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Properties

Label 16.0.33012379009...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 5^{12}\cdot 379^{4}$
Root discriminant $29.51$
Ramified primes $2, 5, 379$
Class number $12$ (GRH)
Class group $[2, 6]$ (GRH)
Galois group 16T1027

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -36, 522, -1528, 2647, -2816, 3302, -1864, 2115, -596, 574, -108, 67, -2, 2, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 2*x^14 - 2*x^13 + 67*x^12 - 108*x^11 + 574*x^10 - 596*x^9 + 2115*x^8 - 1864*x^7 + 3302*x^6 - 2816*x^5 + 2647*x^4 - 1528*x^3 + 522*x^2 - 36*x + 1)
 
gp: K = bnfinit(x^16 + 2*x^14 - 2*x^13 + 67*x^12 - 108*x^11 + 574*x^10 - 596*x^9 + 2115*x^8 - 1864*x^7 + 3302*x^6 - 2816*x^5 + 2647*x^4 - 1528*x^3 + 522*x^2 - 36*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} + 2 x^{14} - 2 x^{13} + 67 x^{12} - 108 x^{11} + 574 x^{10} - 596 x^{9} + 2115 x^{8} - 1864 x^{7} + 3302 x^{6} - 2816 x^{5} + 2647 x^{4} - 1528 x^{3} + 522 x^{2} - 36 x + 1 \)

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:  \(330123790096000000000000=2^{16}\cdot 5^{12}\cdot 379^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.51$
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:  $2, 5, 379$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{5526818992596850338294511} a^{15} - \frac{991482017346103498067055}{5526818992596850338294511} a^{14} - \frac{192253410679301022218208}{5526818992596850338294511} a^{13} + \frac{1078198032254338031999520}{5526818992596850338294511} a^{12} + \frac{1730504791673212370532366}{5526818992596850338294511} a^{11} - \frac{1400640725639071787582923}{5526818992596850338294511} a^{10} + \frac{1683728916043139588753049}{5526818992596850338294511} a^{9} - \frac{2098846801694920006439476}{5526818992596850338294511} a^{8} + \frac{526103784371437872142754}{5526818992596850338294511} a^{7} + \frac{2758330551582066002883473}{5526818992596850338294511} a^{6} + \frac{1771077867086720833123270}{5526818992596850338294511} a^{5} + \frac{474830132466011180065685}{5526818992596850338294511} a^{4} - \frac{426617524364146274646820}{5526818992596850338294511} a^{3} - \frac{1659702280149787851528910}{5526818992596850338294511} a^{2} + \frac{1773020637096734505825150}{5526818992596850338294511} a + \frac{1419742656997146955390907}{5526818992596850338294511}$

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

Class group and class number

$C_{2}\times C_{6}$, which has order $12$ (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:  \( \frac{487117772461740147696672}{5526818992596850338294511} a^{15} + \frac{36989459303499272414620}{5526818992596850338294511} a^{14} + \frac{991314671835712072925488}{5526818992596850338294511} a^{13} - \frac{899470483607681152520812}{5526818992596850338294511} a^{12} + \frac{32594708677119519447549464}{5526818992596850338294511} a^{11} - \frac{50160316571009415115828356}{5526818992596850338294511} a^{10} + \frac{276749433928909252329928712}{5526818992596850338294511} a^{9} - \frac{270878940855293624216859283}{5526818992596850338294511} a^{8} + \frac{1017762477727980997204005208}{5526818992596850338294511} a^{7} - \frac{839037095707248330545342404}{5526818992596850338294511} a^{6} + \frac{1573599753974152855129762364}{5526818992596850338294511} a^{5} - \frac{1277284571534527192870021036}{5526818992596850338294511} a^{4} + \frac{1234521239549055010614719568}{5526818992596850338294511} a^{3} - \frac{684166335784901315196135828}{5526818992596850338294511} a^{2} + \frac{231027435030507418859984900}{5526818992596850338294511} a - \frac{10382479762788039874713384}{5526818992596850338294511} \) (order $10$)
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:  \( 46652.615418 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1027:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 576
The 28 conjugacy class representatives for t16n1027
Character table for t16n1027 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 8.8.22982560000.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 12 sibling: data not computed
Degree 24 siblings: data not computed
Degree 36 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 ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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}$
379Data not computed