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Properties

Label 16.0.30467756702...9696.2
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 $4$
Class group $[2, 2]$
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![2897, 4608, 6172, 4720, 5336, 7040, 7652, 5104, 1174, -496, -316, 160, 136, -16, -4, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^14 - 16*x^13 + 136*x^12 + 160*x^11 - 316*x^10 - 496*x^9 + 1174*x^8 + 5104*x^7 + 7652*x^6 + 7040*x^5 + 5336*x^4 + 4720*x^3 + 6172*x^2 + 4608*x + 2897)
 
gp: K = bnfinit(x^16 - 4*x^14 - 16*x^13 + 136*x^12 + 160*x^11 - 316*x^10 - 496*x^9 + 1174*x^8 + 5104*x^7 + 7652*x^6 + 7040*x^5 + 5336*x^4 + 4720*x^3 + 6172*x^2 + 4608*x + 2897, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{14} - 16 x^{13} + 136 x^{12} + 160 x^{11} - 316 x^{10} - 496 x^{9} + 1174 x^{8} + 5104 x^{7} + 7652 x^{6} + 7040 x^{5} + 5336 x^{4} + 4720 x^{3} + 6172 x^{2} + 4608 x + 2897 \)

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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{6} a^{10} + \frac{1}{6} a^{9} - \frac{1}{6} a^{7} - \frac{1}{6} a^{6} + \frac{1}{6} a^{5} - \frac{1}{2} a^{4} + \frac{1}{6} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{6}$, $\frac{1}{12} a^{11} - \frac{1}{12} a^{10} + \frac{1}{12} a^{9} - \frac{1}{12} a^{8} - \frac{1}{6} a^{7} - \frac{1}{6} a^{5} + \frac{1}{3} a^{4} + \frac{5}{12} a^{3} + \frac{5}{12} a^{2} + \frac{5}{12} a + \frac{1}{12}$, $\frac{1}{12} a^{12} - \frac{1}{4} a^{8} - \frac{1}{6} a^{7} - \frac{1}{6} a^{6} + \frac{1}{6} a^{5} - \frac{1}{4} a^{4} - \frac{1}{6} a^{3} - \frac{1}{6} a^{2} - \frac{1}{2} a + \frac{1}{12}$, $\frac{1}{12} a^{13} - \frac{1}{4} a^{9} - \frac{1}{6} a^{8} - \frac{1}{6} a^{7} + \frac{1}{6} a^{6} - \frac{1}{4} a^{5} - \frac{1}{6} a^{4} - \frac{1}{6} a^{3} - \frac{1}{2} a^{2} + \frac{1}{12} a$, $\frac{1}{1428} a^{14} - \frac{4}{119} a^{13} - \frac{11}{357} a^{12} + \frac{11}{1428} a^{11} - \frac{6}{119} a^{10} + \frac{19}{84} a^{9} - \frac{205}{1428} a^{8} - \frac{5}{34} a^{7} + \frac{185}{1428} a^{6} - \frac{313}{714} a^{5} + \frac{55}{238} a^{4} + \frac{199}{1428} a^{3} - \frac{95}{714} a^{2} - \frac{99}{476} a - \frac{97}{1428}$, $\frac{1}{33593928315086059677204} a^{15} + \frac{1327840535568726157}{16796964157543029838602} a^{14} + \frac{115410877075637239900}{2799494026257171639767} a^{13} + \frac{95850834599317346093}{8398482078771514919301} a^{12} - \frac{15718609260480087757}{1199783154110216417043} a^{11} - \frac{52711751895911867379}{1599710872146955222724} a^{10} - \frac{1754397240975075215369}{11197976105028686559068} a^{9} + \frac{573769914452467706567}{33593928315086059677204} a^{8} - \frac{4288912276389883454141}{33593928315086059677204} a^{7} + \frac{350628590791514195612}{8398482078771514919301} a^{6} - \frac{1652683145614844451851}{8398482078771514919301} a^{5} + \frac{256587612606524463899}{2399566308220432834086} a^{4} - \frac{4120284866956209783334}{8398482078771514919301} a^{3} + \frac{5049496208860118893}{94100639538056189572} a^{2} - \frac{2015673644687772988993}{33593928315086059677204} a + \frac{1426088428617398955119}{33593928315086059677204}$

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

Class group and class number

$C_{2}\times C_{2}$, 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:  \( -\frac{11462221873183071518}{8398482078771514919301} a^{15} - \frac{682787494927165135}{4799132616440865668172} a^{14} + \frac{7913586994981233313}{988056715149589990506} a^{13} + \frac{311174343118287349189}{16796964157543029838602} a^{12} - \frac{3221454077382983078153}{16796964157543029838602} a^{11} - \frac{8721125387886269559209}{33593928315086059677204} a^{10} + \frac{13482855676807499486921}{16796964157543029838602} a^{9} + \frac{3173680040436888377013}{5598988052514343279534} a^{8} - \frac{46062397761785123019403}{16796964157543029838602} a^{7} - \frac{71040095731918714153701}{11197976105028686559068} a^{6} - \frac{3659006798713823742980}{494028357574794995253} a^{5} - \frac{76387297029472233539747}{16796964157543029838602} a^{4} - \frac{37176878835790367213318}{8398482078771514919301} a^{3} - \frac{187587089363202319046939}{33593928315086059677204} a^{2} - \frac{43131934561142816723003}{8398482078771514919301} a - \frac{25941265716177496248835}{5598988052514343279534} \) (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:  \( 145538.831483 \)
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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$Trivial$[\ ]$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$