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Properties

Label 16.8.23630709513...0000.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{8}\cdot 5^{8}\cdot 17^{4}\cdot 19^{2}\cdot 97^{8}$
Root discriminant $91.38$
Ramified primes $2, 5, 17, 19, 97$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1123

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![92416, 0, -105792, 0, -95104, 0, 30716, 0, 25077, 0, 2492, 0, -291, 0, -16, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 16*x^14 - 291*x^12 + 2492*x^10 + 25077*x^8 + 30716*x^6 - 95104*x^4 - 105792*x^2 + 92416)
 
gp: K = bnfinit(x^16 - 16*x^14 - 291*x^12 + 2492*x^10 + 25077*x^8 + 30716*x^6 - 95104*x^4 - 105792*x^2 + 92416, 1)
 

Normalized defining polynomial

\( x^{16} - 16 x^{14} - 291 x^{12} + 2492 x^{10} + 25077 x^{8} + 30716 x^{6} - 95104 x^{4} - 105792 x^{2} + 92416 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(23630709513618089564484100000000=2^{8}\cdot 5^{8}\cdot 17^{4}\cdot 19^{2}\cdot 97^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $91.38$
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:  $2, 5, 17, 19, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{6} a^{8} + \frac{1}{6} a^{6} + \frac{1}{6} a^{4} - \frac{1}{3}$, $\frac{1}{6} a^{9} + \frac{1}{6} a^{7} + \frac{1}{6} a^{5} - \frac{1}{3} a$, $\frac{1}{12} a^{10} - \frac{1}{4} a^{6} - \frac{1}{3} a^{4} + \frac{1}{12} a^{2} - \frac{1}{3}$, $\frac{1}{48} a^{11} - \frac{1}{12} a^{9} + \frac{5}{48} a^{7} - \frac{1}{6} a^{5} - \frac{1}{2} a^{4} - \frac{11}{48} a^{3} - \frac{1}{2} a^{2} + \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{96} a^{12} - \frac{1}{24} a^{10} - \frac{1}{12} a^{9} - \frac{1}{32} a^{8} + \frac{1}{6} a^{7} + \frac{1}{12} a^{6} - \frac{1}{12} a^{5} + \frac{5}{96} a^{4} - \frac{1}{12} a^{2} + \frac{5}{12} a - \frac{1}{3}$, $\frac{1}{192} a^{13} - \frac{1}{24} a^{10} - \frac{1}{64} a^{9} + \frac{11}{48} a^{7} + \frac{1}{8} a^{6} - \frac{11}{192} a^{5} + \frac{1}{6} a^{4} + \frac{11}{48} a^{3} + \frac{11}{24} a^{2} - \frac{1}{2} a + \frac{1}{6}$, $\frac{1}{2289374465846016} a^{14} - \frac{615944856531}{190781205487168} a^{12} + \frac{31174213880751}{763124821948672} a^{10} - \frac{1}{12} a^{9} - \frac{1618295625793}{286171808230752} a^{8} + \frac{1}{6} a^{7} - \frac{173230924221017}{763124821948672} a^{6} - \frac{1}{12} a^{5} - \frac{102318480067501}{286171808230752} a^{4} - \frac{2341398181453}{5961912671474} a^{2} + \frac{5}{12} a - \frac{386841574675}{1882709264676}$, $\frac{1}{9157497863384064} a^{15} - \frac{1}{4578748931692032} a^{14} + \frac{4114078101881}{2289374465846016} a^{13} - \frac{4114078101881}{1144687232923008} a^{12} - \frac{1867961101331}{9157497863384064} a^{11} - \frac{188913244385837}{4578748931692032} a^{10} - \frac{2640291158251}{286171808230752} a^{9} + \frac{2640291158251}{143085904115376} a^{8} - \frac{328911567175883}{9157497863384064} a^{7} + \frac{901255183637387}{4578748931692032} a^{6} + \frac{12422381865121}{71542952057688} a^{5} - \frac{6460469193647}{35771476028844} a^{4} - \frac{52800821453299}{143085904115376} a^{3} + \frac{2341398181453}{11923825342948} a^{2} - \frac{1014411329567}{7530837058704} a + \frac{547327028153}{1255139509784}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $11$
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:  \( 23574232752.7 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1123:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 55 conjugacy class representatives for t16n1123 are not computed
Character table for t16n1123 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{97}) \), \(\Q(\sqrt{485}) \), \(\Q(\sqrt{5}, \sqrt{97})\), 8.8.15990601380625.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ R R ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.4.4.4$x^{4} - 5$$2$$2$$4$$D_{4}$$[2, 2]^{2}$
2.4.4.3$x^{4} + 2 x^{2} + 4 x + 4$$2$$2$$4$$D_{4}$$[2, 2]^{2}$
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$17$17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$19$19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.4.2.2$x^{4} - 19 x^{2} + 722$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
97Data not computed