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Properties

Label 16.4.15808771774...3125.1
Degree $16$
Signature $[4, 6]$
Discriminant $5^{12}\cdot 61\cdot 101^{6}$
Root discriminant $24.40$
Ramified primes $5, 61, 101$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1643

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![11, -38, -69, 618, -1614, 2411, -2226, 1099, 252, -1046, 1109, -769, 386, -147, 41, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 41*x^14 - 147*x^13 + 386*x^12 - 769*x^11 + 1109*x^10 - 1046*x^9 + 252*x^8 + 1099*x^7 - 2226*x^6 + 2411*x^5 - 1614*x^4 + 618*x^3 - 69*x^2 - 38*x + 11)
 
gp: K = bnfinit(x^16 - 8*x^15 + 41*x^14 - 147*x^13 + 386*x^12 - 769*x^11 + 1109*x^10 - 1046*x^9 + 252*x^8 + 1099*x^7 - 2226*x^6 + 2411*x^5 - 1614*x^4 + 618*x^3 - 69*x^2 - 38*x + 11, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 41 x^{14} - 147 x^{13} + 386 x^{12} - 769 x^{11} + 1109 x^{10} - 1046 x^{9} + 252 x^{8} + 1099 x^{7} - 2226 x^{6} + 2411 x^{5} - 1614 x^{4} + 618 x^{3} - 69 x^{2} - 38 x + 11 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(15808771774087158203125=5^{12}\cdot 61\cdot 101^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.40$
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:  $5, 61, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8} + \frac{1}{5} a^{7} - \frac{1}{5} a^{5} - \frac{1}{5} a^{4} - \frac{1}{5} a^{3} + \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{9} - \frac{1}{5} a^{7} - \frac{1}{5} a^{6} + \frac{1}{5} a^{3} + \frac{1}{5} a^{2} - \frac{1}{5}$, $\frac{1}{5} a^{10} - \frac{1}{5} a^{5} + \frac{1}{5}$, $\frac{1}{5} a^{11} - \frac{1}{5} a^{6} + \frac{1}{5} a$, $\frac{1}{25} a^{12} - \frac{1}{25} a^{11} + \frac{1}{25} a^{10} + \frac{9}{25} a^{7} + \frac{6}{25} a^{6} - \frac{1}{25} a^{5} + \frac{1}{5} a^{4} - \frac{2}{5} a^{3} + \frac{6}{25} a^{2} - \frac{11}{25} a - \frac{9}{25}$, $\frac{1}{25} a^{13} + \frac{1}{25} a^{10} - \frac{1}{25} a^{8} + \frac{1}{5} a^{7} + \frac{1}{5} a^{6} - \frac{11}{25} a^{5} + \frac{1}{5} a^{4} + \frac{6}{25} a^{3} - \frac{1}{5} a^{2} - \frac{1}{5} a + \frac{6}{25}$, $\frac{1}{1775} a^{14} - \frac{7}{1775} a^{13} - \frac{24}{1775} a^{12} - \frac{24}{355} a^{11} - \frac{46}{1775} a^{10} + \frac{39}{1775} a^{9} - \frac{18}{1775} a^{8} - \frac{841}{1775} a^{7} - \frac{122}{355} a^{6} - \frac{149}{1775} a^{5} - \frac{294}{1775} a^{4} + \frac{38}{1775} a^{3} - \frac{499}{1775} a^{2} + \frac{151}{355} a + \frac{589}{1775}$, $\frac{1}{1775} a^{15} - \frac{2}{1775} a^{13} - \frac{4}{1775} a^{12} - \frac{21}{355} a^{11} + \frac{72}{1775} a^{10} - \frac{4}{71} a^{9} + \frac{27}{1775} a^{8} - \frac{391}{1775} a^{7} + \frac{96}{355} a^{6} + \frac{83}{1775} a^{5} + \frac{93}{355} a^{4} - \frac{517}{1775} a^{3} + \frac{31}{1775} a^{2} - \frac{32}{71} a - \frac{137}{1775}$

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:  $9$
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:  \( 49730.0794225 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1643:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4096
The 64 conjugacy class representatives for t16n1643 are not computed
Character table for t16n1643 is not computed

Intermediate fields

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

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 $16$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R $16$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$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}$
$61$61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.1.1$x^{2} - 61$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
$101$101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
101.4.0.1$x^{4} - x + 12$$1$$4$$0$$C_4$$[\ ]^{4}$
101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$