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Properties

Label 16.8.42415313391...0000.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{24}\cdot 5^{12}\cdot 11^{4}\cdot 29^{4}$
Root discriminant $39.97$
Ramified primes $2, 5, 11, 29$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $C_2^4.C_2^3.C_2$ (as 16T542)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-409, 2244, 3584, -8906, -7819, 12712, 5006, -5738, -1818, -512, 2066, -262, -374, 126, -6, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 6*x^14 + 126*x^13 - 374*x^12 - 262*x^11 + 2066*x^10 - 512*x^9 - 1818*x^8 - 5738*x^7 + 5006*x^6 + 12712*x^5 - 7819*x^4 - 8906*x^3 + 3584*x^2 + 2244*x - 409)
 
gp: K = bnfinit(x^16 - 4*x^15 - 6*x^14 + 126*x^13 - 374*x^12 - 262*x^11 + 2066*x^10 - 512*x^9 - 1818*x^8 - 5738*x^7 + 5006*x^6 + 12712*x^5 - 7819*x^4 - 8906*x^3 + 3584*x^2 + 2244*x - 409, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 6 x^{14} + 126 x^{13} - 374 x^{12} - 262 x^{11} + 2066 x^{10} - 512 x^{9} - 1818 x^{8} - 5738 x^{7} + 5006 x^{6} + 12712 x^{5} - 7819 x^{4} - 8906 x^{3} + 3584 x^{2} + 2244 x - 409 \)

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:  \(42415313391616000000000000=2^{24}\cdot 5^{12}\cdot 11^{4}\cdot 29^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.97$
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:  $2, 5, 11, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8} - \frac{2}{5} a^{7} - \frac{2}{5} a^{5} - \frac{1}{5} a^{4} + \frac{2}{5} a^{3} + \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{9} + \frac{1}{5} a^{7} - \frac{2}{5} a^{6} - \frac{1}{5} a^{3} + \frac{2}{5} a^{2} + \frac{2}{5}$, $\frac{1}{5} a^{10} + \frac{2}{5} a^{5} - \frac{1}{5}$, $\frac{1}{5} a^{11} + \frac{2}{5} a^{6} - \frac{1}{5} a$, $\frac{1}{5} a^{12} + \frac{2}{5} a^{7} - \frac{1}{5} a^{2}$, $\frac{1}{5} a^{13} - \frac{1}{5} a^{7} - \frac{1}{5} a^{5} + \frac{2}{5} a^{4} + \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{145} a^{14} - \frac{7}{145} a^{13} - \frac{9}{145} a^{12} + \frac{2}{145} a^{11} + \frac{2}{29} a^{10} + \frac{8}{145} a^{9} + \frac{4}{145} a^{8} + \frac{67}{145} a^{7} + \frac{2}{145} a^{6} + \frac{14}{145} a^{5} - \frac{14}{145} a^{4} - \frac{23}{145} a^{3} - \frac{19}{145} a^{2} - \frac{61}{145} a - \frac{8}{29}$, $\frac{1}{1139763255255563682376895} a^{15} - \frac{2174647026183849107104}{1139763255255563682376895} a^{14} + \frac{67978823833281461439109}{1139763255255563682376895} a^{13} + \frac{335030592188940181609}{1139763255255563682376895} a^{12} + \frac{12797691592814665693890}{227952651051112736475379} a^{11} - \frac{3882139504166602133379}{227952651051112736475379} a^{10} - \frac{76284771077848785411}{1598545940049878937415} a^{9} - \frac{45770461634150751170852}{1139763255255563682376895} a^{8} + \frac{5429374150977446702071}{49554924141546247059865} a^{7} + \frac{61218751700779807632514}{227952651051112736475379} a^{6} + \frac{39568608936618028154143}{227952651051112736475379} a^{5} - \frac{359120541026594817714251}{1139763255255563682376895} a^{4} - \frac{426112753914162240063789}{1139763255255563682376895} a^{3} - \frac{397338950566283700368074}{1139763255255563682376895} a^{2} + \frac{91500365907095986064578}{227952651051112736475379} a + \frac{295032543658307093864727}{1139763255255563682376895}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( 2068073.87812 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3.C_2$ (as 16T542):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 40 conjugacy class representatives for $C_2^4.C_2^3.C_2$
Character table for $C_2^4.C_2^3.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.638000.2, 4.4.725.1, 4.4.22000.1, 8.4.37004000000.2, 8.4.1480160000.4, 8.8.407044000000.2

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$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}$
$11$11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$29$29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$