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Properties

Label 16.0.82115019702...0000.4
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 5^{12}\cdot 23^{8}$
Root discriminant $64.14$
Ramified primes $2, 5, 23$
Class number $10800$ (GRH)
Class group $[3, 30, 120]$ (GRH)
Galois group $C_4\times C_2^2$ (as 16T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![12941281, -8526276, 11232934, -5961476, 4429644, -1966724, 1060918, -399988, 170261, -54220, 18698, -4908, 1364, -280, 60, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 60*x^14 - 280*x^13 + 1364*x^12 - 4908*x^11 + 18698*x^10 - 54220*x^9 + 170261*x^8 - 399988*x^7 + 1060918*x^6 - 1966724*x^5 + 4429644*x^4 - 5961476*x^3 + 11232934*x^2 - 8526276*x + 12941281)
 
gp: K = bnfinit(x^16 - 8*x^15 + 60*x^14 - 280*x^13 + 1364*x^12 - 4908*x^11 + 18698*x^10 - 54220*x^9 + 170261*x^8 - 399988*x^7 + 1060918*x^6 - 1966724*x^5 + 4429644*x^4 - 5961476*x^3 + 11232934*x^2 - 8526276*x + 12941281, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 60 x^{14} - 280 x^{13} + 1364 x^{12} - 4908 x^{11} + 18698 x^{10} - 54220 x^{9} + 170261 x^{8} - 399988 x^{7} + 1060918 x^{6} - 1966724 x^{5} + 4429644 x^{4} - 5961476 x^{3} + 11232934 x^{2} - 8526276 x + 12941281 \)

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:  \(82115019702009856000000000000=2^{32}\cdot 5^{12}\cdot 23^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $64.14$
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:  $2, 5, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(920=2^{3}\cdot 5\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{920}(1,·)$, $\chi_{920}(323,·)$, $\chi_{920}(321,·)$, $\chi_{920}(781,·)$, $\chi_{920}(783,·)$, $\chi_{920}(643,·)$, $\chi_{920}(461,·)$, $\chi_{920}(367,·)$, $\chi_{920}(507,·)$, $\chi_{920}(229,·)$, $\chi_{920}(689,·)$, $\chi_{920}(47,·)$, $\chi_{920}(369,·)$, $\chi_{920}(183,·)$, $\chi_{920}(827,·)$, $\chi_{920}(829,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{38} a^{12} - \frac{3}{19} a^{11} + \frac{2}{19} a^{10} - \frac{3}{38} a^{9} + \frac{3}{19} a^{8} + \frac{2}{19} a^{7} - \frac{4}{19} a^{6} - \frac{5}{38} a^{5} - \frac{7}{19} a^{4} + \frac{7}{19} a^{3} - \frac{1}{19} a^{2} + \frac{9}{38} a - \frac{3}{38}$, $\frac{1}{1558} a^{13} + \frac{7}{779} a^{12} - \frac{363}{1558} a^{11} + \frac{305}{1558} a^{10} + \frac{106}{779} a^{9} - \frac{161}{1558} a^{8} + \frac{357}{1558} a^{7} + \frac{215}{1558} a^{6} - \frac{1}{2} a^{4} - \frac{425}{1558} a^{3} + \frac{577}{1558} a^{2} - \frac{773}{1558} a - \frac{1}{38}$, $\frac{1}{876006318477422} a^{14} - \frac{7}{876006318477422} a^{13} + \frac{701641793255}{79636938043402} a^{12} - \frac{46308358354739}{876006318477422} a^{11} - \frac{101568010452462}{438003159238711} a^{10} + \frac{126163911726761}{876006318477422} a^{9} + \frac{17860967219725}{79636938043402} a^{8} - \frac{300245332975465}{876006318477422} a^{7} - \frac{179134126042325}{438003159238711} a^{6} - \frac{1860477696199}{21366007767742} a^{5} + \frac{257356362704103}{876006318477422} a^{4} - \frac{76881674407189}{876006318477422} a^{3} + \frac{12868428929755}{46105595709338} a^{2} - \frac{104546529103284}{438003159238711} a - \frac{2259374549332}{10683003883871}$, $\frac{1}{18116686672431564382} a^{15} + \frac{10333}{18116686672431564382} a^{14} - \frac{56105532167598}{823485757837798381} a^{13} - \frac{150545768522022057}{18116686672431564382} a^{12} + \frac{1920343560003767275}{18116686672431564382} a^{11} - \frac{1971118697762098348}{9058343336215782191} a^{10} - \frac{403362391977522421}{1646971515675596762} a^{9} - \frac{53882024322042250}{476754912432409589} a^{8} - \frac{6779198933086549855}{18116686672431564382} a^{7} - \frac{3721426375739699378}{9058343336215782191} a^{6} - \frac{1050168919410102445}{18116686672431564382} a^{5} - \frac{631656461456070354}{9058343336215782191} a^{4} - \frac{4215934739325212881}{9058343336215782191} a^{3} - \frac{122099372527481999}{953509824864819178} a^{2} - \frac{2481494925753109393}{18116686672431564382} a - \frac{230252429046355}{40170036967697482}$

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

Class group and class number

$C_{3}\times C_{30}\times C_{120}$, which has order $10800$ (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:  \( -1 \) (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:  \( 7114.135357253273 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_4$ (as 16T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 16
The 16 conjugacy class representatives for $C_4\times C_2^2$
Character table for $C_4\times C_2^2$

Intermediate fields

\(\Q(\sqrt{-230}) \), \(\Q(\sqrt{-115}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-46}) \), \(\Q(\sqrt{-23}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{2}, \sqrt{-115})\), \(\Q(\sqrt{5}, \sqrt{-46})\), \(\Q(\sqrt{10}, \sqrt{-23})\), \(\Q(\sqrt{5}, \sqrt{-23})\), \(\Q(\sqrt{10}, \sqrt{-46})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{-23})\), \(\Q(\zeta_{20})^+\), 4.0.4232000.2, 4.0.1058000.1, 4.4.8000.1, 8.0.716392960000.2, 8.0.286557184000000.25, 8.0.286557184000000.34, 8.0.1119364000000.1, 8.0.17909824000000.12, \(\Q(\zeta_{40})^+\), 8.0.286557184000000.40

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

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$2$2.8.16.3$x^{8} + 2 x^{6} + 6 x^{4} + 4 x^{2} + 8 x + 28$$4$$2$$16$$C_4\times C_2$$[2, 3]^{2}$
2.8.16.3$x^{8} + 2 x^{6} + 6 x^{4} + 4 x^{2} + 8 x + 28$$4$$2$$16$$C_4\times C_2$$[2, 3]^{2}$
$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}$
$23$23.8.4.1$x^{8} + 11638 x^{4} - 12167 x^{2} + 33860761$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
23.8.4.1$x^{8} + 11638 x^{4} - 12167 x^{2} + 33860761$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$