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Properties

Label 16.10.1042864758...0000.1
Degree $16$
Signature $[10, 3]$
Discriminant $-\,2^{8}\cdot 3^{10}\cdot 5^{12}\cdot 41^{4}$
Root discriminant $23.78$
Ramified primes $2, 3, 5, 41$
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, -128, 392, -504, -96, 1304, -2016, 1368, 13, -780, 575, -95, -98, 59, -6, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 6*x^14 + 59*x^13 - 98*x^12 - 95*x^11 + 575*x^10 - 780*x^9 + 13*x^8 + 1368*x^7 - 2016*x^6 + 1304*x^5 - 96*x^4 - 504*x^3 + 392*x^2 - 128*x + 16)
 
gp: K = bnfinit(x^16 - 4*x^15 - 6*x^14 + 59*x^13 - 98*x^12 - 95*x^11 + 575*x^10 - 780*x^9 + 13*x^8 + 1368*x^7 - 2016*x^6 + 1304*x^5 - 96*x^4 - 504*x^3 + 392*x^2 - 128*x + 16, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 6 x^{14} + 59 x^{13} - 98 x^{12} - 95 x^{11} + 575 x^{10} - 780 x^{9} + 13 x^{8} + 1368 x^{7} - 2016 x^{6} + 1304 x^{5} - 96 x^{4} - 504 x^{3} + 392 x^{2} - 128 x + 16 \)

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:  \(-10428647580562500000000=-\,2^{8}\cdot 3^{10}\cdot 5^{12}\cdot 41^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.78$
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:  $2, 3, 5, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{132} a^{12} + \frac{4}{33} a^{11} + \frac{1}{66} a^{10} + \frac{47}{132} a^{9} + \frac{5}{22} a^{8} + \frac{25}{132} a^{7} - \frac{1}{4} a^{6} - \frac{8}{33} a^{5} + \frac{65}{132} a^{4} - \frac{5}{11} a^{3} + \frac{14}{33} a^{2} - \frac{5}{11} a - \frac{14}{33}$, $\frac{1}{132} a^{13} + \frac{5}{66} a^{11} + \frac{5}{44} a^{10} - \frac{31}{66} a^{9} - \frac{59}{132} a^{8} - \frac{37}{132} a^{7} - \frac{8}{33} a^{6} + \frac{49}{132} a^{5} - \frac{1}{3} a^{4} - \frac{10}{33} a^{3} - \frac{8}{33} a^{2} - \frac{5}{33} a - \frac{7}{33}$, $\frac{1}{264} a^{14} - \frac{13}{264} a^{11} + \frac{25}{132} a^{10} + \frac{131}{264} a^{9} - \frac{73}{264} a^{8} + \frac{19}{44} a^{7} - \frac{17}{264} a^{6} - \frac{5}{11} a^{5} - \frac{5}{44} a^{4} + \frac{5}{33} a^{3} - \frac{13}{66} a^{2} - \frac{1}{3} a + \frac{4}{33}$, $\frac{1}{264} a^{15} - \frac{1}{264} a^{12} - \frac{1}{12} a^{11} + \frac{23}{264} a^{10} - \frac{37}{264} a^{9} - \frac{9}{44} a^{8} - \frac{113}{264} a^{7} + \frac{1}{22} a^{6} - \frac{3}{44} a^{5} - \frac{13}{33} a^{4} + \frac{5}{66} a^{3} - \frac{19}{66} a^{2} + \frac{13}{33} a + \frac{5}{11}$

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:  \( 154992.464824 \) (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

\(\Q(\sqrt{5}) \), 4.4.9225.1, \(\Q(\zeta_{15})^+\), 4.4.5125.1, 8.8.2127515625.1

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 ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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$2.8.8.9$x^{8} + 6 x^{6} + 4 x^{5} + 16$$2$$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$$[\ ]_{2}^{4}$
3.8.6.3$x^{8} - 3 x^{4} + 18$$4$$2$$6$$C_8:C_2$$[\ ]_{4}^{4}$
5Data not computed
$41$41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$