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Properties

Label 16.0.29630251663...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 5^{12}\cdot 41^{4}$
Root discriminant $33.84$
Ramified primes $2, 5, 41$
Class number $16$ (GRH)
Class group $[2, 2, 4]$ (GRH)
Galois group $C_2^4.C_2^3$ (as 16T217)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![10721, -14312, 25462, -22416, 22768, -16660, 12686, -8408, 5193, -2596, 1034, -288, 88, -52, 28, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 28*x^14 - 52*x^13 + 88*x^12 - 288*x^11 + 1034*x^10 - 2596*x^9 + 5193*x^8 - 8408*x^7 + 12686*x^6 - 16660*x^5 + 22768*x^4 - 22416*x^3 + 25462*x^2 - 14312*x + 10721)
 
gp: K = bnfinit(x^16 - 8*x^15 + 28*x^14 - 52*x^13 + 88*x^12 - 288*x^11 + 1034*x^10 - 2596*x^9 + 5193*x^8 - 8408*x^7 + 12686*x^6 - 16660*x^5 + 22768*x^4 - 22416*x^3 + 25462*x^2 - 14312*x + 10721, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 28 x^{14} - 52 x^{13} + 88 x^{12} - 288 x^{11} + 1034 x^{10} - 2596 x^{9} + 5193 x^{8} - 8408 x^{7} + 12686 x^{6} - 16660 x^{5} + 22768 x^{4} - 22416 x^{3} + 25462 x^{2} - 14312 x + 10721 \)

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:  \(2963025166336000000000000=2^{32}\cdot 5^{12}\cdot 41^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.84$
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:  $2, 5, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a$, $\frac{1}{1918} a^{14} - \frac{156}{959} a^{13} - \frac{15}{274} a^{12} + \frac{117}{959} a^{11} + \frac{141}{959} a^{10} - \frac{435}{1918} a^{9} + \frac{201}{1918} a^{8} - \frac{134}{959} a^{7} + \frac{473}{959} a^{6} - \frac{449}{1918} a^{5} - \frac{123}{274} a^{4} + \frac{139}{959} a^{3} + \frac{507}{1918} a^{2} - \frac{659}{1918} a + \frac{146}{959}$, $\frac{1}{2309669918236057165981598} a^{15} - \frac{45734461909994373004}{1154834959118028582990799} a^{14} + \frac{38538716383539480664663}{1154834959118028582990799} a^{13} - \frac{385644693048659877551149}{2309669918236057165981598} a^{12} - \frac{658015215189709162204}{19573473883356416660861} a^{11} - \frac{268852886062179052535445}{1154834959118028582990799} a^{10} - \frac{180387969291017853746655}{2309669918236057165981598} a^{9} + \frac{24897638722868576305223}{329952845462293880854514} a^{8} - \frac{367430386068976161384777}{1154834959118028582990799} a^{7} - \frac{252661900997497775573658}{1154834959118028582990799} a^{6} - \frac{1044398152702121922492955}{2309669918236057165981598} a^{5} - \frac{928875274377951481277923}{2309669918236057165981598} a^{4} - \frac{625402005875506444217369}{2309669918236057165981598} a^{3} - \frac{248807153467040501620506}{1154834959118028582990799} a^{2} - \frac{609518925138378043281543}{2309669918236057165981598} a + \frac{29339177883769269528657}{1154834959118028582990799}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4}$, which has order $16$ (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:  \( 27845.058004 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3$ (as 16T217):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 32 conjugacy class representatives for $C_2^4.C_2^3$
Character table for $C_2^4.C_2^3$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{2}, \sqrt{5})\), 8.4.2624000000.4, 8.0.1721344000000.34, 8.4.1679360000.4

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$2$2.8.16.3$x^{8} + 2 x^{6} + 6 x^{4} + 4 x^{2} + 8 x + 28$$4$$2$$16$$C_4\times C_2$$[2, 3]^{2}$
2.8.16.3$x^{8} + 2 x^{6} + 6 x^{4} + 4 x^{2} + 8 x + 28$$4$$2$$16$$C_4\times C_2$$[2, 3]^{2}$
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$41$41.2.1.1$x^{2} - 41$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.1$x^{2} - 41$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.1.1$x^{2} - 41$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.1$x^{2} - 41$$2$$1$$1$$C_2$$[\ ]_{2}$