↑

Properties

Label 16.0.16383506625...6256.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{40}\cdot 3^{8}\cdot 17^{6}\cdot 97^{2}$
Root discriminant $50.22$
Ramified primes $2, 3, 17, 97$
Class number $928$ (GRH)
Class group $[2, 2, 2, 116]$ (GRH)
Galois group $C_2^3.C_2^4.C_2$ (as 16T595)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![97081, -209824, 292616, -260368, 258952, -181520, 150072, -67552, 50410, -14592, 10632, -1456, 1112, -48, 56, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 56*x^14 - 48*x^13 + 1112*x^12 - 1456*x^11 + 10632*x^10 - 14592*x^9 + 50410*x^8 - 67552*x^7 + 150072*x^6 - 181520*x^5 + 258952*x^4 - 260368*x^3 + 292616*x^2 - 209824*x + 97081)
 
gp: K = bnfinit(x^16 + 56*x^14 - 48*x^13 + 1112*x^12 - 1456*x^11 + 10632*x^10 - 14592*x^9 + 50410*x^8 - 67552*x^7 + 150072*x^6 - 181520*x^5 + 258952*x^4 - 260368*x^3 + 292616*x^2 - 209824*x + 97081, 1)
 

Normalized defining polynomial

\( x^{16} + 56 x^{14} - 48 x^{13} + 1112 x^{12} - 1456 x^{11} + 10632 x^{10} - 14592 x^{9} + 50410 x^{8} - 67552 x^{7} + 150072 x^{6} - 181520 x^{5} + 258952 x^{4} - 260368 x^{3} + 292616 x^{2} - 209824 x + 97081 \)

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:  \(1638350662595178231031136256=2^{40}\cdot 3^{8}\cdot 17^{6}\cdot 97^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $50.22$
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:  $2, 3, 17, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{8} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a + \frac{1}{8}$, $\frac{1}{8} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} + \frac{1}{8} a$, $\frac{1}{8} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} + \frac{1}{8} a^{2}$, $\frac{1}{8} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} + \frac{1}{8} a^{3} - \frac{1}{2}$, $\frac{1}{136} a^{12} + \frac{5}{136} a^{11} - \frac{7}{136} a^{10} - \frac{1}{68} a^{9} - \frac{1}{17} a^{8} + \frac{4}{17} a^{6} + \frac{3}{17} a^{5} + \frac{65}{136} a^{4} + \frac{33}{136} a^{3} + \frac{1}{136} a^{2} + \frac{13}{68} a - \frac{3}{17}$, $\frac{1}{136} a^{13} + \frac{1}{68} a^{11} - \frac{1}{136} a^{10} + \frac{1}{68} a^{9} + \frac{3}{68} a^{8} + \frac{4}{17} a^{7} - \frac{55}{136} a^{5} - \frac{5}{34} a^{4} + \frac{3}{68} a^{3} - \frac{13}{136} a^{2} - \frac{9}{68} a - \frac{25}{68}$, $\frac{1}{26384} a^{14} - \frac{7}{13192} a^{13} + \frac{9}{26384} a^{12} + \frac{547}{13192} a^{11} + \frac{681}{26384} a^{10} - \frac{579}{13192} a^{9} + \frac{1235}{26384} a^{8} + \frac{414}{1649} a^{7} - \frac{10303}{26384} a^{6} - \frac{147}{776} a^{5} - \frac{1639}{26384} a^{4} + \frac{6595}{13192} a^{3} + \frac{2313}{26384} a^{2} - \frac{4415}{13192} a - \frac{2069}{26384}$, $\frac{1}{493549780388813887596298582544} a^{15} + \frac{2642978499155858058425589}{493549780388813887596298582544} a^{14} - \frac{1487353613907864588599172209}{493549780388813887596298582544} a^{13} - \frac{758996828970514070511002153}{493549780388813887596298582544} a^{12} - \frac{2652435496065301420078666623}{493549780388813887596298582544} a^{11} - \frac{20183051616084722281008038113}{493549780388813887596298582544} a^{10} + \frac{9167120641633475107503023069}{493549780388813887596298582544} a^{9} - \frac{889865298185243009110936825}{29032340022871405152723446032} a^{8} - \frac{226407015119385900848505147679}{493549780388813887596298582544} a^{7} + \frac{13376982947742876461270257117}{493549780388813887596298582544} a^{6} + \frac{132037961865200618669157841887}{493549780388813887596298582544} a^{5} + \frac{224405183267684755084169581951}{493549780388813887596298582544} a^{4} - \frac{56289928349264001363617403847}{493549780388813887596298582544} a^{3} + \frac{28468517465887746375639188495}{493549780388813887596298582544} a^{2} - \frac{132000725566314229254129944531}{493549780388813887596298582544} a + \frac{103665300291761777736098271}{493549780388813887596298582544}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{116}$, which has order $928$ (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:  \( 9072.35800888 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_2^4.C_2$ (as 16T595):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 40 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

\(\Q(\sqrt{6}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{3}) \), 4.4.9792.1, 4.4.4352.1, \(\Q(\sqrt{2}, \sqrt{3})\), 8.8.1534132224.1

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 R R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
2Data not computed
3Data not computed
$17$17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.8.6.2$x^{8} + 85 x^{4} + 2601$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$97$97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.1.1$x^{2} - 97$$2$$1$$1$$C_2$$[\ ]_{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.1.1$x^{2} - 97$$2$$1$$1$$C_2$$[\ ]_{2}$