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Properties

Label 16.4.15231185783...001.31
Degree $16$
Signature $[4, 6]$
Discriminant $17^{14}\cdot 67^{6}$
Root discriminant $57.73$
Ramified primes $17, 67$
Class number $8$ (GRH)
Class group $[2, 4]$ (GRH)
Galois group $C_2^3.C_4$ (as 16T41)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![42176, -316912, -185436, 229926, 74499, -37023, -197, 19634, 4023, -3327, 1348, 611, -207, 54, 17, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 17*x^14 + 54*x^13 - 207*x^12 + 611*x^11 + 1348*x^10 - 3327*x^9 + 4023*x^8 + 19634*x^7 - 197*x^6 - 37023*x^5 + 74499*x^4 + 229926*x^3 - 185436*x^2 - 316912*x + 42176)
 
gp: K = bnfinit(x^16 - 3*x^15 + 17*x^14 + 54*x^13 - 207*x^12 + 611*x^11 + 1348*x^10 - 3327*x^9 + 4023*x^8 + 19634*x^7 - 197*x^6 - 37023*x^5 + 74499*x^4 + 229926*x^3 - 185436*x^2 - 316912*x + 42176, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 17 x^{14} + 54 x^{13} - 207 x^{12} + 611 x^{11} + 1348 x^{10} - 3327 x^{9} + 4023 x^{8} + 19634 x^{7} - 197 x^{6} - 37023 x^{5} + 74499 x^{4} + 229926 x^{3} - 185436 x^{2} - 316912 x + 42176 \)

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:  \(15231185783695887615175635001=17^{14}\cdot 67^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $57.73$
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:  $17, 67$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{4}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{5}$, $\frac{1}{16} a^{12} - \frac{1}{8} a^{10} + \frac{1}{16} a^{9} - \frac{1}{8} a^{8} - \frac{1}{4} a^{7} + \frac{3}{16} a^{6} + \frac{1}{8} a^{4} + \frac{3}{16} a^{3} - \frac{3}{8} a^{2} - \frac{1}{4} a$, $\frac{1}{1072} a^{13} + \frac{3}{1072} a^{12} + \frac{45}{536} a^{11} - \frac{37}{1072} a^{10} - \frac{11}{1072} a^{9} + \frac{17}{536} a^{8} - \frac{153}{1072} a^{7} + \frac{217}{1072} a^{6} - \frac{69}{536} a^{5} - \frac{231}{1072} a^{4} + \frac{5}{16} a^{3} + \frac{111}{536} a^{2} + \frac{121}{268} a - \frac{28}{67}$, $\frac{1}{2144} a^{14} - \frac{1}{2144} a^{13} - \frac{7}{268} a^{12} + \frac{139}{2144} a^{11} - \frac{131}{2144} a^{10} + \frac{53}{536} a^{9} - \frac{21}{2144} a^{8} - \frac{243}{2144} a^{7} + \frac{25}{268} a^{6} - \frac{215}{2144} a^{5} - \frac{81}{2144} a^{4} - \frac{45}{536} a^{3} - \frac{21}{67} a^{2} + \frac{37}{268} a - \frac{11}{67}$, $\frac{1}{6000106440478837933756974353984} a^{15} - \frac{671225127198439858826654475}{3000053220239418966878487176992} a^{14} - \frac{2311402408014851013271983213}{6000106440478837933756974353984} a^{13} - \frac{26182460884254824866510692231}{6000106440478837933756974353984} a^{12} + \frac{93193521753072780286640415129}{3000053220239418966878487176992} a^{11} - \frac{506127244135527835678145316683}{6000106440478837933756974353984} a^{10} + \frac{645811608176087983959167369897}{6000106440478837933756974353984} a^{9} - \frac{305564056862374786939715557363}{3000053220239418966878487176992} a^{8} - \frac{328656038363551099856853463583}{6000106440478837933756974353984} a^{7} - \frac{750867269150886767684872285549}{6000106440478837933756974353984} a^{6} - \frac{211365118992883497302748841}{20134585370734355482405954208} a^{5} - \frac{416721257164164800532159886393}{6000106440478837933756974353984} a^{4} - \frac{1336328016994226344748021867107}{3000053220239418966878487176992} a^{3} + \frac{327400107533512914487345023025}{1500026610119709483439243588496} a^{2} - \frac{12833587925947830815676831761}{93751663132481842714952724281} a + \frac{22023506740805514371483479512}{93751663132481842714952724281}$

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

Class group and class number

$C_{2}\times C_{4}$, which has order $8$ (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:  \( 71792452.1584 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_4$ (as 16T41):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $C_2^3.C_4$
Character table for $C_2^3.C_4$

Intermediate fields

\(\Q(\sqrt{17}) \), 4.2.329171.1, 4.4.4913.1, 4.2.19363.1, 8.4.1842010303097.1 x2, 8.4.108353547241.1

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

Sibling fields

Galois closure: data not computed
Degree 8 siblings: data not computed
Degree 16 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$
$67$$\Q_{67}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{67}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{67}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{67}$$x + 4$$1$$1$$0$Trivial$[\ ]$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$