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Properties

Label 16.0.11574317056...0000.7
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 5^{12}\cdot 41^{4}$
Root discriminant $23.93$
Ramified primes $2, 5, 41$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^4.C_2^3$ (as 16T268)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![496, 672, -1032, -8320, 20480, -16792, 444, 8020, -3801, -1520, 1876, -374, -200, 100, -3, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 - 3*x^14 + 100*x^13 - 200*x^12 - 374*x^11 + 1876*x^10 - 1520*x^9 - 3801*x^8 + 8020*x^7 + 444*x^6 - 16792*x^5 + 20480*x^4 - 8320*x^3 - 1032*x^2 + 672*x + 496)
 
gp: K = bnfinit(x^16 - 6*x^15 - 3*x^14 + 100*x^13 - 200*x^12 - 374*x^11 + 1876*x^10 - 1520*x^9 - 3801*x^8 + 8020*x^7 + 444*x^6 - 16792*x^5 + 20480*x^4 - 8320*x^3 - 1032*x^2 + 672*x + 496, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} - 3 x^{14} + 100 x^{13} - 200 x^{12} - 374 x^{11} + 1876 x^{10} - 1520 x^{9} - 3801 x^{8} + 8020 x^{7} + 444 x^{6} - 16792 x^{5} + 20480 x^{4} - 8320 x^{3} - 1032 x^{2} + 672 x + 496 \)

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:  \(11574317056000000000000=2^{24}\cdot 5^{12}\cdot 41^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.93$
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:  $2, 5, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{12} a^{12} - \frac{1}{6} a^{11} - \frac{1}{12} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} + \frac{1}{3} a^{5} - \frac{5}{12} a^{4} - \frac{1}{2} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{12} a^{13} + \frac{1}{12} a^{11} - \frac{1}{6} a^{10} - \frac{1}{2} a^{8} + \frac{1}{3} a^{6} + \frac{1}{4} a^{5} + \frac{1}{6} a^{4} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{24} a^{14} - \frac{1}{24} a^{12} + \frac{1}{12} a^{11} + \frac{1}{12} a^{10} + \frac{1}{4} a^{9} - \frac{1}{2} a^{8} + \frac{1}{6} a^{7} - \frac{3}{8} a^{6} + \frac{1}{4} a^{5} - \frac{1}{12} a^{4} - \frac{1}{6} a^{3} - \frac{1}{2} a^{2} + \frac{1}{3}$, $\frac{1}{34690021324536} a^{15} - \frac{36979022601}{5781670220756} a^{14} + \frac{402581851027}{11563340441512} a^{13} - \frac{16127051189}{1445417555189} a^{12} + \frac{87271619306}{4336252665567} a^{11} - \frac{926453143451}{5781670220756} a^{10} - \frac{399100529327}{1445417555189} a^{9} - \frac{1266917213365}{4336252665567} a^{8} - \frac{4384804008659}{11563340441512} a^{7} + \frac{348599772829}{8672505331134} a^{6} - \frac{520207499543}{4336252665567} a^{5} - \frac{1316518861597}{8672505331134} a^{4} - \frac{1209287127021}{2890835110378} a^{3} - \frac{69000663971}{4336252665567} a^{2} + \frac{1173231010025}{4336252665567} a + \frac{694403794126}{1445417555189}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( -\frac{3872851358149}{8672505331134} a^{15} + \frac{65031526386805}{34690021324536} a^{14} + \frac{82100434036529}{17345010662268} a^{13} - \frac{1254505275705485}{34690021324536} a^{12} + \frac{207331121331763}{8672505331134} a^{11} + \frac{3658608301485941}{17345010662268} a^{10} - \frac{2641307798764673}{5781670220756} a^{9} - \frac{1307097575863927}{8672505331134} a^{8} + \frac{6200612197152473}{4336252665567} a^{7} - \frac{11434417107281439}{11563340441512} a^{6} - \frac{2896371701032776}{1445417555189} a^{5} + \frac{22389115028065175}{5781670220756} a^{4} - \frac{9159680884802132}{4336252665567} a^{3} - \frac{176997946363827}{1445417555189} a^{2} + \frac{330680939035162}{1445417555189} a + \frac{523674803447468}{4336252665567} \) (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:  \( 60573.9234845 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3$ (as 16T268):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 29 conjugacy class representatives for $C_2^4.C_2^3$
Character table for $C_2^4.C_2^3$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), 4.0.8000.2, \(\Q(\zeta_{5})\), \(\Q(\sqrt{2}, \sqrt{5})\), 8.0.64000000.2

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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
$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}$
41Data not computed