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Properties

Label 16.0.13887098173...7872.3
Degree $16$
Signature $[0, 8]$
Discriminant $2^{67}\cdot 7^{14}\cdot 193^{4}$
Root discriminant $372.75$
Ramified primes $2, 7, 193$
Class number $190299648$ (GRH)
Class group $[2, 2, 2, 2, 2, 4, 1486716]$ (GRH)
Galois group 16T1230

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![135973824098, 0, 197267724080, 0, 45900154748, 0, 4457348896, 0, 217602126, 0, 5577768, 0, 73388, 0, 448, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 448*x^14 + 73388*x^12 + 5577768*x^10 + 217602126*x^8 + 4457348896*x^6 + 45900154748*x^4 + 197267724080*x^2 + 135973824098)
 
gp: K = bnfinit(x^16 + 448*x^14 + 73388*x^12 + 5577768*x^10 + 217602126*x^8 + 4457348896*x^6 + 45900154748*x^4 + 197267724080*x^2 + 135973824098, 1)
 

Normalized defining polynomial

\( x^{16} + 448 x^{14} + 73388 x^{12} + 5577768 x^{10} + 217602126 x^{8} + 4457348896 x^{6} + 45900154748 x^{4} + 197267724080 x^{2} + 135973824098 \)

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:  \(138870981735380017471819830544433646927872=2^{67}\cdot 7^{14}\cdot 193^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $372.75$
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:  $2, 7, 193$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{7} a^{8}$, $\frac{1}{7} a^{9}$, $\frac{1}{4053} a^{10} - \frac{131}{4053} a^{8} + \frac{62}{579} a^{6} - \frac{73}{579} a^{4} - \frac{106}{579} a^{2} - \frac{1}{3}$, $\frac{1}{4053} a^{11} - \frac{131}{4053} a^{9} + \frac{62}{579} a^{7} - \frac{73}{579} a^{5} - \frac{106}{579} a^{3} - \frac{1}{3} a$, $\frac{1}{2346687} a^{12} + \frac{62}{2346687} a^{10} + \frac{4069}{782229} a^{8} - \frac{158333}{335241} a^{6} - \frac{100466}{335241} a^{4} - \frac{599}{1737} a^{2} - \frac{4}{9}$, $\frac{1}{2346687} a^{13} + \frac{62}{2346687} a^{11} + \frac{4069}{782229} a^{9} - \frac{158333}{335241} a^{7} - \frac{100466}{335241} a^{5} - \frac{599}{1737} a^{3} - \frac{4}{9} a$, $\frac{1}{9446784241225496390342583669} a^{14} + \frac{11346198768317713868}{449846868629785542397265889} a^{12} - \frac{158525368242884910025036}{1349540605889356627191797667} a^{10} + \frac{21184123825666031192093185}{1349540605889356627191797667} a^{8} + \frac{267027689026367143003152155}{1349540605889356627191797667} a^{6} + \frac{405814096850353914002297}{998919767497673299179717} a^{4} + \frac{695239726481616482797}{1725250030220506561623} a^{2} - \frac{11445919928184310588}{26817357982702174533}$, $\frac{1}{9446784241225496390342583669} a^{15} + \frac{11346198768317713868}{449846868629785542397265889} a^{13} - \frac{158525368242884910025036}{1349540605889356627191797667} a^{11} + \frac{21184123825666031192093185}{1349540605889356627191797667} a^{9} + \frac{267027689026367143003152155}{1349540605889356627191797667} a^{7} + \frac{405814096850353914002297}{998919767497673299179717} a^{5} + \frac{695239726481616482797}{1725250030220506561623} a^{3} - \frac{11445919928184310588}{26817357982702174533} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{4}\times C_{1486716}$, which has order $190299648$ (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:  \( 1200202.11347 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1230:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 58 conjugacy class representatives for t16n1230 are not computed
Character table for t16n1230 is not computed

Intermediate fields

\(\Q(\sqrt{7}) \), \(\Q(\sqrt{14}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{2}, \sqrt{7})\), 8.8.3947645370368.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$7$7.8.7.2$x^{8} - 7$$8$$1$$7$$D_{8}$$[\ ]_{8}^{2}$
7.8.7.2$x^{8} - 7$$8$$1$$7$$D_{8}$$[\ ]_{8}^{2}$
193Data not computed