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Properties

Label 16.0.93492089436...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{48}\cdot 3^{12}\cdot 5^{4}$
Root discriminant $27.27$
Ramified primes $2, 3, 5$
Class number $4$
Class group $[2, 2]$
Galois group $(C_2^2\times C_4):C_2$ (as 16T54)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![94, 424, 1048, 1552, 1544, 1000, 424, -16, -242, -216, -20, 72, 40, -8, -8, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^14 - 8*x^13 + 40*x^12 + 72*x^11 - 20*x^10 - 216*x^9 - 242*x^8 - 16*x^7 + 424*x^6 + 1000*x^5 + 1544*x^4 + 1552*x^3 + 1048*x^2 + 424*x + 94)
 
gp: K = bnfinit(x^16 - 8*x^14 - 8*x^13 + 40*x^12 + 72*x^11 - 20*x^10 - 216*x^9 - 242*x^8 - 16*x^7 + 424*x^6 + 1000*x^5 + 1544*x^4 + 1552*x^3 + 1048*x^2 + 424*x + 94, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{14} - 8 x^{13} + 40 x^{12} + 72 x^{11} - 20 x^{10} - 216 x^{9} - 242 x^{8} - 16 x^{7} + 424 x^{6} + 1000 x^{5} + 1544 x^{4} + 1552 x^{3} + 1048 x^{2} + 424 x + 94 \)

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:  \(93492089436304834560000=2^{48}\cdot 3^{12}\cdot 5^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.27$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7} a^{12} - \frac{1}{7} a^{11} + \frac{1}{7} a^{10} + \frac{2}{7} a^{9} + \frac{1}{7} a^{7} - \frac{3}{7} a^{6} - \frac{2}{7} a^{4} - \frac{1}{7} a^{2} - \frac{3}{7} a - \frac{3}{7}$, $\frac{1}{427} a^{13} + \frac{8}{427} a^{12} - \frac{78}{427} a^{11} + \frac{102}{427} a^{10} - \frac{108}{427} a^{9} - \frac{83}{427} a^{8} + \frac{55}{427} a^{7} - \frac{153}{427} a^{6} + \frac{110}{427} a^{5} + \frac{38}{427} a^{4} - \frac{92}{427} a^{3} - \frac{26}{427} a^{2} - \frac{58}{427} a - \frac{174}{427}$, $\frac{1}{427} a^{14} - \frac{20}{427} a^{12} + \frac{177}{427} a^{11} + \frac{52}{427} a^{10} + \frac{171}{427} a^{9} - \frac{135}{427} a^{8} - \frac{44}{427} a^{7} + \frac{114}{427} a^{6} + \frac{12}{427} a^{5} - \frac{213}{427} a^{4} - \frac{144}{427} a^{3} + \frac{4}{61} a^{2} - \frac{76}{427} a + \frac{172}{427}$, $\frac{1}{534942828343} a^{15} + \frac{466130486}{534942828343} a^{14} + \frac{361207089}{534942828343} a^{13} + \frac{1484365374}{48631166213} a^{12} - \frac{47434484420}{534942828343} a^{11} - \frac{210704438498}{534942828343} a^{10} - \frac{29966983451}{76420404049} a^{9} - \frac{123504468070}{534942828343} a^{8} - \frac{234201049903}{534942828343} a^{7} + \frac{202928417367}{534942828343} a^{6} + \frac{229926066570}{534942828343} a^{5} + \frac{1737202405}{534942828343} a^{4} + \frac{168996788007}{534942828343} a^{3} + \frac{134128279365}{534942828343} a^{2} - \frac{137647210454}{534942828343} a - \frac{732749849}{8769554563}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 34258.1672484 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2^2\times C_4):C_2$ (as 16T54):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $(C_2^2\times C_4):C_2$
Character table for $(C_2^2\times C_4):C_2$

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{6}) \), 4.0.27648.1 x2, 4.0.13824.1 x2, \(\Q(\sqrt{2}, \sqrt{3})\), 8.0.3057647616.7, 8.0.19110297600.3, 8.8.8493465600.1

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

Sibling fields

Galois closure: data not computed
Degree 16 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 R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
2Data not computed
3Data not computed
$5$5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$