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Properties

Label 16.0.11819246862...024.15
Degree $16$
Signature $[0, 8]$
Discriminant $2^{54}\cdot 3^{8}$
Root discriminant $17.97$
Ramified primes $2, 3$
Class number $2$
Class group $[2]$
Galois group $C_2^2 : C_4$ (as 16T10)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![196, -1344, 4608, -8416, 10144, -9568, 7808, -5392, 3044, -1696, 1024, -496, 192, -80, 32, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 32*x^14 - 80*x^13 + 192*x^12 - 496*x^11 + 1024*x^10 - 1696*x^9 + 3044*x^8 - 5392*x^7 + 7808*x^6 - 9568*x^5 + 10144*x^4 - 8416*x^3 + 4608*x^2 - 1344*x + 196)
 
gp: K = bnfinit(x^16 - 8*x^15 + 32*x^14 - 80*x^13 + 192*x^12 - 496*x^11 + 1024*x^10 - 1696*x^9 + 3044*x^8 - 5392*x^7 + 7808*x^6 - 9568*x^5 + 10144*x^4 - 8416*x^3 + 4608*x^2 - 1344*x + 196, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 32 x^{14} - 80 x^{13} + 192 x^{12} - 496 x^{11} + 1024 x^{10} - 1696 x^{9} + 3044 x^{8} - 5392 x^{7} + 7808 x^{6} - 9568 x^{5} + 10144 x^{4} - 8416 x^{3} + 4608 x^{2} - 1344 x + 196 \)

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:  \(118192468620711297024=2^{54}\cdot 3^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.97$
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:  $2, 3$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{8} a^{8} - \frac{1}{2} a^{6} + \frac{1}{4}$, $\frac{1}{8} a^{9} - \frac{1}{2} a^{7} + \frac{1}{4} a$, $\frac{1}{8} a^{10} + \frac{1}{4} a^{2}$, $\frac{1}{16} a^{11} - \frac{1}{2} a^{6} + \frac{1}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{4592} a^{12} - \frac{3}{2296} a^{11} - \frac{73}{2296} a^{10} + \frac{59}{1148} a^{9} - \frac{15}{287} a^{8} + \frac{277}{574} a^{7} - \frac{71}{287} a^{6} - \frac{108}{287} a^{5} + \frac{1081}{2296} a^{4} - \frac{279}{1148} a^{3} - \frac{247}{1148} a^{2} - \frac{263}{574} a - \frac{6}{41}$, $\frac{1}{4592} a^{13} + \frac{15}{656} a^{11} - \frac{33}{2296} a^{10} + \frac{1}{164} a^{9} + \frac{101}{2296} a^{8} - \frac{101}{287} a^{7} + \frac{40}{287} a^{6} + \frac{489}{2296} a^{5} - \frac{120}{287} a^{4} + \frac{1037}{2296} a^{3} - \frac{573}{1148} a^{2} + \frac{30}{287} a - \frac{21}{164}$, $\frac{1}{4592} a^{14} - \frac{5}{2296} a^{11} - \frac{5}{164} a^{10} + \frac{13}{574} a^{9} + \frac{25}{2296} a^{8} - \frac{9}{287} a^{7} - \frac{715}{2296} a^{6} + \frac{27}{287} a^{5} + \frac{9}{574} a^{4} - \frac{265}{1148} a^{3} - \frac{31}{574} a^{2} - \frac{11}{41} a + \frac{19}{164}$, $\frac{1}{1017267858544} a^{15} + \frac{11544075}{254316964636} a^{14} + \frac{5418347}{254316964636} a^{13} - \frac{14687313}{508633929272} a^{12} - \frac{3579991041}{127158482318} a^{11} - \frac{26656622803}{508633929272} a^{10} + \frac{1828123963}{508633929272} a^{9} + \frac{15861479345}{254316964636} a^{8} - \frac{8151290645}{72661989896} a^{7} - \frac{48088823563}{127158482318} a^{6} - \frac{28353205883}{63579241159} a^{5} + \frac{99312510791}{254316964636} a^{4} + \frac{14138332521}{63579241159} a^{3} - \frac{17299645621}{36330994948} a^{2} + \frac{17086984061}{36330994948} a - \frac{2416149665}{18165497474}$

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:  \( \frac{9864760933}{1017267858544} a^{15} - \frac{71265853503}{1017267858544} a^{14} + \frac{130808620283}{508633929272} a^{13} - \frac{21248760675}{36330994948} a^{12} + \frac{731547702955}{508633929272} a^{11} - \frac{1910859423587}{508633929272} a^{10} + \frac{3644433206097}{508633929272} a^{9} - \frac{5744868267235}{508633929272} a^{8} + \frac{10935527119121}{508633929272} a^{7} - \frac{18638804431643}{508633929272} a^{6} + \frac{12556196009033}{254316964636} a^{5} - \frac{3700401953001}{63579241159} a^{4} + \frac{14652775533567}{254316964636} a^{3} - \frac{10479954196211}{254316964636} a^{2} + \frac{4245289755881}{254316964636} a - \frac{86180409917}{36330994948} \) (order $8$)
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:  \( 7434.5048088 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:C_4$ (as 16T10):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $C_2^2 : C_4$
Character table for $C_2^2 : C_4$

Intermediate fields

\(\Q(\sqrt{-2}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-1}) \), \(\Q(\zeta_{8})\), 4.2.1024.1 x2, 4.0.512.1 x2, 4.0.18432.1 x2, 4.2.18432.3 x2, 4.0.18432.2, 4.4.18432.1, 8.0.4194304.1, 8.0.1358954496.9, 8.0.1358954496.4, 8.0.5435817984.7 x2, 8.4.5435817984.2 x2

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

Sibling fields

Degree 8 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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$3.8.4.1$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}$