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Properties

Label 16.0.28525864220...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 3^{8}\cdot 5^{8}\cdot 19^{8}$
Root discriminant $33.76$
Ramified primes $2, 3, 5, 19$
Class number $8$ (GRH)
Class group $[8]$ (GRH)
Galois group $C_2\times S_4$ (as 16T61)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5776, -34352, 109440, -166896, 128200, -9936, -59040, 37448, 560, -8620, 2632, 352, -362, 72, 6, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 6*x^14 + 72*x^13 - 362*x^12 + 352*x^11 + 2632*x^10 - 8620*x^9 + 560*x^8 + 37448*x^7 - 59040*x^6 - 9936*x^5 + 128200*x^4 - 166896*x^3 + 109440*x^2 - 34352*x + 5776)
 
gp: K = bnfinit(x^16 - 6*x^15 + 6*x^14 + 72*x^13 - 362*x^12 + 352*x^11 + 2632*x^10 - 8620*x^9 + 560*x^8 + 37448*x^7 - 59040*x^6 - 9936*x^5 + 128200*x^4 - 166896*x^3 + 109440*x^2 - 34352*x + 5776, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 6 x^{14} + 72 x^{13} - 362 x^{12} + 352 x^{11} + 2632 x^{10} - 8620 x^{9} + 560 x^{8} + 37448 x^{7} - 59040 x^{6} - 9936 x^{5} + 128200 x^{4} - 166896 x^{3} + 109440 x^{2} - 34352 x + 5776 \)

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:  \(2852586422067225600000000=2^{16}\cdot 3^{8}\cdot 5^{8}\cdot 19^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.76$
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:  $2, 3, 5, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{4} a^{8}$, $\frac{1}{4} a^{9}$, $\frac{1}{4} a^{10}$, $\frac{1}{4} a^{11}$, $\frac{1}{80} a^{12} + \frac{1}{40} a^{11} + \frac{1}{40} a^{9} + \frac{1}{10} a^{8} - \frac{1}{5} a^{7} - \frac{1}{20} a^{6} - \frac{1}{5} a^{5} - \frac{1}{5} a^{4} + \frac{3}{10} a^{2} + \frac{1}{5} a + \frac{3}{10}$, $\frac{1}{80} a^{13} - \frac{1}{20} a^{11} + \frac{1}{40} a^{10} + \frac{1}{20} a^{9} + \frac{1}{10} a^{8} - \frac{3}{20} a^{7} - \frac{1}{10} a^{6} + \frac{1}{5} a^{5} - \frac{1}{10} a^{4} + \frac{3}{10} a^{3} - \frac{2}{5} a^{2} - \frac{1}{10} a + \frac{2}{5}$, $\frac{1}{160} a^{14} + \frac{1}{16} a^{11} - \frac{1}{10} a^{10} + \frac{1}{10} a^{9} - \frac{1}{8} a^{8} + \frac{1}{20} a^{7} - \frac{1}{5} a^{5} - \frac{1}{4} a^{4} - \frac{1}{5} a^{3} - \frac{9}{20} a^{2} + \frac{1}{10} a - \frac{2}{5}$, $\frac{1}{9655348602822563732239520} a^{15} - \frac{7381345683118451142631}{4827674301411281866119760} a^{14} - \frac{3587864303682252880173}{603459287676410233264970} a^{13} - \frac{10788668756131262109093}{4827674301411281866119760} a^{12} + \frac{198234365660281186763071}{2413837150705640933059880} a^{11} + \frac{277550036797805777998}{3110614884929949656005} a^{10} + \frac{1252782818316679232591}{2413837150705640933059880} a^{9} - \frac{3947878726633887807442}{301729643838205116632485} a^{8} - \frac{3146801472611566210887}{301729643838205116632485} a^{7} - \frac{51293232906243318989192}{301729643838205116632485} a^{6} - \frac{121522494696528624640711}{1206918575352820466529940} a^{5} - \frac{46852315368710420882606}{301729643838205116632485} a^{4} + \frac{113753083989125521972711}{241383715070564093305988} a^{3} + \frac{153183655148221433209}{15880507570431848243815} a^{2} + \frac{2777179643427884519354}{15880507570431848243815} a - \frac{1393969999017671659369}{15880507570431848243815}$

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

Class group and class number

$C_{8}$, which has order $8$ (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:  \( -\frac{343982983}{9709726205680} a^{15} + \frac{1136873817}{3883890482272} a^{14} - \frac{2866581917}{4854863102840} a^{13} - \frac{24731954579}{9709726205680} a^{12} + \frac{182943612111}{9709726205680} a^{11} - \frac{42276611437}{1213715775710} a^{10} - \frac{453592959797}{4854863102840} a^{9} + \frac{2577017222191}{4854863102840} a^{8} - \frac{1168338324967}{2427431551420} a^{7} - \frac{4506903605299}{2427431551420} a^{6} + \frac{1191024379665}{242743155142} a^{5} - \frac{4064224304597}{2427431551420} a^{4} - \frac{2142108374057}{242743155142} a^{3} + \frac{34301610383841}{2427431551420} a^{2} - \frac{2133840947871}{242743155142} a + \frac{3015242656297}{1213715775710} \) (order $6$)
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:  \( 269457.876984 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times S_4$ (as 16T61):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 48
The 10 conjugacy class representatives for $C_2\times S_4$
Character table for $C_2\times S_4$

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{5}) \), 4.4.259920.1, \(\Q(\sqrt{-3}, \sqrt{5})\), 8.8.1688960160000.3, 8.0.67558406400.5, 8.0.1688960160000.60

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

Sibling fields

Degree 6 siblings: data not computed
Degree 8 siblings: data not computed
Degree 12 siblings: data not computed
Degree 24 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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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$2.8.8.11$x^{8} + 20 x^{2} + 4$$4$$2$$8$$S_4$$[4/3, 4/3]_{3}^{2}$
2.8.8.11$x^{8} + 20 x^{2} + 4$$4$$2$$8$$S_4$$[4/3, 4/3]_{3}^{2}$
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$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}$
5.4.2.1$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}$
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.3.2.3$x^{3} - 304$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.3$x^{3} - 304$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.3$x^{3} - 304$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.3$x^{3} - 304$$3$$1$$2$$C_3$$[\ ]_{3}$