↑

Properties

Label 16.0.19361901958...5625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{8}\cdot 13^{4}\cdot 19^{2}\cdot 29^{8}\cdot 31^{2}$
Root discriminant $50.75$
Ramified primes $5, 13, 19, 29, 31$
Class number $1344$ (GRH)
Class group $[2, 2, 336]$ (GRH)
Galois group $C_2^3.C_2^4.C_2$ (as 16T608)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1254661, -2465349, 4766774, -4706434, 3869799, -2228506, 1146942, -470475, 193101, -65833, 23130, -6320, 1790, -336, 71, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 71*x^14 - 336*x^13 + 1790*x^12 - 6320*x^11 + 23130*x^10 - 65833*x^9 + 193101*x^8 - 470475*x^7 + 1146942*x^6 - 2228506*x^5 + 3869799*x^4 - 4706434*x^3 + 4766774*x^2 - 2465349*x + 1254661)
 
gp: K = bnfinit(x^16 - 7*x^15 + 71*x^14 - 336*x^13 + 1790*x^12 - 6320*x^11 + 23130*x^10 - 65833*x^9 + 193101*x^8 - 470475*x^7 + 1146942*x^6 - 2228506*x^5 + 3869799*x^4 - 4706434*x^3 + 4766774*x^2 - 2465349*x + 1254661, 1)
 

Normalized defining polynomial

\( x^{16} - 7 x^{15} + 71 x^{14} - 336 x^{13} + 1790 x^{12} - 6320 x^{11} + 23130 x^{10} - 65833 x^{9} + 193101 x^{8} - 470475 x^{7} + 1146942 x^{6} - 2228506 x^{5} + 3869799 x^{4} - 4706434 x^{3} + 4766774 x^{2} - 2465349 x + 1254661 \)

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:  \(1936190195826058295484765625=5^{8}\cdot 13^{4}\cdot 19^{2}\cdot 29^{8}\cdot 31^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $50.75$
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:  $5, 13, 19, 29, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{39} a^{13} - \frac{2}{39} a^{12} - \frac{7}{39} a^{11} + \frac{16}{39} a^{10} - \frac{5}{13} a^{9} - \frac{10}{39} a^{8} + \frac{10}{39} a^{7} - \frac{7}{39} a^{6} + \frac{11}{39} a^{5} - \frac{2}{13} a^{4} + \frac{17}{39} a^{3} - \frac{16}{39} a^{2} + \frac{4}{13} a + \frac{2}{39}$, $\frac{1}{5421} a^{14} + \frac{53}{5421} a^{13} + \frac{4}{417} a^{12} + \frac{593}{5421} a^{11} + \frac{2516}{5421} a^{10} - \frac{1940}{5421} a^{9} + \frac{110}{5421} a^{8} - \frac{1004}{5421} a^{7} - \frac{220}{1807} a^{6} + \frac{178}{1807} a^{5} - \frac{781}{5421} a^{4} - \frac{1421}{5421} a^{3} - \frac{1232}{5421} a^{2} - \frac{815}{1807} a + \frac{587}{1807}$, $\frac{1}{884263225681596165666931851962598689079} a^{15} + \frac{34304565703693114550606844480481324}{884263225681596165666931851962598689079} a^{14} + \frac{2501351734792782155780324229526278883}{294754408560532055222310617320866229693} a^{13} - \frac{115428275737378909164478001391751230596}{884263225681596165666931851962598689079} a^{12} + \frac{128377741650344232764038261938795168329}{294754408560532055222310617320866229693} a^{11} + \frac{11494850732993347048623523937871486923}{884263225681596165666931851962598689079} a^{10} + \frac{188486649776935693672229265775114801814}{884263225681596165666931851962598689079} a^{9} - \frac{156620373413145024095312315121762261436}{884263225681596165666931851962598689079} a^{8} + \frac{131901553454188026319551786139290274778}{884263225681596165666931851962598689079} a^{7} + \frac{91978346149585745550570521328495305027}{294754408560532055222310617320866229693} a^{6} - \frac{3202320655769970485052188634231020516}{22673416043117850401716201332374325361} a^{5} - \frac{305757869973934407832557805294002527965}{884263225681596165666931851962598689079} a^{4} - \frac{214796660587769593200202854908989349203}{884263225681596165666931851962598689079} a^{3} + \frac{303315732331220486344469982716249511629}{884263225681596165666931851962598689079} a^{2} - \frac{18427956770614377284484303740301854710}{68020248129353551205148603997122976083} a - \frac{267530829587235897564368865638514329297}{884263225681596165666931851962598689079}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{336}$, which has order $1344$ (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:  \( 3793.72993285 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_2^4.C_2$ (as 16T608):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 34 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

\(\Q(\sqrt{29}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{145}) \), 4.4.4205.1 x2, 4.4.725.1 x2, \(\Q(\sqrt{5}, \sqrt{29})\), 8.8.442050625.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ R R ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$13$13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$19$19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
19.4.2.2$x^{4} - 19 x^{2} + 722$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$29$29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$31$31.4.0.1$x^{4} - 2 x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
31.4.0.1$x^{4} - 2 x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
31.4.2.2$x^{4} - 31 x^{2} + 11532$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
31.4.0.1$x^{4} - 2 x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$