Properties

Label 16.0.6884340550074368.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 17^{7}$
Root discriminant $9.77$
Ramified primes $2, 17$
Class number $1$
Class group Trivial
Galois Group $C_4.D_4:C_4$ (as 16T289)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -2, -6, 20, -14, -30, 112, -202, 259, -266, 232, -174, 110, -56, 22, -6, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 22*x^14 - 56*x^13 + 110*x^12 - 174*x^11 + 232*x^10 - 266*x^9 + 259*x^8 - 202*x^7 + 112*x^6 - 30*x^5 - 14*x^4 + 20*x^3 - 6*x^2 - 2*x + 1)
gp: K = bnfinit(x^16 - 6*x^15 + 22*x^14 - 56*x^13 + 110*x^12 - 174*x^11 + 232*x^10 - 266*x^9 + 259*x^8 - 202*x^7 + 112*x^6 - 30*x^5 - 14*x^4 + 20*x^3 - 6*x^2 - 2*x + 1, 1)

Normalized defining polynomial

\(x^{16} \) \(\mathstrut -\mathstrut 6 x^{15} \) \(\mathstrut +\mathstrut 22 x^{14} \) \(\mathstrut -\mathstrut 56 x^{13} \) \(\mathstrut +\mathstrut 110 x^{12} \) \(\mathstrut -\mathstrut 174 x^{11} \) \(\mathstrut +\mathstrut 232 x^{10} \) \(\mathstrut -\mathstrut 266 x^{9} \) \(\mathstrut +\mathstrut 259 x^{8} \) \(\mathstrut -\mathstrut 202 x^{7} \) \(\mathstrut +\mathstrut 112 x^{6} \) \(\mathstrut -\mathstrut 30 x^{5} \) \(\mathstrut -\mathstrut 14 x^{4} \) \(\mathstrut +\mathstrut 20 x^{3} \) \(\mathstrut -\mathstrut 6 x^{2} \) \(\mathstrut -\mathstrut 2 x \) \(\mathstrut +\mathstrut 1 \)

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:  \(6884340550074368=2^{24}\cdot 17^{7}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $9.77$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{2}$, $\frac{1}{298} a^{15} + \frac{31}{298} a^{14} - \frac{23}{298} a^{13} - \frac{13}{298} a^{12} - \frac{73}{298} a^{11} - \frac{22}{149} a^{10} - \frac{55}{298} a^{9} - \frac{33}{149} a^{8} - \frac{97}{298} a^{7} - \frac{33}{149} a^{6} - \frac{95}{298} a^{5} - \frac{59}{149} a^{4} + \frac{45}{149} a^{3} - \frac{77}{298} a^{2} - \frac{12}{149} a - \frac{145}{298}$

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:  \( -\frac{413}{2} a^{15} + 1123 a^{14} - 3912 a^{13} + \frac{18731}{2} a^{12} - \frac{34901}{2} a^{11} + \frac{52239}{2} a^{10} - \frac{66437}{2} a^{9} + 36242 a^{8} - \frac{66179}{2} a^{7} + \frac{46169}{2} a^{6} - \frac{20241}{2} a^{5} + 478 a^{4} + 3178 a^{3} - \frac{4703}{2} a^{2} - \frac{163}{2} a + \frac{737}{2} \) (order $4$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{51471}{298} a^{15} - \frac{280011}{298} a^{14} + \frac{487812}{149} a^{13} - \frac{1168142}{149} a^{12} + \frac{4354177}{298} a^{11} - \frac{3259486}{149} a^{10} + \frac{8292839}{298} a^{9} - \frac{9050295}{298} a^{8} + \frac{8266525}{298} a^{7} - \frac{2885924}{149} a^{6} + \frac{2536117}{298} a^{5} - \frac{126541}{298} a^{4} - \frac{394860}{149} a^{3} + \frac{586001}{298} a^{2} + \frac{20021}{298} a - \frac{91967}{298} \),  \( \frac{35019}{149} a^{15} - \frac{190447}{149} a^{14} + \frac{1326959}{298} a^{13} - \frac{1588541}{149} a^{12} + \frac{5920527}{298} a^{11} - \frac{8863021}{298} a^{10} + \frac{5636897}{149} a^{9} - \frac{6151133}{149} a^{8} + \frac{11235165}{298} a^{7} - \frac{7842547}{298} a^{6} + \frac{1721911}{149} a^{5} - \frac{84806}{149} a^{4} - \frac{1074911}{298} a^{3} + \frac{796981}{298} a^{2} + \frac{27373}{298} a - \frac{62415}{149} \),  \( \frac{35235}{298} a^{15} - \frac{191797}{298} a^{14} + \frac{334210}{149} a^{13} - \frac{1601183}{298} a^{12} + \frac{2984949}{298} a^{11} - \frac{2235221}{149} a^{10} + \frac{5688491}{298} a^{9} - \frac{3105119}{149} a^{8} + \frac{5675079}{298} a^{7} - \frac{1983000}{149} a^{6} + \frac{1746687}{298} a^{5} - \frac{45909}{149} a^{4} - \frac{269773}{149} a^{3} + \frac{402197}{298} a^{2} + \frac{12749}{298} a - \frac{63041}{298} \),  \( \frac{24039}{298} a^{15} - \frac{65381}{149} a^{14} + \frac{455535}{298} a^{13} - \frac{545367}{149} a^{12} + \frac{1016292}{149} a^{11} - \frac{1521496}{149} a^{10} + \frac{1935476}{149} a^{9} - \frac{2112384}{149} a^{8} + \frac{1929509}{149} a^{7} - \frac{1347418}{149} a^{6} + \frac{592433}{149} a^{5} - \frac{30366}{149} a^{4} - \frac{367255}{298} a^{3} + \frac{136496}{149} a^{2} + \frac{9379}{298} a - \frac{21356}{149} \),  \( \frac{27396}{149} a^{15} - \frac{149024}{149} a^{14} + \frac{1038407}{298} a^{13} - \frac{2486439}{298} a^{12} + \frac{4633691}{298} a^{11} - \frac{3468436}{149} a^{10} + \frac{8823745}{298} a^{9} - \frac{4814360}{149} a^{8} + \frac{8793539}{298} a^{7} - \frac{3068974}{149} a^{6} + \frac{2694591}{298} a^{5} - \frac{65733}{149} a^{4} - \frac{842321}{298} a^{3} + \frac{312354}{149} a^{2} + \frac{10463}{149} a - \frac{97755}{298} \),  \( \frac{667}{298} a^{15} - \frac{3759}{298} a^{14} + \frac{13267}{298} a^{13} - \frac{16181}{149} a^{12} + \frac{30561}{149} a^{11} - \frac{92971}{298} a^{10} + \frac{119765}{298} a^{9} - \frac{66413}{149} a^{8} + \frac{61893}{149} a^{7} - \frac{89765}{298} a^{6} + \frac{43021}{298} a^{5} - \frac{3146}{149} a^{4} - \frac{9553}{298} a^{3} + \frac{4344}{149} a^{2} - \frac{256}{149} a - \frac{603}{149} \),  \( \frac{63209}{298} a^{15} - \frac{343763}{298} a^{14} + \frac{1197499}{298} a^{13} - \frac{2866891}{298} a^{12} + \frac{2670887}{149} a^{11} - \frac{7995451}{298} a^{10} + \frac{5084239}{149} a^{9} - \frac{5547018}{149} a^{8} + \frac{5064474}{149} a^{7} - \frac{7066417}{298} a^{6} + \frac{1549001}{149} a^{5} - \frac{73318}{149} a^{4} - \frac{972533}{298} a^{3} + \frac{359682}{149} a^{2} + \frac{25283}{298} a - \frac{112959}{298} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 23.5886655901 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$C_4.D_4:C_4$ (as 16T289):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 128
The 44 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 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 $16$ ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ $16$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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$17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.8.7.2$x^{8} - 153$$8$$1$$7$$C_8$$[\ ]_{8}$