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Properties

Label 16.0.25376112779...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{28}\cdot 5^{12}\cdot 61^{2}\cdot 101^{4}$
Root discriminant $59.60$
Ramified primes $2, 5, 61, 101$
Class number $8$ (GRH)
Class group $[2, 4]$ (GRH)
Galois group 16T1161

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![198301, 244956, -832, -73952, 85328, 50868, -18944, 24, 5738, -5628, 2144, 1296, -472, 44, 32, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 32*x^14 + 44*x^13 - 472*x^12 + 1296*x^11 + 2144*x^10 - 5628*x^9 + 5738*x^8 + 24*x^7 - 18944*x^6 + 50868*x^5 + 85328*x^4 - 73952*x^3 - 832*x^2 + 244956*x + 198301)
 
gp: K = bnfinit(x^16 - 8*x^15 + 32*x^14 + 44*x^13 - 472*x^12 + 1296*x^11 + 2144*x^10 - 5628*x^9 + 5738*x^8 + 24*x^7 - 18944*x^6 + 50868*x^5 + 85328*x^4 - 73952*x^3 - 832*x^2 + 244956*x + 198301, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 32 x^{14} + 44 x^{13} - 472 x^{12} + 1296 x^{11} + 2144 x^{10} - 5628 x^{9} + 5738 x^{8} + 24 x^{7} - 18944 x^{6} + 50868 x^{5} + 85328 x^{4} - 73952 x^{3} - 832 x^{2} + 244956 x + 198301 \)

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:  \(25376112779001856000000000000=2^{28}\cdot 5^{12}\cdot 61^{2}\cdot 101^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $59.60$
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:  $2, 5, 61, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{4} a^{2} + \frac{1}{4}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} + \frac{1}{4} a$, $\frac{1}{8} a^{8} - \frac{1}{4} a^{4} + \frac{1}{8}$, $\frac{1}{8} a^{9} - \frac{1}{4} a^{5} + \frac{1}{8} a$, $\frac{1}{8} a^{10} - \frac{1}{4} a^{4} - \frac{1}{8} a^{2} + \frac{1}{4}$, $\frac{1}{16} a^{11} - \frac{1}{16} a^{10} - \frac{1}{16} a^{9} - \frac{1}{16} a^{8} - \frac{1}{8} a^{7} - \frac{1}{8} a^{6} + \frac{1}{8} a^{5} - \frac{1}{8} a^{4} + \frac{1}{16} a^{3} + \frac{3}{16} a^{2} - \frac{1}{16} a + \frac{3}{16}$, $\frac{1}{16} a^{12} - \frac{1}{16} a^{8} - \frac{1}{16} a^{4} + \frac{1}{16}$, $\frac{1}{16} a^{13} - \frac{1}{16} a^{9} - \frac{1}{16} a^{5} + \frac{1}{16} a$, $\frac{1}{1952} a^{14} - \frac{25}{976} a^{13} + \frac{25}{1952} a^{12} + \frac{17}{976} a^{11} - \frac{41}{1952} a^{10} - \frac{7}{244} a^{9} + \frac{115}{1952} a^{8} + \frac{13}{488} a^{7} - \frac{241}{1952} a^{6} - \frac{177}{976} a^{5} + \frac{95}{1952} a^{4} + \frac{1}{16} a^{3} - \frac{279}{1952} a^{2} - \frac{217}{488} a - \frac{187}{1952}$, $\frac{1}{1859225136805019589379715288288} a^{15} - \frac{43421244001208900070046295}{232403142100627448672464411036} a^{14} - \frac{28648140891557690334822786933}{1859225136805019589379715288288} a^{13} - \frac{25627688372633831502656211685}{929612568402509794689857644144} a^{12} + \frac{278066973125479439154162653}{30479100603360976875077299808} a^{11} - \frac{3200383627436465349409801769}{232403142100627448672464411036} a^{10} + \frac{77197539769477449503216543571}{1859225136805019589379715288288} a^{9} + \frac{219142305822803277895059953}{929612568402509794689857644144} a^{8} + \frac{25770625671658718604577898275}{1859225136805019589379715288288} a^{7} + \frac{16491531138509577993470235885}{232403142100627448672464411036} a^{6} - \frac{378318322377494162076659134495}{1859225136805019589379715288288} a^{5} + \frac{174668436261147981909001055485}{929612568402509794689857644144} a^{4} - \frac{346456247347591475289029396717}{1859225136805019589379715288288} a^{3} + \frac{35422484545453778569949285455}{232403142100627448672464411036} a^{2} - \frac{451661817123385006897371721151}{1859225136805019589379715288288} a - \frac{411555292106162818301645396425}{929612568402509794689857644144}$

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:  $7$
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:  \( 4532667.33365 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1161:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 43 conjugacy class representatives for t16n1161
Character table for t16n1161 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{20})^+\), 8.0.6464000000.11

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
2Data not computed
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
61Data not computed
$101$$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.1.2$x^{2} + 202$$2$$1$$1$$C_2$$[\ ]_{2}$
101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.1.2$x^{2} + 202$$2$$1$$1$$C_2$$[\ ]_{2}$
101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$