Properties

Label 16.0.4919707275390625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 67^{4}$
Root discriminant $9.57$
Ramified primes $5, 67$
Class number $1$
Class group Trivial
Galois Group $C_4\times S_4$ (as 16T181)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -4, 15, -36, 73, -118, 159, -182, 176, -149, 112, -77, 50, -29, 14, -5, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 14*x^14 - 29*x^13 + 50*x^12 - 77*x^11 + 112*x^10 - 149*x^9 + 176*x^8 - 182*x^7 + 159*x^6 - 118*x^5 + 73*x^4 - 36*x^3 + 15*x^2 - 4*x + 1)
gp: K = bnfinit(x^16 - 5*x^15 + 14*x^14 - 29*x^13 + 50*x^12 - 77*x^11 + 112*x^10 - 149*x^9 + 176*x^8 - 182*x^7 + 159*x^6 - 118*x^5 + 73*x^4 - 36*x^3 + 15*x^2 - 4*x + 1, 1)

Normalized defining polynomial

\(x^{16} \) \(\mathstrut -\mathstrut 5 x^{15} \) \(\mathstrut +\mathstrut 14 x^{14} \) \(\mathstrut -\mathstrut 29 x^{13} \) \(\mathstrut +\mathstrut 50 x^{12} \) \(\mathstrut -\mathstrut 77 x^{11} \) \(\mathstrut +\mathstrut 112 x^{10} \) \(\mathstrut -\mathstrut 149 x^{9} \) \(\mathstrut +\mathstrut 176 x^{8} \) \(\mathstrut -\mathstrut 182 x^{7} \) \(\mathstrut +\mathstrut 159 x^{6} \) \(\mathstrut -\mathstrut 118 x^{5} \) \(\mathstrut +\mathstrut 73 x^{4} \) \(\mathstrut -\mathstrut 36 x^{3} \) \(\mathstrut +\mathstrut 15 x^{2} \) \(\mathstrut -\mathstrut 4 x \) \(\mathstrut +\mathstrut 1 \)

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:  \(4919707275390625=5^{12}\cdot 67^{4}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $9.57$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $5, 67$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{19} a^{15} - \frac{1}{19} a^{14} - \frac{9}{19} a^{13} - \frac{8}{19} a^{12} - \frac{1}{19} a^{11} - \frac{5}{19} a^{10} - \frac{3}{19} a^{9} - \frac{9}{19} a^{8} + \frac{7}{19} a^{7} - \frac{2}{19} a^{6} - \frac{1}{19} a^{5} - \frac{8}{19} a^{4} + \frac{3}{19} a^{3} - \frac{5}{19} a^{2} - \frac{5}{19} a - \frac{5}{19}$

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:  \( \frac{217}{19} a^{15} - \frac{1015}{19} a^{14} + \frac{2626}{19} a^{13} - \frac{5042}{19} a^{12} + \frac{8162}{19} a^{11} - \frac{12010}{19} a^{10} + \frac{17057}{19} a^{9} - \frac{21846}{19} a^{8} + \frac{24053}{19} a^{7} - \frac{22588}{19} a^{6} + \frac{17035}{19} a^{5} - \frac{10419}{19} a^{4} + \frac{5040}{19} a^{3} - \frac{1541}{19} a^{2} + \frac{435}{19} a + \frac{36}{19} \) (order $10$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{195}{19} a^{15} - \frac{974}{19} a^{14} + \frac{2653}{19} a^{13} - \frac{5303}{19} a^{12} + \frac{8830}{19} a^{11} - \frac{13211}{19} a^{10} + \frac{18871}{19} a^{9} - \frac{24650}{19} a^{8} + \frac{28060}{19} a^{7} - \frac{27427}{19} a^{6} + \frac{21959}{19} a^{5} - \frac{14290}{19} a^{4} + \frac{7463}{19} a^{3} - \frac{2742}{19} a^{2} + \frac{792}{19} a - \frac{82}{19} \),  \( \frac{183}{19} a^{15} - \frac{1000}{19} a^{14} + \frac{2894}{19} a^{13} - \frac{6024}{19} a^{12} + \frac{10305}{19} a^{11} - \frac{15697}{19} a^{10} + \frac{22612}{19} a^{9} - \frac{30128}{19} a^{8} + \frac{35329}{19} a^{7} - \frac{35725}{19} a^{6} + \frac{30160}{19} a^{5} - \frac{20901}{19} a^{4} + \frac{11816}{19} a^{3} - \frac{5038}{19} a^{2} + \frac{1593}{19} a - \frac{326}{19} \),  \( a \),  \( \frac{22}{19} a^{15} - \frac{41}{19} a^{14} - \frac{27}{19} a^{13} + \frac{261}{19} a^{12} - \frac{668}{19} a^{11} + \frac{1201}{19} a^{10} - \frac{1814}{19} a^{9} + \frac{2804}{19} a^{8} - \frac{4007}{19} a^{7} + \frac{4839}{19} a^{6} - \frac{4924}{19} a^{5} + \frac{3871}{19} a^{4} - \frac{2423}{19} a^{3} + \frac{1201}{19} a^{2} - \frac{357}{19} a + \frac{118}{19} \),  \( \frac{99}{19} a^{15} - \frac{422}{19} a^{14} + \frac{1009}{19} a^{13} - \frac{1837}{19} a^{12} + \frac{2884}{19} a^{11} - \frac{4181}{19} a^{10} + \frac{5916}{19} a^{9} - \frac{7332}{19} a^{8} + \frac{7666}{19} a^{7} - \frac{6886}{19} a^{6} + \frac{4803}{19} a^{5} - \frac{2749}{19} a^{4} + \frac{1209}{19} a^{3} - \frac{191}{19} a^{2} + \frac{56}{19} a + \frac{75}{19} \),  \( \frac{105}{19} a^{15} - \frac{542}{19} a^{14} + \frac{1506}{19} a^{13} - \frac{3025}{19} a^{12} + \frac{5025}{19} a^{11} - \frac{7498}{19} a^{10} + \frac{10686}{19} a^{9} - \frac{13998}{19} a^{8} + \frac{15954}{19} a^{7} - \frac{15467}{19} a^{6} + \frac{12245}{19} a^{5} - \frac{7813}{19} a^{4} + \frac{3906}{19} a^{3} - \frac{1380}{19} a^{2} + \frac{349}{19} a - \frac{31}{19} \),  \( \frac{46}{19} a^{15} - \frac{65}{19} a^{14} - \frac{167}{19} a^{13} + \frac{829}{19} a^{12} - \frac{1927}{19} a^{11} + \frac{3380}{19} a^{10} - \frac{5116}{19} a^{9} + \frac{7775}{19} a^{8} - \frac{10869}{19} a^{7} + \frac{12847}{19} a^{6} - \frac{12947}{19} a^{5} + \frac{10367}{19} a^{4} - \frac{6645}{19} a^{3} + \frac{3437}{19} a^{2} - \frac{1123}{19} a + \frac{359}{19} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 48.4508350282 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$C_4\times S_4$ (as 16T181):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 96
The 20 conjugacy class representatives for $C_4\times S_4$
Character table for $C_4\times S_4$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 4.2.8375.1, 8.4.70140625.1

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

Sibling fields

Degree 12 siblings: data not computed
Degree 16 sibling: data not computed
Degree 24 siblings: data not computed
Degree 32 sibling: 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 ${\href{/LocalNumberField/2.12.0.1}{12} }{,}\,{\href{/LocalNumberField/2.4.0.1}{4} }$ ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
5Data not computed
$67$67.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
67.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
67.8.4.1$x^{8} + 17956 x^{4} - 300763 x^{2} + 80604484$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$