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Properties

Label 16.8.17877675240...0000.2
Degree $16$
Signature $[8, 4]$
Discriminant $2^{16}\cdot 3^{12}\cdot 5^{14}\cdot 29^{2}$
Root discriminant $28.40$
Ramified primes $2, 3, 5, 29$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1162

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 22, 110, -10, 95, 236, -288, 90, -260, -240, 42, -76, -10, 20, -10, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 10*x^14 + 20*x^13 - 10*x^12 - 76*x^11 + 42*x^10 - 240*x^9 - 260*x^8 + 90*x^7 - 288*x^6 + 236*x^5 + 95*x^4 - 10*x^3 + 110*x^2 + 22*x + 1)
 
gp: K = bnfinit(x^16 - 2*x^15 - 10*x^14 + 20*x^13 - 10*x^12 - 76*x^11 + 42*x^10 - 240*x^9 - 260*x^8 + 90*x^7 - 288*x^6 + 236*x^5 + 95*x^4 - 10*x^3 + 110*x^2 + 22*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 10 x^{14} + 20 x^{13} - 10 x^{12} - 76 x^{11} + 42 x^{10} - 240 x^{9} - 260 x^{8} + 90 x^{7} - 288 x^{6} + 236 x^{5} + 95 x^{4} - 10 x^{3} + 110 x^{2} + 22 x + 1 \)

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:  \(178776752400000000000000=2^{16}\cdot 3^{12}\cdot 5^{14}\cdot 29^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.40$
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:  $2, 3, 5, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{30} a^{8} + \frac{7}{15} a^{7} + \frac{1}{15} a^{6} + \frac{2}{5} a^{5} - \frac{1}{3} a^{4} - \frac{2}{5} a^{3} - \frac{13}{30} a^{2} - \frac{2}{15} a + \frac{1}{30}$, $\frac{1}{30} a^{9} - \frac{7}{15} a^{7} + \frac{7}{15} a^{6} + \frac{1}{15} a^{5} + \frac{4}{15} a^{4} + \frac{1}{6} a^{3} - \frac{1}{15} a^{2} - \frac{1}{10} a - \frac{7}{15}$, $\frac{1}{30} a^{10} - \frac{2}{15} a^{5} - \frac{1}{2} a^{4} + \frac{1}{3} a^{3} - \frac{1}{6} a^{2} - \frac{1}{3} a + \frac{7}{15}$, $\frac{1}{30} a^{11} - \frac{2}{15} a^{6} - \frac{1}{2} a^{5} + \frac{1}{3} a^{4} - \frac{1}{6} a^{3} - \frac{1}{3} a^{2} + \frac{7}{15} a$, $\frac{1}{30} a^{12} - \frac{2}{15} a^{7} - \frac{1}{2} a^{6} + \frac{1}{3} a^{5} - \frac{1}{6} a^{4} - \frac{1}{3} a^{3} + \frac{7}{15} a^{2}$, $\frac{1}{30} a^{13} + \frac{11}{30} a^{7} - \frac{2}{5} a^{6} + \frac{13}{30} a^{5} + \frac{1}{3} a^{4} - \frac{2}{15} a^{3} + \frac{4}{15} a^{2} + \frac{7}{15} a + \frac{2}{15}$, $\frac{1}{30} a^{14} + \frac{7}{15} a^{7} - \frac{3}{10} a^{6} - \frac{1}{15} a^{5} - \frac{7}{15} a^{4} - \frac{1}{3} a^{3} + \frac{7}{30} a^{2} - \frac{2}{5} a - \frac{11}{30}$, $\frac{1}{1456983870} a^{15} + \frac{2171327}{1456983870} a^{14} + \frac{1117864}{145698387} a^{13} - \frac{20437427}{1456983870} a^{12} - \frac{52199}{242830645} a^{11} - \frac{12138322}{728491935} a^{10} - \frac{7248067}{485661290} a^{9} - \frac{4537687}{728491935} a^{8} + \frac{108973363}{485661290} a^{7} - \frac{228134747}{728491935} a^{6} - \frac{71404424}{145698387} a^{5} + \frac{476564179}{1456983870} a^{4} + \frac{191792744}{728491935} a^{3} - \frac{4636907}{291396774} a^{2} + \frac{83674244}{242830645} a + \frac{14162833}{1456983870}$

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:  $11$
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:  \( 537175.15876 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1162:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 49 conjugacy class representatives for t16n1162
Character table for t16n1162 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{15})^+\), 8.8.14580000000.1

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 ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$2$2.8.8.5$x^{8} + 2 x^{7} + 8 x^{2} + 16$$2$$4$$8$$C_8:C_2$$[2, 2]^{4}$
2.8.8.5$x^{8} + 2 x^{7} + 8 x^{2} + 16$$2$$4$$8$$C_8:C_2$$[2, 2]^{4}$
$3$3.8.6.3$x^{8} - 3 x^{4} + 18$$4$$2$$6$$C_8:C_2$$[\ ]_{4}^{4}$
3.8.6.3$x^{8} - 3 x^{4} + 18$$4$$2$$6$$C_8:C_2$$[\ ]_{4}^{4}$
5Data not computed
$29$$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{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$$[\ ]^{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$