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Properties

Label 16.0.24681786277...1824.5
Degree $16$
Signature $[0, 8]$
Discriminant $2^{62}\cdot 3^{8}\cdot 13^{8}$
Root discriminant $91.63$
Ramified primes $2, 3, 13$
Class number $129312$ (GRH)
Class group $[3, 43104]$ (GRH)
Galois group $C_8\times C_2$ (as 16T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![387186238, -188732656, 232760200, -94452624, 63853488, -21769840, 10419088, -2984864, 1104105, -262712, 77596, -14872, 3510, -504, 92, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 92*x^14 - 504*x^13 + 3510*x^12 - 14872*x^11 + 77596*x^10 - 262712*x^9 + 1104105*x^8 - 2984864*x^7 + 10419088*x^6 - 21769840*x^5 + 63853488*x^4 - 94452624*x^3 + 232760200*x^2 - 188732656*x + 387186238)
 
gp: K = bnfinit(x^16 - 8*x^15 + 92*x^14 - 504*x^13 + 3510*x^12 - 14872*x^11 + 77596*x^10 - 262712*x^9 + 1104105*x^8 - 2984864*x^7 + 10419088*x^6 - 21769840*x^5 + 63853488*x^4 - 94452624*x^3 + 232760200*x^2 - 188732656*x + 387186238, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 92 x^{14} - 504 x^{13} + 3510 x^{12} - 14872 x^{11} + 77596 x^{10} - 262712 x^{9} + 1104105 x^{8} - 2984864 x^{7} + 10419088 x^{6} - 21769840 x^{5} + 63853488 x^{4} - 94452624 x^{3} + 232760200 x^{2} - 188732656 x + 387186238 \)

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:  \(24681786277054131674683564621824=2^{62}\cdot 3^{8}\cdot 13^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $91.63$
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:  $2, 3, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1248=2^{5}\cdot 3\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{1248}(1,·)$, $\chi_{1248}(389,·)$, $\chi_{1248}(781,·)$, $\chi_{1248}(77,·)$, $\chi_{1248}(1169,·)$, $\chi_{1248}(469,·)$, $\chi_{1248}(857,·)$, $\chi_{1248}(157,·)$, $\chi_{1248}(1093,·)$, $\chi_{1248}(545,·)$, $\chi_{1248}(937,·)$, $\chi_{1248}(625,·)$, $\chi_{1248}(1013,·)$, $\chi_{1248}(233,·)$, $\chi_{1248}(313,·)$, $\chi_{1248}(701,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{17} a^{12} - \frac{6}{17} a^{11} + \frac{5}{17} a^{10} - \frac{4}{17} a^{9} + \frac{4}{17} a^{8} - \frac{7}{17} a^{7} - \frac{6}{17} a^{6} + \frac{5}{17} a^{5} - \frac{7}{17} a^{4} - \frac{2}{17} a^{3} + \frac{8}{17} a^{2} - \frac{8}{17} a - \frac{5}{17}$, $\frac{1}{17} a^{13} + \frac{3}{17} a^{11} - \frac{8}{17} a^{10} - \frac{3}{17} a^{9} + \frac{3}{17} a^{7} + \frac{3}{17} a^{6} + \frac{6}{17} a^{5} + \frac{7}{17} a^{4} - \frac{4}{17} a^{3} + \frac{6}{17} a^{2} - \frac{2}{17} a + \frac{4}{17}$, $\frac{1}{2237572821431101393} a^{14} - \frac{7}{2237572821431101393} a^{13} + \frac{38164717908479502}{2237572821431101393} a^{12} - \frac{228988307450876921}{2237572821431101393} a^{11} + \frac{297261205961115221}{2237572821431101393} a^{10} + \frac{612753455160795504}{2237572821431101393} a^{9} - \frac{442475825484439393}{2237572821431101393} a^{8} + \frac{49656831875541489}{2237572821431101393} a^{7} - \frac{742974079506376565}{2237572821431101393} a^{6} + \frac{434840756296417252}{2237572821431101393} a^{5} + \frac{910624353189522987}{2237572821431101393} a^{4} - \frac{29285134595538478}{131621930672417729} a^{3} + \frac{287946957518331502}{2237572821431101393} a^{2} - \frac{718962777344356446}{2237572821431101393} a - \frac{237859525581739614}{2237572821431101393}$, $\frac{1}{10603448124935667609872081} a^{15} + \frac{2369401}{10603448124935667609872081} a^{14} + \frac{103476856535660520648389}{10603448124935667609872081} a^{13} - \frac{270305639147702655264638}{10603448124935667609872081} a^{12} + \frac{1892646986526967639158815}{10603448124935667609872081} a^{11} + \frac{1882173203512614871894625}{10603448124935667609872081} a^{10} - \frac{4552431857392919101721576}{10603448124935667609872081} a^{9} + \frac{2764635026363880875506079}{10603448124935667609872081} a^{8} + \frac{3583331087844845325869672}{10603448124935667609872081} a^{7} - \frac{1157601436451245465393824}{10603448124935667609872081} a^{6} + \frac{3187347991530546185779684}{10603448124935667609872081} a^{5} + \frac{3229148600922464767759415}{10603448124935667609872081} a^{4} + \frac{5217969199335626649295253}{10603448124935667609872081} a^{3} - \frac{4757472539818891925664840}{10603448124935667609872081} a^{2} + \frac{100087788792506231828146}{623732242643274565286593} a - \frac{3233397870476920710136230}{10603448124935667609872081}$

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

Class group and class number

$C_{3}\times C_{43104}$, which has order $129312$ (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:  \( 15753.94986242651 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_8$ (as 16T5):

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 $C_8\times C_2$
Character table for $C_8\times C_2$

Intermediate fields

\(\Q(\sqrt{-78}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-39}) \), \(\Q(\sqrt{2}, \sqrt{-39})\), \(\Q(\zeta_{16})^+\), 4.0.3115008.1, 8.0.9703274840064.5, 8.0.4968076718112768.8, \(\Q(\zeta_{32})^+\)

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 R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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$2.8.31.6$x^{8} + 16 x^{7} + 28 x^{4} + 16 x^{3} + 2$$8$$1$$31$$C_8$$[3, 4, 5]$
2.8.31.6$x^{8} + 16 x^{7} + 28 x^{4} + 16 x^{3} + 2$$8$$1$$31$$C_8$$[3, 4, 5]$
3Data not computed
13Data not computed