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Properties

Label 16.8.33118016836...2400.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{50}\cdot 3^{10}\cdot 5^{2}\cdot 109^{8}$
Root discriminant $221.31$
Ramified primes $2, 3, 5, 109$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1220

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![27225, 0, 317160, 0, -542088, 0, -117840, 0, 58666, 0, 6152, 0, -648, 0, -16, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 16*x^14 - 648*x^12 + 6152*x^10 + 58666*x^8 - 117840*x^6 - 542088*x^4 + 317160*x^2 + 27225)
 
gp: K = bnfinit(x^16 - 16*x^14 - 648*x^12 + 6152*x^10 + 58666*x^8 - 117840*x^6 - 542088*x^4 + 317160*x^2 + 27225, 1)
 

Normalized defining polynomial

\( x^{16} - 16 x^{14} - 648 x^{12} + 6152 x^{10} + 58666 x^{8} - 117840 x^{6} - 542088 x^{4} + 317160 x^{2} + 27225 \)

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:  \(33118016836326980261932928527132262400=2^{50}\cdot 3^{10}\cdot 5^{2}\cdot 109^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $221.31$
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:  $2, 3, 5, 109$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{24} a^{8} + \frac{1}{6} a^{6} + \frac{1}{6} a^{4} - \frac{1}{2} a^{2} + \frac{3}{8}$, $\frac{1}{24} a^{9} + \frac{1}{6} a^{7} + \frac{1}{6} a^{5} - \frac{1}{2} a^{3} + \frac{3}{8} a$, $\frac{1}{24} a^{10} - \frac{1}{2} a^{6} - \frac{1}{6} a^{4} + \frac{3}{8} a^{2} - \frac{1}{2}$, $\frac{1}{24} a^{11} - \frac{1}{2} a^{7} - \frac{1}{6} a^{5} + \frac{3}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{120} a^{12} - \frac{1}{60} a^{10} - \frac{1}{120} a^{8} - \frac{4}{15} a^{6} + \frac{13}{120} a^{4} - \frac{3}{20} a^{2} - \frac{3}{8}$, $\frac{1}{1320} a^{13} + \frac{13}{1320} a^{11} + \frac{1}{330} a^{9} - \frac{23}{55} a^{7} - \frac{19}{40} a^{5} - \frac{21}{440} a^{3} + \frac{7}{22} a$, $\frac{1}{295689519136045320} a^{14} + \frac{72316602853607}{295689519136045320} a^{12} + \frac{200007025411091}{49281586522674220} a^{10} + \frac{4637749790560489}{295689519136045320} a^{8} + \frac{64858362993705}{137850591671816} a^{6} + \frac{129437122182958439}{295689519136045320} a^{4} - \frac{6128979448298087}{49281586522674220} a^{2} - \frac{454553738497731}{1792057691733608}$, $\frac{1}{591379038272090640} a^{15} - \frac{1}{591379038272090640} a^{14} + \frac{72316602853607}{591379038272090640} a^{13} - \frac{72316602853607}{591379038272090640} a^{12} - \frac{11120354478202009}{591379038272090640} a^{11} + \frac{11120354478202009}{591379038272090640} a^{10} + \frac{4637749790560489}{591379038272090640} a^{9} - \frac{4637749790560489}{591379038272090640} a^{8} - \frac{4066932842203}{275701183343632} a^{7} + \frac{4066932842203}{275701183343632} a^{6} - \frac{38990270143470887}{197126346090696880} a^{5} + \frac{38990270143470887}{197126346090696880} a^{4} - \frac{49219148788601839}{197126346090696880} a^{3} + \frac{49219148788601839}{197126346090696880} a^{2} + \frac{441475107369073}{3584115383467216} a - \frac{441475107369073}{3584115383467216}$

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:  $11$
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:  \( 13436675245200 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1220:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 55 conjugacy class representatives for t16n1220 are not computed
Character table for t16n1220 is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{654}) \), \(\Q(\sqrt{218}) \), 4.4.251136.1, 4.4.62784.1, \(\Q(\sqrt{3}, \sqrt{218})\), 8.8.2997304961531904.1

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 R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
2Data not computed
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
$5$5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$109$109.2.1.2$x^{2} + 654$$2$$1$$1$$C_2$$[\ ]_{2}$
109.2.1.2$x^{2} + 654$$2$$1$$1$$C_2$$[\ ]_{2}$
109.4.2.1$x^{4} + 1199 x^{2} + 427716$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
109.4.2.1$x^{4} + 1199 x^{2} + 427716$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
109.4.2.1$x^{4} + 1199 x^{2} + 427716$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$