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Properties

Label 16.0.17612050268...0000.4
Degree $16$
Signature $[0, 8]$
Discriminant $2^{36}\cdot 3^{8}\cdot 5^{8}$
Root discriminant $18.42$
Ramified primes $2, 3, 5$
Class number $4$
Class group $[4]$
Galois group $C_2^2 \times D_4$ (as 16T25)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![10, -40, 116, -252, 484, -824, 1254, -1636, 1775, -1556, 1090, -612, 284, -112, 36, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 36*x^14 - 112*x^13 + 284*x^12 - 612*x^11 + 1090*x^10 - 1556*x^9 + 1775*x^8 - 1636*x^7 + 1254*x^6 - 824*x^5 + 484*x^4 - 252*x^3 + 116*x^2 - 40*x + 10)
 
gp: K = bnfinit(x^16 - 8*x^15 + 36*x^14 - 112*x^13 + 284*x^12 - 612*x^11 + 1090*x^10 - 1556*x^9 + 1775*x^8 - 1636*x^7 + 1254*x^6 - 824*x^5 + 484*x^4 - 252*x^3 + 116*x^2 - 40*x + 10, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 36 x^{14} - 112 x^{13} + 284 x^{12} - 612 x^{11} + 1090 x^{10} - 1556 x^{9} + 1775 x^{8} - 1636 x^{7} + 1254 x^{6} - 824 x^{5} + 484 x^{4} - 252 x^{3} + 116 x^{2} - 40 x + 10 \)

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:  \(176120502681600000000=2^{36}\cdot 3^{8}\cdot 5^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.42$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{5699} a^{14} - \frac{7}{5699} a^{13} + \frac{979}{5699} a^{12} - \frac{84}{5699} a^{11} + \frac{1313}{5699} a^{10} + \frac{687}{5699} a^{9} + \frac{1046}{5699} a^{8} + \frac{2454}{5699} a^{7} + \frac{604}{5699} a^{6} - \frac{623}{5699} a^{5} - \frac{1168}{5699} a^{4} - \frac{1146}{5699} a^{3} + \frac{1043}{5699} a^{2} + \frac{600}{5699} a - \frac{60}{5699}$, $\frac{1}{2741219} a^{15} + \frac{233}{2741219} a^{14} - \frac{724474}{2741219} a^{13} + \frac{63221}{210863} a^{12} + \frac{1234933}{2741219} a^{11} + \frac{1353025}{2741219} a^{10} + \frac{80441}{2741219} a^{9} + \frac{1159635}{2741219} a^{8} + \frac{179236}{2741219} a^{7} - \frac{300185}{2741219} a^{6} - \frac{1102421}{2741219} a^{5} - \frac{1369975}{2741219} a^{4} + \frac{512465}{2741219} a^{3} + \frac{28082}{66859} a^{2} + \frac{525773}{2741219} a + \frac{965828}{2741219}$

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

Class group and class number

$C_{4}$, which has order $4$

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:  \( \frac{5424}{5699} a^{14} - \frac{37968}{5699} a^{13} + \frac{152501}{5699} a^{12} - \frac{421422}{5699} a^{11} + \frac{983889}{5699} a^{10} - \frac{1961314}{5699} a^{9} + \frac{3074760}{5699} a^{8} - \frac{3615534}{5699} a^{7} + \frac{3258999}{5699} a^{6} - \frac{2353332}{5699} a^{5} + \frac{1478097}{5699} a^{4} - \frac{824650}{5699} a^{3} + \frac{425550}{5699} a^{2} - \frac{165000}{5699} a + \frac{56393}{5699} \) (order $4$)
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:  \( 8352.98291929 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times D_4$ (as 16T25):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_2^2 \times D_4$
Character table for $C_2^2 \times D_4$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{15}) \), 4.0.11520.1, 4.0.320.1, 4.0.2880.1, 4.0.1280.1, \(\Q(\sqrt{6}, \sqrt{10})\), \(\Q(i, \sqrt{6})\), \(\Q(\sqrt{6}, \sqrt{-10})\), \(\Q(i, \sqrt{10})\), \(\Q(\sqrt{-6}, \sqrt{10})\), \(\Q(i, \sqrt{15})\), \(\Q(\sqrt{-6}, \sqrt{-10})\), 8.0.3317760000.5, 8.0.530841600.4, 8.0.530841600.2, 8.0.13271040000.2, 8.0.163840000.2, 8.0.3317760000.11, 8.0.207360000.3

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: data not computed
Degree 16 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 R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$2.8.18.55$x^{8} + 2 x^{6} + 4 x^{3} + 10$$8$$1$$18$$D_4\times C_2$$[2, 2, 3]^{2}$
2.8.18.55$x^{8} + 2 x^{6} + 4 x^{3} + 10$$8$$1$$18$$D_4\times C_2$$[2, 2, 3]^{2}$
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
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}$