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Properties

Label 16.8.62125350736...5625.1
Degree $16$
Signature $[8, 4]$
Discriminant $5^{12}\cdot 11^{2}\cdot 101^{6}\cdot 4451^{2}$
Root discriminant $72.79$
Ramified primes $5, 11, 101, 4451$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $C_2^4.C_2^3.C_2$ (as 16T707)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![241811, 1831867, -1780634, 427733, 354290, -385678, 398, 64964, -8222, -1404, 2738, -794, -156, 119, -18, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 18*x^14 + 119*x^13 - 156*x^12 - 794*x^11 + 2738*x^10 - 1404*x^9 - 8222*x^8 + 64964*x^7 + 398*x^6 - 385678*x^5 + 354290*x^4 + 427733*x^3 - 1780634*x^2 + 1831867*x + 241811)
 
gp: K = bnfinit(x^16 - 4*x^15 - 18*x^14 + 119*x^13 - 156*x^12 - 794*x^11 + 2738*x^10 - 1404*x^9 - 8222*x^8 + 64964*x^7 + 398*x^6 - 385678*x^5 + 354290*x^4 + 427733*x^3 - 1780634*x^2 + 1831867*x + 241811, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 18 x^{14} + 119 x^{13} - 156 x^{12} - 794 x^{11} + 2738 x^{10} - 1404 x^{9} - 8222 x^{8} + 64964 x^{7} + 398 x^{6} - 385678 x^{5} + 354290 x^{4} + 427733 x^{3} - 1780634 x^{2} + 1831867 x + 241811 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(621253507360730723174072265625=5^{12}\cdot 11^{2}\cdot 101^{6}\cdot 4451^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $72.79$
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:  $5, 11, 101, 4451$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} - \frac{1}{5} a^{11} + \frac{1}{5} a^{8} - \frac{1}{5} a^{6} + \frac{1}{5} a^{4} - \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{13} - \frac{1}{5} a^{11} + \frac{1}{5} a^{9} + \frac{1}{5} a^{8} - \frac{1}{5} a^{7} - \frac{1}{5} a^{6} + \frac{1}{5} a^{5} + \frac{1}{5} a^{4} - \frac{1}{5} a^{2} + \frac{1}{5}$, $\frac{1}{25} a^{14} - \frac{2}{25} a^{12} + \frac{6}{25} a^{11} - \frac{4}{25} a^{10} + \frac{6}{25} a^{9} - \frac{2}{25} a^{8} + \frac{4}{25} a^{7} - \frac{8}{25} a^{6} + \frac{1}{25} a^{5} - \frac{6}{25} a^{4} + \frac{9}{25} a^{3} + \frac{2}{5} a^{2} - \frac{8}{25} a - \frac{1}{25}$, $\frac{1}{17245675874412633906765236431718017346502025} a^{15} + \frac{11866004429285021251797759719550609830741}{1567788715855693991524112402883456122409275} a^{14} - \frac{118439795500420707143006391779905845634082}{17245675874412633906765236431718017346502025} a^{13} - \frac{1581522385777368730276380451119397339258241}{17245675874412633906765236431718017346502025} a^{12} + \frac{2030695672098240333797432651409576094815177}{17245675874412633906765236431718017346502025} a^{11} + \frac{3314980258306546582332721104979153055568502}{17245675874412633906765236431718017346502025} a^{10} + \frac{2031459284471912584336547334392357977382224}{17245675874412633906765236431718017346502025} a^{9} - \frac{241454257317410554192678065522495953700298}{17245675874412633906765236431718017346502025} a^{8} - \frac{4075436896180763826476872426536020861931074}{17245675874412633906765236431718017346502025} a^{7} + \frac{6780262888627267577781066928997427481248218}{17245675874412633906765236431718017346502025} a^{6} - \frac{1131737839891224282917962010964260849788587}{3449135174882526781353047286343603469300405} a^{5} - \frac{1767778975892736989899719778264201212059847}{17245675874412633906765236431718017346502025} a^{4} - \frac{5493886472918633822068907482730339464599281}{17245675874412633906765236431718017346502025} a^{3} + \frac{5427743296831907044170800876209211342632557}{17245675874412633906765236431718017346502025} a^{2} - \frac{4146841385122686295024138196709905292106789}{17245675874412633906765236431718017346502025} a - \frac{6768920758217830438049546917895516878982826}{17245675874412633906765236431718017346502025}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  $11$
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:  \( 552775535.441 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3.C_2$ (as 16T707):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 31 conjugacy class representatives for $C_2^4.C_2^3.C_2$
Character table for $C_2^4.C_2^3.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.2525.1, 8.8.16098453125.1, 8.4.312156975625.1, 8.4.788196363453125.2

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
$101$101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.1.2$x^{2} + 202$$2$$1$$1$$C_2$$[\ ]_{2}$
101.2.1.2$x^{2} + 202$$2$$1$$1$$C_2$$[\ ]_{2}$
101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
4451Data not computed