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Properties

Label 16.8.53479750597...0000.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{32}\cdot 5^{10}\cdot 11^{8}\cdot 29^{6}$
Root discriminant $128.24$
Ramified primes $2, 5, 11, 29$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^3.C_2^4.C_2$ (as 16T456)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5427475, -5479100, -53694970, 22356840, 98870734, 2291740, -8160964, -759000, -333886, -124520, -18048, -3080, -159, 0, -54, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 54*x^14 - 159*x^12 - 3080*x^11 - 18048*x^10 - 124520*x^9 - 333886*x^8 - 759000*x^7 - 8160964*x^6 + 2291740*x^5 + 98870734*x^4 + 22356840*x^3 - 53694970*x^2 - 5479100*x + 5427475)
 
gp: K = bnfinit(x^16 - 54*x^14 - 159*x^12 - 3080*x^11 - 18048*x^10 - 124520*x^9 - 333886*x^8 - 759000*x^7 - 8160964*x^6 + 2291740*x^5 + 98870734*x^4 + 22356840*x^3 - 53694970*x^2 - 5479100*x + 5427475, 1)
 

Normalized defining polynomial

\( x^{16} - 54 x^{14} - 159 x^{12} - 3080 x^{11} - 18048 x^{10} - 124520 x^{9} - 333886 x^{8} - 759000 x^{7} - 8160964 x^{6} + 2291740 x^{5} + 98870734 x^{4} + 22356840 x^{3} - 53694970 x^{2} - 5479100 x + 5427475 \)

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:  \(5347975059777049895895040000000000=2^{32}\cdot 5^{10}\cdot 11^{8}\cdot 29^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $128.24$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{32245} a^{14} - \frac{521}{6449} a^{13} - \frac{3554}{32245} a^{12} + \frac{2306}{6449} a^{11} - \frac{9204}{32245} a^{10} + \frac{1998}{6449} a^{9} - \frac{9938}{32245} a^{8} + \frac{593}{6449} a^{7} - \frac{4006}{32245} a^{6} + \frac{389}{6449} a^{5} + \frac{11186}{32245} a^{4} + \frac{694}{6449} a^{3} - \frac{13986}{32245} a^{2} + \frac{802}{6449} a - \frac{1913}{6449}$, $\frac{1}{83648779937863693113447274540190398421154856794150718184293505} a^{15} + \frac{105317659591880689681065300836825451658397477368321132647}{16729755987572738622689454908038079684230971358830143636858701} a^{14} - \frac{146587924226406135701653174872059149592813857556425549423661}{561401207636669081298303855974432204168824542242622269693245} a^{13} - \frac{40949961130880107092600406713327307119833496509896752224097}{112280241527333816259660771194886440833764908448524453938649} a^{12} - \frac{34799897935221289886775243824011906900246854141762866147833394}{83648779937863693113447274540190398421154856794150718184293505} a^{11} + \frac{749057520341632415842298452113336249450029980534402222529990}{16729755987572738622689454908038079684230971358830143636858701} a^{10} - \frac{9820536517922358708088486587556651440691787991783599462057408}{83648779937863693113447274540190398421154856794150718184293505} a^{9} - \frac{546589584762401566131281632212302598146908885255308867117059}{16729755987572738622689454908038079684230971358830143636858701} a^{8} + \frac{39109986771318663736616826236146577654989838255167274944229389}{83648779937863693113447274540190398421154856794150718184293505} a^{7} - \frac{3870775095399453809221150927798159671447911127616610000470933}{16729755987572738622689454908038079684230971358830143636858701} a^{6} - \frac{39977041847638982792284974516910475450212136624442796087432429}{83648779937863693113447274540190398421154856794150718184293505} a^{5} + \frac{5766548442956012532837398854400559863592456215332106376015840}{16729755987572738622689454908038079684230971358830143636858701} a^{4} - \frac{28339405058900900132861318409926160053814369607539008494513406}{83648779937863693113447274540190398421154856794150718184293505} a^{3} - \frac{2074231043821711643512690896335738867266917511364408398876385}{16729755987572738622689454908038079684230971358830143636858701} a^{2} + \frac{1993558086771197812409614777257054237161001128787023158577164}{16729755987572738622689454908038079684230971358830143636858701} a - \frac{231558678307152216663276926105970720638055757940382771063034}{16729755987572738622689454908038079684230971358830143636858701}$

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:  $11$
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:  \( 49349732781.0 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_2^4.C_2$ (as 16T456):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 46 conjugacy class representatives for $C_2^3.C_2^4.C_2$
Character table for $C_2^3.C_2^4.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{110}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{22}) \), 4.4.5614400.1, 4.4.725.1, \(\Q(\sqrt{5}, \sqrt{22})\), 8.8.31521487360000.3

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$11$11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$29$29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.3.3$x^{4} + 58$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.3.4$x^{4} + 232$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$