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Properties

Label 16.0.48802261431...0625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{8}\cdot 13^{4}\cdot 29^{6}\cdot 271181^{2}$
Root discriminant $71.70$
Ramified primes $5, 13, 29, 271181$
Class number $9960$ (GRH)
Class group $[2, 4980]$ (GRH)
Galois group 16T1177

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![24393721, 0, 5888892, 0, 3286212, 0, 844707, 0, 138918, 0, 13986, 0, 948, 0, 39, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 39*x^14 + 948*x^12 + 13986*x^10 + 138918*x^8 + 844707*x^6 + 3286212*x^4 + 5888892*x^2 + 24393721)
 
gp: K = bnfinit(x^16 + 39*x^14 + 948*x^12 + 13986*x^10 + 138918*x^8 + 844707*x^6 + 3286212*x^4 + 5888892*x^2 + 24393721, 1)
 

Normalized defining polynomial

\( x^{16} + 39 x^{14} + 948 x^{12} + 13986 x^{10} + 138918 x^{8} + 844707 x^{6} + 3286212 x^{4} + 5888892 x^{2} + 24393721 \)

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:  \(488022614316879794822906640625=5^{8}\cdot 13^{4}\cdot 29^{6}\cdot 271181^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $71.70$
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:  $5, 13, 29, 271181$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{8} a^{7} + \frac{1}{8} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{4} + \frac{1}{8} a^{3} + \frac{3}{8} a^{2} + \frac{3}{8} a - \frac{3}{8}$, $\frac{1}{16} a^{12} + \frac{1}{16} a^{10} - \frac{3}{16} a^{8} + \frac{3}{16} a^{6} - \frac{1}{16} a^{4} - \frac{1}{2} a^{3} + \frac{3}{8} a^{2} + \frac{1}{16}$, $\frac{1}{352} a^{13} - \frac{1}{32} a^{12} - \frac{19}{352} a^{11} + \frac{3}{32} a^{10} - \frac{51}{352} a^{9} + \frac{3}{32} a^{8} + \frac{15}{352} a^{7} + \frac{1}{32} a^{6} + \frac{31}{352} a^{5} + \frac{1}{32} a^{4} - \frac{71}{176} a^{3} + \frac{7}{16} a^{2} - \frac{107}{352} a - \frac{5}{32}$, $\frac{1}{233479141875068178496} a^{14} + \frac{523268836324223351}{29184892734383522312} a^{12} + \frac{1624677790211733423}{58369785468767044624} a^{10} - \frac{1263096843980581641}{116739570937534089248} a^{8} + \frac{13903400020979850265}{58369785468767044624} a^{6} + \frac{43353526116445208951}{233479141875068178496} a^{4} - \frac{23220873007166494437}{233479141875068178496} a^{2} - \frac{1}{2} a + \frac{5372534329019188533}{21225376534097107136}$, $\frac{1}{209664269403811224289408} a^{15} - \frac{1}{466958283750136356992} a^{14} + \frac{94888022585001578}{1638002104717275189761} a^{13} - \frac{523268836324223351}{58369785468767044624} a^{12} - \frac{2619046031154090463275}{52416067350952806072352} a^{11} + \frac{12967768576980027733}{116739570937534089248} a^{10} + \frac{321097309267619232987}{104832134701905612144704} a^{9} - \frac{57106688624786462983}{233479141875068178496} a^{8} - \frac{11236209456067177466413}{52416067350952806072352} a^{7} + \frac{689046346211910891}{116739570937534089248} a^{6} + \frac{27607158127708300963439}{209664269403811224289408} a^{5} + \frac{190125615758622969545}{466958283750136356992} a^{4} + \frac{76886930998293701212859}{209664269403811224289408} a^{3} + \frac{198330229413467628309}{466958283750136356992} a^{2} - \frac{5870741691594168232257}{209664269403811224289408} a - \frac{10678878462543465317}{42450753068194214272}$

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

Class group and class number

$C_{2}\times C_{4980}$, which has order $9960$ (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:  \( 10968.6213178 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1177:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 76 conjugacy class representatives for t16n1177 are not computed
Character table for t16n1177 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.725.1, 8.8.2576088125.1, 8.0.142539513125.1, 8.0.698586153825625.2

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$13$13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$29$29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.3.3$x^{4} + 58$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.3.3$x^{4} + 58$$4$$1$$3$$C_4$$[\ ]_{4}$
271181Data not computed