}{3181751646084745390645} a^{9} + \frac{18952726765869183712}{3181751646084745390645} a^{8} + \frac{1182356011273308759808}{3181751646084745390645} a^{7} - \frac{542194278006763272594}{3181751646084745390645} a^{6} + \frac{1072160771139179350088}{3181751646084745390645} a^{5} - \frac{917853734424217380593}{3181751646084745390645} a^{4} - \frac{971277889758550029311}{3181751646084745390645} a^{3} - \frac{784464313392222779794}{3181751646084745390645} a^{2} + \frac{6267305548313236995}{636350329216949078129} a - \frac{1490395518925703700012}{3181751646084745390645}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( \frac{33011176497484065292}{3181751646084745390645} a^{15} - \frac{11446399803138798422}{3181751646084745390645} a^{14} + \frac{190501846509102311487}{3181751646084745390645} a^{13} - \frac{137748543845063942648}{636350329216949078129} a^{12} + \frac{2677566340901059146071}{3181751646084745390645} a^{11} - \frac{13993261431261005611119}{3181751646084745390645} a^{10} + \frac{61493495482927145842616}{3181751646084745390645} a^{9} - \frac{208591066494423572627301}{3181751646084745390645} a^{8} + \frac{526690371816070871393421}{3181751646084745390645} a^{7} - \frac{1000001322569513139701863}{3181751646084745390645} a^{6} + \frac{1455220385607484114602881}{3181751646084745390645} a^{5} - \frac{1621961667683844009179356}{3181751646084745390645} a^{4} + \frac{1366920733674495665225298}{3181751646084745390645} a^{3} - \frac{859966079730071231906963}{3181751646084745390645} a^{2} + \frac{72077231665152418483674}{636350329216949078129} a - \frac{99380105881128538131279}{3181751646084745390645} \) (order $16$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 259949.690161 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3$ (as 16T203):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 41 conjugacy class representatives for $C_2^4.C_2^3$
Character table for $C_2^4.C_2^3$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), 4.0.2048.2, \(\Q(\zeta_{16})^+\), \(\Q(\zeta_{8})\), \(\Q(\zeta_{16})\)

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 R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$3$3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$17$$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$