↑

Properties

Label 16.0.13804460977...5625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{10}\cdot 29^{8}\cdot 41^{4}$
Root discriminant $37.26$
Ramified primes $5, 29, 41$
Class number $104$ (GRH)
Class group $[2, 52]$ (GRH)
Galois group $D_4.D_4$ (as 16T175)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![15151, -26360, 42543, -81341, 122135, -126586, 109939, -70521, 42209, -18505, 8219, -2460, 846, -160, 45, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 45*x^14 - 160*x^13 + 846*x^12 - 2460*x^11 + 8219*x^10 - 18505*x^9 + 42209*x^8 - 70521*x^7 + 109939*x^6 - 126586*x^5 + 122135*x^4 - 81341*x^3 + 42543*x^2 - 26360*x + 15151)
 
gp: K = bnfinit(x^16 - 4*x^15 + 45*x^14 - 160*x^13 + 846*x^12 - 2460*x^11 + 8219*x^10 - 18505*x^9 + 42209*x^8 - 70521*x^7 + 109939*x^6 - 126586*x^5 + 122135*x^4 - 81341*x^3 + 42543*x^2 - 26360*x + 15151, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 45 x^{14} - 160 x^{13} + 846 x^{12} - 2460 x^{11} + 8219 x^{10} - 18505 x^{9} + 42209 x^{8} - 70521 x^{7} + 109939 x^{6} - 126586 x^{5} + 122135 x^{4} - 81341 x^{3} + 42543 x^{2} - 26360 x + 15151 \)

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:  \(13804460977881721884765625=5^{10}\cdot 29^{8}\cdot 41^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.26$
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:  $5, 29, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{205} a^{12} - \frac{3}{205} a^{11} - \frac{9}{205} a^{10} + \frac{7}{41} a^{9} + \frac{21}{205} a^{8} + \frac{9}{41} a^{7} - \frac{3}{205} a^{6} - \frac{67}{205} a^{5} + \frac{14}{205} a^{4} + \frac{7}{205} a^{3} + \frac{48}{205} a^{2} - \frac{89}{205} a - \frac{99}{205}$, $\frac{1}{205} a^{13} - \frac{18}{205} a^{11} + \frac{8}{205} a^{10} - \frac{79}{205} a^{9} - \frac{97}{205} a^{8} - \frac{73}{205} a^{7} - \frac{76}{205} a^{6} + \frac{18}{205} a^{5} + \frac{49}{205} a^{4} + \frac{69}{205} a^{3} + \frac{11}{41} a^{2} + \frac{44}{205} a - \frac{92}{205}$, $\frac{1}{205} a^{14} - \frac{46}{205} a^{11} - \frac{36}{205} a^{10} - \frac{2}{5} a^{9} + \frac{20}{41} a^{8} - \frac{86}{205} a^{7} - \frac{36}{205} a^{6} + \frac{73}{205} a^{5} - \frac{89}{205} a^{4} - \frac{24}{205} a^{3} + \frac{88}{205} a^{2} - \frac{54}{205} a + \frac{63}{205}$, $\frac{1}{8590796793353286979334275} a^{15} - \frac{3953775299539832931729}{1718159358670657395866855} a^{14} + \frac{1230713235707248783583}{1718159358670657395866855} a^{13} + \frac{520371055599572325851}{1718159358670657395866855} a^{12} - \frac{2371137894916359000243704}{8590796793353286979334275} a^{11} + \frac{1775187674219766206528524}{8590796793353286979334275} a^{10} - \frac{30900519439226099621919}{1718159358670657395866855} a^{9} - \frac{591085774463630984803416}{1718159358670657395866855} a^{8} - \frac{3488715813419877192315441}{8590796793353286979334275} a^{7} + \frac{510518044288893042004349}{1718159358670657395866855} a^{6} + \frac{14471016076588632041489}{209531629106177731203275} a^{5} + \frac{472871175652189585161897}{1718159358670657395866855} a^{4} + \frac{210546371742159994203573}{1718159358670657395866855} a^{3} - \frac{2033118558268506396785901}{8590796793353286979334275} a^{2} - \frac{2267807022987775710679276}{8590796793353286979334275} a + \frac{3133228124841919342329556}{8590796793353286979334275}$

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

Class group and class number

$C_{2}\times C_{52}$, which has order $104$ (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:  \( 3793.72993285 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_4.D_4$ (as 16T175):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 16 conjugacy class representatives for $D_4.D_4$
Character table for $D_4.D_4$

Intermediate fields

\(\Q(\sqrt{29}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{145}) \), 4.4.4205.1 x2, 4.4.725.1 x2, \(\Q(\sqrt{5}, \sqrt{29})\), 8.8.442050625.1

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$5$5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$29$29.4.2.1$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$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$41$41.4.2.2$x^{4} - 41 x^{2} + 20172$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.4.2.2$x^{4} - 41 x^{2} + 20172$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$