↑

Properties

Label 16.0.43527014496...0625.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{8}\cdot 5^{8}\cdot 19^{8}$
Root discriminant $16.88$
Ramified primes $3, 5, 19$
Class number $2$
Class group $[2]$
Galois group $D_4\times C_2$ (as 16T9)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, -17, 68, 271, -582, 862, -480, 85, 268, -194, 86, 2, -16, 10, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 10*x^14 - 16*x^13 + 2*x^12 + 86*x^11 - 194*x^10 + 268*x^9 + 85*x^8 - 480*x^7 + 862*x^6 - 582*x^5 + 271*x^4 + 68*x^3 - 17*x^2 + 1)
 
gp: K = bnfinit(x^16 - 4*x^15 + 10*x^14 - 16*x^13 + 2*x^12 + 86*x^11 - 194*x^10 + 268*x^9 + 85*x^8 - 480*x^7 + 862*x^6 - 582*x^5 + 271*x^4 + 68*x^3 - 17*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 10 x^{14} - 16 x^{13} + 2 x^{12} + 86 x^{11} - 194 x^{10} + 268 x^{9} + 85 x^{8} - 480 x^{7} + 862 x^{6} - 582 x^{5} + 271 x^{4} + 68 x^{3} - 17 x^{2} + 1 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(43527014496875390625=3^{8}\cdot 5^{8}\cdot 19^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.88$
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:  $3, 5, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{6} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{2} a^{4} - \frac{1}{3} a^{3} - \frac{1}{6} a^{2} - \frac{1}{2} a + \frac{1}{6}$, $\frac{1}{6} a^{9} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{6} a^{5} - \frac{1}{3} a^{4} + \frac{1}{6} a^{3} + \frac{1}{6} a^{2} + \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{6} a^{10} + \frac{1}{3} a^{7} - \frac{1}{6} a^{6} + \frac{1}{3} a^{5} + \frac{1}{6} a^{4} - \frac{1}{6} a^{3} - \frac{1}{2} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{6} a^{11} - \frac{1}{2} a^{7} + \frac{1}{3} a^{6} - \frac{1}{6} a^{5} - \frac{1}{6} a^{4} + \frac{1}{6} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{6} a^{12} + \frac{1}{3} a^{7} - \frac{1}{6} a^{6} - \frac{1}{6} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{6} a^{2} + \frac{1}{6} a - \frac{1}{2}$, $\frac{1}{6} a^{13} - \frac{1}{2} a^{7} - \frac{1}{6} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{6} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{3}$, $\frac{1}{2622} a^{14} + \frac{169}{2622} a^{13} + \frac{50}{1311} a^{12} - \frac{175}{2622} a^{11} - \frac{5}{437} a^{10} + \frac{67}{874} a^{9} + \frac{29}{2622} a^{8} + \frac{409}{874} a^{7} - \frac{131}{2622} a^{6} + \frac{583}{1311} a^{5} - \frac{1}{3} a^{4} - \frac{13}{69} a^{3} - \frac{413}{2622} a^{2} + \frac{533}{1311} a - \frac{389}{1311}$, $\frac{1}{231277545258} a^{15} + \frac{14539561}{115638772629} a^{14} + \frac{3201864265}{231277545258} a^{13} + \frac{954500681}{77092515086} a^{12} + \frac{18531745505}{231277545258} a^{11} + \frac{2327580621}{38546257543} a^{10} + \frac{15999439019}{231277545258} a^{9} + \frac{5951824933}{115638772629} a^{8} - \frac{48358100188}{115638772629} a^{7} - \frac{14450102017}{115638772629} a^{6} - \frac{106203427777}{231277545258} a^{5} - \frac{2726146855}{12172502382} a^{4} + \frac{36048283699}{77092515086} a^{3} + \frac{8310804329}{231277545258} a^{2} + \frac{1498655479}{77092515086} a - \frac{34058883331}{115638772629}$

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

Class group and class number

$C_{2}$, which has order $2$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{4005796}{1675924241} a^{15} + \frac{339545318}{5027772723} a^{14} - \frac{1297710080}{5027772723} a^{13} + \frac{1051002470}{1675924241} a^{12} - \frac{4803016742}{5027772723} a^{11} - \frac{271151872}{5027772723} a^{10} + \frac{9122772828}{1675924241} a^{9} - \frac{60742872703}{5027772723} a^{8} + \frac{79321004752}{5027772723} a^{7} + \frac{9142818742}{1675924241} a^{6} - \frac{2651077154}{88206539} a^{5} + \frac{4597477514}{88206539} a^{4} - \frac{60922380384}{1675924241} a^{3} + \frac{85112969255}{5027772723} a^{2} + \frac{10136908840}{5027772723} a - \frac{5463596}{88206539} \) (order $6$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 2590.08235462 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times D_4$ (as 16T9):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $D_4\times C_2$
Character table for $D_4\times C_2$

Intermediate fields

\(\Q(\sqrt{285}) \), \(\Q(\sqrt{-95}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{57}) \), \(\Q(\sqrt{-19}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-3}, \sqrt{-95})\), \(\Q(\sqrt{5}, \sqrt{57})\), \(\Q(\sqrt{-15}, \sqrt{-19})\), \(\Q(\sqrt{5}, \sqrt{-19})\), \(\Q(\sqrt{-15}, \sqrt{57})\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\sqrt{-3}, \sqrt{-19})\), 4.2.475.1 x2, 4.2.4275.1 x2, 4.0.1805.1 x2, 4.0.16245.1 x2, 8.0.6597500625.1, 8.4.6597500625.1 x2, 8.0.6597500625.2 x2, 8.0.81450625.1, 8.0.6597500625.3, 8.0.18275625.1 x2, 8.0.263900025.1 x2

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

Sibling fields

Degree 8 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ R R ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$19$19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$