Properties

Label 16.0.7771775074107392.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 17^{9}$
Root discriminant $9.84$
Ramified primes $2, 17$
Class number $1$
Class group Trivial
Galois Group $C_4.D_4:C_4$ (as 16T260)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 2, -2, -4, 6, 2, -10, 6, 1, 2, -3, -10, 26, -28, 17, -6, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 17*x^14 - 28*x^13 + 26*x^12 - 10*x^11 - 3*x^10 + 2*x^9 + x^8 + 6*x^7 - 10*x^6 + 2*x^5 + 6*x^4 - 4*x^3 - 2*x^2 + 2*x + 1)
gp: K = bnfinit(x^16 - 6*x^15 + 17*x^14 - 28*x^13 + 26*x^12 - 10*x^11 - 3*x^10 + 2*x^9 + x^8 + 6*x^7 - 10*x^6 + 2*x^5 + 6*x^4 - 4*x^3 - 2*x^2 + 2*x + 1, 1)

Normalized defining polynomial

\(x^{16} \) \(\mathstrut -\mathstrut 6 x^{15} \) \(\mathstrut +\mathstrut 17 x^{14} \) \(\mathstrut -\mathstrut 28 x^{13} \) \(\mathstrut +\mathstrut 26 x^{12} \) \(\mathstrut -\mathstrut 10 x^{11} \) \(\mathstrut -\mathstrut 3 x^{10} \) \(\mathstrut +\mathstrut 2 x^{9} \) \(\mathstrut +\mathstrut x^{8} \) \(\mathstrut +\mathstrut 6 x^{7} \) \(\mathstrut -\mathstrut 10 x^{6} \) \(\mathstrut +\mathstrut 2 x^{5} \) \(\mathstrut +\mathstrut 6 x^{4} \) \(\mathstrut -\mathstrut 4 x^{3} \) \(\mathstrut -\mathstrut 2 x^{2} \) \(\mathstrut +\mathstrut 2 x \) \(\mathstrut +\mathstrut 1 \)

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:  \(7771775074107392=2^{16}\cdot 17^{9}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $9.84$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $2, 17$
magma: PrimeDivisors(Discriminant(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{67} a^{15} - \frac{29}{67} a^{14} + \frac{14}{67} a^{13} - \frac{15}{67} a^{12} - \frac{31}{67} a^{11} + \frac{33}{67} a^{10} - \frac{25}{67} a^{9} - \frac{26}{67} a^{8} - \frac{4}{67} a^{7} + \frac{31}{67} a^{6} + \frac{14}{67} a^{5} + \frac{15}{67} a^{4} - \frac{4}{67} a^{3} + \frac{21}{67} a^{2} - \frac{16}{67} a - \frac{32}{67}$

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

Class group and class number

Trivial Abelian group, order $1$

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:  \( -\frac{44}{67} a^{15} + \frac{137}{67} a^{14} - \frac{80}{67} a^{13} - \frac{546}{67} a^{12} + \frac{1699}{67} a^{11} - \frac{2390}{67} a^{10} + \frac{1971}{67} a^{9} - \frac{1067}{67} a^{8} + \frac{779}{67} a^{7} - \frac{694}{67} a^{6} + \frac{54}{67} a^{5} + \frac{546}{67} a^{4} - \frac{427}{67} a^{3} - \frac{53}{67} a^{2} + \frac{168}{67} a + \frac{68}{67} \) (order $4$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{69}{67} a^{15} - \frac{393}{67} a^{14} + \frac{1100}{67} a^{13} - \frac{1839}{67} a^{12} + \frac{1881}{67} a^{11} - \frac{1140}{67} a^{10} + \frac{419}{67} a^{9} - \frac{253}{67} a^{8} + \frac{126}{67} a^{7} + \frac{330}{67} a^{6} - \frac{508}{67} a^{5} + \frac{97}{67} a^{4} + \frac{260}{67} a^{3} - \frac{226}{67} a^{2} - \frac{32}{67} a + \frac{3}{67} \),  \( \frac{115}{67} a^{15} - \frac{655}{67} a^{14} + \frac{1878}{67} a^{13} - \frac{3266}{67} a^{12} + \frac{3604}{67} a^{11} - \frac{2570}{67} a^{10} + \frac{1346}{67} a^{9} - \frac{980}{67} a^{8} + \frac{679}{67} a^{7} + \frac{215}{67} a^{6} - \frac{802}{67} a^{5} + \frac{318}{67} a^{4} + \frac{277}{67} a^{3} - \frac{265}{67} a^{2} - \frac{98}{67} a + \frac{72}{67} \),  \( \frac{9}{67} a^{15} + \frac{7}{67} a^{14} - \frac{142}{67} a^{13} + \frac{468}{67} a^{12} - \frac{748}{67} a^{11} + \frac{565}{67} a^{10} - \frac{24}{67} a^{9} - \frac{301}{67} a^{8} + \frac{98}{67} a^{7} + \frac{11}{67} a^{6} + \frac{193}{67} a^{5} - \frac{200}{67} a^{4} - \frac{103}{67} a^{3} + \frac{256}{67} a^{2} - \frac{77}{67} a - \frac{87}{67} \),  \( \frac{111}{67} a^{15} - \frac{606}{67} a^{14} + \frac{1621}{67} a^{13} - \frac{2536}{67} a^{12} + \frac{2254}{67} a^{11} - \frac{893}{67} a^{10} - \frac{162}{67} a^{9} + \frac{129}{67} a^{8} - \frac{42}{67} a^{7} + \frac{560}{67} a^{6} - \frac{657}{67} a^{5} + \frac{57}{67} a^{4} + \frac{494}{67} a^{3} - \frac{349}{67} a^{2} - \frac{101}{67} a + \frac{66}{67} \),  \( \frac{84}{67} a^{15} - \frac{426}{67} a^{14} + \frac{1042}{67} a^{13} - \frac{1394}{67} a^{12} + \frac{813}{67} a^{11} + \frac{293}{67} a^{10} - \frac{827}{67} a^{9} + \frac{563}{67} a^{8} - \frac{470}{67} a^{7} + \frac{728}{67} a^{6} - \frac{499}{67} a^{5} - \frac{147}{67} a^{4} + \frac{401}{67} a^{3} - \frac{179}{67} a^{2} - \frac{138}{67} a + \frac{59}{67} \),  \( \frac{108}{67} a^{15} - \frac{720}{67} a^{14} + \frac{2316}{67} a^{13} - \frac{4568}{67} a^{12} + \frac{5831}{67} a^{11} - \frac{4878}{67} a^{10} + \frac{2727}{67} a^{9} - \frac{1334}{67} a^{8} + \frac{774}{67} a^{7} + \frac{266}{67} a^{6} - \frac{1302}{67} a^{5} + \frac{1084}{67} a^{4} - \frac{30}{67} a^{3} - \frac{479}{67} a^{2} + \frac{148}{67} a + \frac{95}{67} \),  \( \frac{59}{67} a^{15} - \frac{438}{67} a^{14} + \frac{1496}{67} a^{13} - \frac{3096}{67} a^{12} + \frac{4067}{67} a^{11} - \frac{3346}{67} a^{10} + \frac{1607}{67} a^{9} - \frac{529}{67} a^{8} + \frac{367}{67} a^{7} + \frac{221}{67} a^{6} - \frac{1050}{67} a^{5} + \frac{952}{67} a^{4} + \frac{32}{67} a^{3} - \frac{436}{67} a^{2} + \frac{128}{67} a + \frac{122}{67} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 25.2889722607 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$C_4.D_4:C_4$ (as 16T260):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 128
The 32 conjugacy class representatives for $C_4.D_4:C_4$
Character table for $C_4.D_4:C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), 4.0.272.1, 8.0.1257728.1

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

Sibling fields

Degree 16 sibling: 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 $16$ ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
2Data not computed
$17$17.4.2.2$x^{4} - 17 x^{2} + 867$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.8.7.4$x^{8} - 12393$$8$$1$$7$$C_8$$[\ ]_{8}$