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Properties

Label 16.8.65574646090...9441.1
Degree $16$
Signature $[8, 4]$
Discriminant $17^{12}\cdot 103^{4}$
Root discriminant $26.67$
Ramified primes $17, 103$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1031

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -22, -3, 123, 75, -157, -327, 9, 375, 163, -189, -125, 57, 37, -11, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 11*x^14 + 37*x^13 + 57*x^12 - 125*x^11 - 189*x^10 + 163*x^9 + 375*x^8 + 9*x^7 - 327*x^6 - 157*x^5 + 75*x^4 + 123*x^3 - 3*x^2 - 22*x - 1)
 
gp: K = bnfinit(x^16 - 4*x^15 - 11*x^14 + 37*x^13 + 57*x^12 - 125*x^11 - 189*x^10 + 163*x^9 + 375*x^8 + 9*x^7 - 327*x^6 - 157*x^5 + 75*x^4 + 123*x^3 - 3*x^2 - 22*x - 1, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 11 x^{14} + 37 x^{13} + 57 x^{12} - 125 x^{11} - 189 x^{10} + 163 x^{9} + 375 x^{8} + 9 x^{7} - 327 x^{6} - 157 x^{5} + 75 x^{4} + 123 x^{3} - 3 x^{2} - 22 x - 1 \)

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:  \(65574646090400599969441=17^{12}\cdot 103^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.67$
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:  $17, 103$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{7243146351862} a^{15} + \frac{645556703790}{3621573175931} a^{14} + \frac{1652058848911}{3621573175931} a^{13} - \frac{1015101926799}{3621573175931} a^{12} + \frac{1734705778129}{3621573175931} a^{11} + \frac{157303002754}{3621573175931} a^{10} - \frac{318894744799}{3621573175931} a^{9} + \frac{572346656847}{3621573175931} a^{8} - \frac{1094113244194}{3621573175931} a^{7} - \frac{322303773060}{3621573175931} a^{6} + \frac{1336861973529}{3621573175931} a^{5} + \frac{1434860639380}{3621573175931} a^{4} + \frac{1671647528967}{3621573175931} a^{3} + \frac{852071067828}{3621573175931} a^{2} - \frac{1416125841382}{3621573175931} a - \frac{1346420894407}{7243146351862}$

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:  \( 335452.042865 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1031:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 576
The 13 conjugacy class representatives for t16n1031
Character table for t16n1031

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, 8.4.15063262913.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 8 sibling: data not computed
Degree 12 siblings: data not computed
Degree 16 sibling: data not computed
Degree 18 siblings: data not computed
Degree 24 siblings: data not computed
Degree 32 sibling: data not computed
Degree 36 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$17$17.8.6.1$x^{8} - 119 x^{4} + 23409$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
17.8.6.1$x^{8} - 119 x^{4} + 23409$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$103$$\Q_{103}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{103}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{103}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{103}$$x + 2$$1$$1$$0$Trivial$[\ ]$
103.3.2.2$x^{3} + 206$$3$$1$$2$$C_3$$[\ ]_{3}$
103.3.0.1$x^{3} - x + 29$$1$$3$$0$$C_3$$[\ ]^{3}$
103.3.2.2$x^{3} + 206$$3$$1$$2$$C_3$$[\ ]_{3}$
103.3.0.1$x^{3} - x + 29$$1$$3$$0$$C_3$$[\ ]^{3}$