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Properties

Label 16.8.23630709513...0000.2
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![-1424, 4200, 41772, 20384, -78458, -99694, 39775, 62047, -32065, -9371, 9399, -1302, -581, 217, -9, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 - 9*x^14 + 217*x^13 - 581*x^12 - 1302*x^11 + 9399*x^10 - 9371*x^9 - 32065*x^8 + 62047*x^7 + 39775*x^6 - 99694*x^5 - 78458*x^4 + 20384*x^3 + 41772*x^2 + 4200*x - 1424)
 
gp: K = bnfinit(x^16 - 7*x^15 - 9*x^14 + 217*x^13 - 581*x^12 - 1302*x^11 + 9399*x^10 - 9371*x^9 - 32065*x^8 + 62047*x^7 + 39775*x^6 - 99694*x^5 - 78458*x^4 + 20384*x^3 + 41772*x^2 + 4200*x - 1424, 1)
 

Normalized defining polynomial

\( x^{16} - 7 x^{15} - 9 x^{14} + 217 x^{13} - 581 x^{12} - 1302 x^{11} + 9399 x^{10} - 9371 x^{9} - 32065 x^{8} + 62047 x^{7} + 39775 x^{6} - 99694 x^{5} - 78458 x^{4} + 20384 x^{3} + 41772 x^{2} + 4200 x - 1424 \)

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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{6} a^{12} - \frac{1}{6} a^{10} + \frac{1}{6} a^{8} + \frac{1}{6} a^{7} - \frac{1}{3} a^{6} + \frac{1}{6} a^{5} - \frac{1}{3} a^{4} - \frac{1}{2} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{204} a^{13} + \frac{1}{102} a^{12} - \frac{1}{204} a^{11} + \frac{7}{51} a^{10} - \frac{95}{204} a^{9} - \frac{11}{68} a^{8} - \frac{6}{17} a^{7} + \frac{1}{68} a^{6} + \frac{1}{34} a^{5} + \frac{83}{204} a^{4} - \frac{35}{102} a^{3} + \frac{11}{102} a^{2} + \frac{19}{51} a + \frac{5}{51}$, $\frac{1}{65892} a^{14} + \frac{2}{867} a^{13} + \frac{571}{65892} a^{12} - \frac{3269}{16473} a^{11} - \frac{30847}{65892} a^{10} - \frac{7583}{21964} a^{9} + \frac{11497}{32946} a^{8} - \frac{9403}{65892} a^{7} - \frac{7541}{32946} a^{6} + \frac{30631}{65892} a^{5} + \frac{13211}{32946} a^{4} - \frac{6565}{32946} a^{3} + \frac{2827}{32946} a^{2} + \frac{6476}{16473} a + \frac{2042}{16473}$, $\frac{1}{62349109519886851424397090648} a^{15} + \frac{273390975336918226831019}{62349109519886851424397090648} a^{14} - \frac{112770957798567248891183497}{62349109519886851424397090648} a^{13} + \frac{3863332868916589375225397183}{62349109519886851424397090648} a^{12} - \frac{200871945632614666846238389}{3667594677640403024964534744} a^{11} + \frac{5191511007251644012977135479}{15587277379971712856099272662} a^{10} - \frac{4635764068799389489434452841}{20783036506628950474799030216} a^{9} + \frac{17233184899037236309334184841}{62349109519886851424397090648} a^{8} - \frac{16419031973729410767649558631}{62349109519886851424397090648} a^{7} - \frac{7882048170138614450616277767}{20783036506628950474799030216} a^{6} - \frac{6073240639319421713540196393}{20783036506628950474799030216} a^{5} - \frac{5532442817963989526728271609}{31174554759943425712198545324} a^{4} - \frac{827074380545014377512288587}{31174554759943425712198545324} a^{3} + \frac{18134076182344831709421187}{53935215847652985661243158} a^{2} + \frac{526120649942153577565903551}{5195759126657237618699757554} a + \frac{3728810973218284918495747670}{7793638689985856428049636331}$

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:  \( 16508295896.4 \) (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

\(\Q(\sqrt{97}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{485}) \), \(\Q(\sqrt{5}, \sqrt{97})\), 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 ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ 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}$
2.4.4.4$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}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$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}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
19.4.2.2$x^{4} - 19 x^{2} + 722$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
97Data not computed