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Properties

Label 16.0.23373022359...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{46}\cdot 3^{12}\cdot 5^{4}$
Root discriminant $25.01$
Ramified primes $2, 3, 5$
Class number $4$
Class group $[2, 2]$
Galois group $C_2^4:C_2^2$ (as 16T119)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![94, -344, 224, 168, -148, 488, 76, -216, 352, -24, -68, 88, -22, 0, 8, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 8*x^14 - 22*x^12 + 88*x^11 - 68*x^10 - 24*x^9 + 352*x^8 - 216*x^7 + 76*x^6 + 488*x^5 - 148*x^4 + 168*x^3 + 224*x^2 - 344*x + 94)
 
gp: K = bnfinit(x^16 - 4*x^15 + 8*x^14 - 22*x^12 + 88*x^11 - 68*x^10 - 24*x^9 + 352*x^8 - 216*x^7 + 76*x^6 + 488*x^5 - 148*x^4 + 168*x^3 + 224*x^2 - 344*x + 94, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 8 x^{14} - 22 x^{12} + 88 x^{11} - 68 x^{10} - 24 x^{9} + 352 x^{8} - 216 x^{7} + 76 x^{6} + 488 x^{5} - 148 x^{4} + 168 x^{3} + 224 x^{2} - 344 x + 94 \)

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:  \(23373022359076208640000=2^{46}\cdot 3^{12}\cdot 5^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.01$
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 not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{9} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{57} a^{13} + \frac{1}{57} a^{12} + \frac{1}{19} a^{11} + \frac{8}{57} a^{10} - \frac{3}{19} a^{9} - \frac{3}{19} a^{8} - \frac{6}{19} a^{7} + \frac{25}{57} a^{6} + \frac{1}{19} a^{5} + \frac{8}{57} a^{4} + \frac{28}{57} a^{3} + \frac{2}{19} a^{2} + \frac{2}{19} a - \frac{2}{19}$, $\frac{1}{6555} a^{14} - \frac{8}{1311} a^{13} - \frac{29}{345} a^{12} + \frac{436}{6555} a^{11} + \frac{312}{2185} a^{10} + \frac{683}{6555} a^{9} - \frac{173}{1311} a^{8} - \frac{987}{2185} a^{7} - \frac{309}{2185} a^{6} - \frac{52}{437} a^{5} - \frac{632}{2185} a^{4} - \frac{781}{6555} a^{3} - \frac{1011}{2185} a^{2} - \frac{305}{1311} a - \frac{583}{2185}$, $\frac{1}{16579842899595} a^{15} - \frac{83227390}{1105322859973} a^{14} - \frac{73726746541}{16579842899595} a^{13} + \frac{306760284702}{5526614299865} a^{12} + \frac{1985677512391}{16579842899595} a^{11} + \frac{2368012134493}{16579842899595} a^{10} - \frac{543421173910}{3315968579919} a^{9} + \frac{2586515267729}{16579842899595} a^{8} - \frac{590376167349}{5526614299865} a^{7} + \frac{1482634778857}{3315968579919} a^{6} + \frac{635887357973}{5526614299865} a^{5} + \frac{860040238774}{16579842899595} a^{4} + \frac{6730654944667}{16579842899595} a^{3} - \frac{514754884376}{3315968579919} a^{2} - \frac{2663294051688}{5526614299865} a - \frac{64041040}{3713290683}$

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:  \( 19636.7572221 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4:C_2^2$ (as 16T119):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 22 conjugacy class representatives for $C_2^4:C_2^2$
Character table for $C_2^4:C_2^2$ is not computed

Intermediate fields

\(\Q(\sqrt{6}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{2}) \), 4.0.27648.1 x2, 4.0.13824.1 x2, \(\Q(\sqrt{2}, \sqrt{3})\), 8.4.530841600.1, 8.4.76441190400.2, 8.0.3057647616.7

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 R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$3$3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
$5$5.4.0.1$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$$[\ ]_{2}^{2}$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$