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Properties

Label 16.0.65044924825...3184.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 17^{6}\cdot 89^{4}$
Root discriminant $35.55$
Ramified primes $2, 17, 89$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^3.C_2^4.C_2$ (as 16T657)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![31688, 158848, 272884, -211004, 118945, -81008, 70802, -31988, 12608, -8112, 5248, -1864, 296, 4, 2, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 2*x^14 + 4*x^13 + 296*x^12 - 1864*x^11 + 5248*x^10 - 8112*x^9 + 12608*x^8 - 31988*x^7 + 70802*x^6 - 81008*x^5 + 118945*x^4 - 211004*x^3 + 272884*x^2 + 158848*x + 31688)
 
gp: K = bnfinit(x^16 - 4*x^15 + 2*x^14 + 4*x^13 + 296*x^12 - 1864*x^11 + 5248*x^10 - 8112*x^9 + 12608*x^8 - 31988*x^7 + 70802*x^6 - 81008*x^5 + 118945*x^4 - 211004*x^3 + 272884*x^2 + 158848*x + 31688, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 2 x^{14} + 4 x^{13} + 296 x^{12} - 1864 x^{11} + 5248 x^{10} - 8112 x^{9} + 12608 x^{8} - 31988 x^{7} + 70802 x^{6} - 81008 x^{5} + 118945 x^{4} - 211004 x^{3} + 272884 x^{2} + 158848 x + 31688 \)

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:  \(6504492482542510154973184=2^{32}\cdot 17^{6}\cdot 89^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $35.55$
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:  $2, 17, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{11} + \frac{1}{4} a^{9} - \frac{1}{2} a^{6} + \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{9} - \frac{1}{2} a^{8} + \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{24} a^{14} - \frac{1}{8} a^{13} + \frac{1}{24} a^{11} + \frac{1}{8} a^{10} + \frac{1}{4} a^{9} + \frac{7}{24} a^{8} + \frac{7}{24} a^{7} + \frac{5}{12} a^{6} + \frac{3}{8} a^{5} + \frac{1}{8} a^{4} - \frac{1}{12} a^{3} - \frac{1}{6} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{8713380706801529387921786774811463327992} a^{15} + \frac{51991227350363697428946034101507995353}{2904460235600509795973928924937154442664} a^{14} + \frac{21510391809149013960230826147208715993}{363057529450063724496741115617144305333} a^{13} - \frac{424953908576497195484135912292588375389}{8713380706801529387921786774811463327992} a^{12} + \frac{123157838153958973535849115604294226907}{2904460235600509795973928924937154442664} a^{11} + \frac{130614147172488747863806848245078914177}{1452230117800254897986964462468577221332} a^{10} + \frac{3400448757252898582240772970605632496665}{8713380706801529387921786774811463327992} a^{9} + \frac{3224487482233343273622370108639935440833}{8713380706801529387921786774811463327992} a^{8} - \frac{1268772433874275874389083091179978559207}{4356690353400764693960893387405731663996} a^{7} - \frac{520159846836833213789574720765570417727}{2904460235600509795973928924937154442664} a^{6} - \frac{930587739136391388544277870585009785953}{2904460235600509795973928924937154442664} a^{5} - \frac{817109930573780335868221698799318406701}{4356690353400764693960893387405731663996} a^{4} - \frac{801578874503193970748832148628756619377}{4356690353400764693960893387405731663996} a^{3} - \frac{202221689070583960166788689019520728066}{1089172588350191173490223346851432915999} a^{2} - \frac{493781357387450089772955001059801297985}{1089172588350191173490223346851432915999} a + \frac{5890250397466289909943919392896071167}{363057529450063724496741115617144305333}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( -\frac{44250165121311042654210995770}{5300961168217724371749348298517197} a^{15} + \frac{1500109596565883227268588752121}{42407689345741794973994786388137576} a^{14} - \frac{1506355984694204418359797537541}{42407689345741794973994786388137576} a^{13} - \frac{592918956813176638438513444155}{21203844672870897486997393194068788} a^{12} - \frac{101512456778332791937867009397841}{42407689345741794973994786388137576} a^{11} + \frac{690552447818196378215222700615089}{42407689345741794973994786388137576} a^{10} - \frac{1075537189181812195896658283179437}{21203844672870897486997393194068788} a^{9} + \frac{3627724168958449196050907064289295}{42407689345741794973994786388137576} a^{8} - \frac{5251148284788877125721072249489703}{42407689345741794973994786388137576} a^{7} + \frac{3067331690432805018711758478978263}{10601922336435448743498696597034394} a^{6} - \frac{30778553947940628409904399698018677}{42407689345741794973994786388137576} a^{5} + \frac{39227199661091396691243853155353569}{42407689345741794973994786388137576} a^{4} - \frac{23876760831599099100725073394927547}{21203844672870897486997393194068788} a^{3} + \frac{9547885549388264792323342070584572}{5300961168217724371749348298517197} a^{2} - \frac{17412656744100105393119740702518116}{5300961168217724371749348298517197} a - \frac{7997486894026496719404740367805436}{5300961168217724371749348298517197} \) (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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 3903757.22482 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_2^4.C_2$ (as 16T657):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 34 conjugacy class representatives for $C_2^3.C_2^4.C_2$
Character table for $C_2^3.C_2^4.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), 4.4.4352.1, 4.0.1088.2, \(\Q(\zeta_{8})\), 8.0.18939904.2

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$2$2.4.8.2$x^{4} + 6 x^{2} + 1$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.8.2$x^{4} + 6 x^{2} + 1$$4$$1$$8$$C_2^2$$[2, 3]$
2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
$17$17.4.3.4$x^{4} + 459$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.3.4$x^{4} + 459$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
89Data not computed