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Properties

Label 16.10.8410853036...0000.3
Degree $16$
Signature $[10, 3]$
Discriminant $-\,2^{8}\cdot 3^{8}\cdot 5^{12}\cdot 29^{5}$
Root discriminant $23.46$
Ramified primes $2, 3, 5, 29$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1616

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16, 112, 8, -280, -320, -172, 224, 260, -1, 65, -34, -74, 45, 15, -13, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 13*x^14 + 15*x^13 + 45*x^12 - 74*x^11 - 34*x^10 + 65*x^9 - x^8 + 260*x^7 + 224*x^6 - 172*x^5 - 320*x^4 - 280*x^3 + 8*x^2 + 112*x + 16)
 
gp: K = bnfinit(x^16 - x^15 - 13*x^14 + 15*x^13 + 45*x^12 - 74*x^11 - 34*x^10 + 65*x^9 - x^8 + 260*x^7 + 224*x^6 - 172*x^5 - 320*x^4 - 280*x^3 + 8*x^2 + 112*x + 16, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 13 x^{14} + 15 x^{13} + 45 x^{12} - 74 x^{11} - 34 x^{10} + 65 x^{9} - x^{8} + 260 x^{7} + 224 x^{6} - 172 x^{5} - 320 x^{4} - 280 x^{3} + 8 x^{2} + 112 x + 16 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-8410853036812500000000=-\,2^{8}\cdot 3^{8}\cdot 5^{12}\cdot 29^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.46$
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:  $2, 3, 5, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{220} a^{12} + \frac{39}{220} a^{11} - \frac{47}{220} a^{10} + \frac{9}{220} a^{9} - \frac{81}{220} a^{8} + \frac{6}{55} a^{7} - \frac{23}{55} a^{6} - \frac{67}{220} a^{5} + \frac{31}{220} a^{4} - \frac{9}{110} a^{3} - \frac{21}{110} a^{2} + \frac{17}{55} a - \frac{24}{55}$, $\frac{1}{220} a^{13} - \frac{7}{55} a^{11} - \frac{7}{55} a^{10} - \frac{51}{110} a^{9} - \frac{7}{220} a^{8} - \frac{19}{110} a^{7} - \frac{109}{220} a^{6} + \frac{1}{55} a^{5} + \frac{93}{220} a^{4} - \frac{1}{2} a^{3} + \frac{14}{55} a^{2} - \frac{27}{55} a + \frac{1}{55}$, $\frac{1}{440} a^{14} - \frac{1}{440} a^{13} - \frac{1}{440} a^{12} + \frac{63}{440} a^{11} - \frac{23}{440} a^{10} - \frac{51}{220} a^{9} + \frac{101}{220} a^{8} + \frac{137}{440} a^{7} - \frac{61}{440} a^{6} + \frac{1}{11} a^{5} - \frac{17}{55} a^{4} - \frac{5}{22} a^{3} + \frac{3}{10} a^{2} - \frac{4}{55} a - \frac{2}{5}$, $\frac{1}{1042369618840} a^{15} - \frac{722147621}{1042369618840} a^{14} + \frac{1889264269}{1042369618840} a^{13} + \frac{11271727}{7188755992} a^{12} - \frac{15183487303}{208473923768} a^{11} - \frac{4590649883}{521184809420} a^{10} - \frac{92835339}{104236961884} a^{9} + \frac{75333712635}{208473923768} a^{8} - \frac{283387960733}{1042369618840} a^{7} + \frac{30555232603}{521184809420} a^{6} - \frac{21336615675}{52118480942} a^{5} + \frac{8553452911}{521184809420} a^{4} + \frac{12431408107}{130296202355} a^{3} + \frac{56168839611}{260592404710} a^{2} - \frac{9166358995}{26059240471} a + \frac{24525691501}{130296202355}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $12$
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:  \( 137619.756723 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1616:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4096
The 94 conjugacy class representatives for t16n1616 are not computed
Character table for t16n1616 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.32625.1, \(\Q(\zeta_{15})^+\), 4.4.725.1, 8.8.1064390625.1

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 ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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$2.8.8.8$x^{8} + 4 x^{5} + 8 x^{2} + 48$$2$$4$$8$$((C_8 : C_2):C_2):C_2$$[2, 2, 2, 2]^{4}$
2.8.0.1$x^{8} + x^{4} + x^{3} + x + 1$$1$$8$$0$$C_8$$[\ ]^{8}$
$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}$
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$29$29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.3.3$x^{4} + 58$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$