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Properties

Label 16.8.54648796976...7536.2
Degree $16$
Signature $[8, 4]$
Discriminant $2^{24}\cdot 3^{4}\cdot 7^{8}\cdot 17^{8}$
Root discriminant $40.61$
Ramified primes $2, 3, 7, 17$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $\GL(2,Z/4)$ (as 16T193)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-588, -7056, -9352, 36568, -44358, 26564, -10332, -582, 2815, -1564, 726, -138, -47, 48, -14, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 14*x^14 + 48*x^13 - 47*x^12 - 138*x^11 + 726*x^10 - 1564*x^9 + 2815*x^8 - 582*x^7 - 10332*x^6 + 26564*x^5 - 44358*x^4 + 36568*x^3 - 9352*x^2 - 7056*x - 588)
 
gp: K = bnfinit(x^16 - 2*x^15 - 14*x^14 + 48*x^13 - 47*x^12 - 138*x^11 + 726*x^10 - 1564*x^9 + 2815*x^8 - 582*x^7 - 10332*x^6 + 26564*x^5 - 44358*x^4 + 36568*x^3 - 9352*x^2 - 7056*x - 588, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 14 x^{14} + 48 x^{13} - 47 x^{12} - 138 x^{11} + 726 x^{10} - 1564 x^{9} + 2815 x^{8} - 582 x^{7} - 10332 x^{6} + 26564 x^{5} - 44358 x^{4} + 36568 x^{3} - 9352 x^{2} - 7056 x - 588 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(54648796976762560274497536=2^{24}\cdot 3^{4}\cdot 7^{8}\cdot 17^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $40.61$
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:  $2, 3, 7, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5}$, $\frac{1}{14} a^{12} - \frac{1}{14} a^{11} - \frac{3}{14} a^{10} - \frac{1}{7} a^{9} - \frac{1}{14} a^{8} - \frac{1}{7} a^{7} - \frac{1}{7} a^{6} - \frac{1}{14} a^{5} + \frac{2}{7} a^{4} - \frac{1}{7} a^{3} + \frac{2}{7} a^{2}$, $\frac{1}{14} a^{13} + \frac{3}{14} a^{11} + \frac{1}{7} a^{10} - \frac{3}{14} a^{9} - \frac{3}{14} a^{8} - \frac{2}{7} a^{7} - \frac{3}{14} a^{6} - \frac{2}{7} a^{5} - \frac{5}{14} a^{4} + \frac{1}{7} a^{3} + \frac{2}{7} a^{2}$, $\frac{1}{28} a^{14} - \frac{1}{28} a^{13} - \frac{5}{28} a^{11} + \frac{1}{7} a^{10} - \frac{1}{28} a^{9} - \frac{5}{28} a^{8} + \frac{3}{7} a^{6} - \frac{3}{7} a^{5} - \frac{3}{7} a^{4} + \frac{2}{7} a^{3} + \frac{3}{7} a^{2}$, $\frac{1}{47889822606525982073893463548} a^{15} - \frac{392844489661060410719957815}{47889822606525982073893463548} a^{14} + \frac{23191162228727973490924329}{3420701614751855862420961682} a^{13} + \frac{37802866020442078829834917}{47889822606525982073893463548} a^{12} - \frac{641566913327589668146853523}{23944911303262991036946731774} a^{11} + \frac{2765272750211234080208371395}{47889822606525982073893463548} a^{10} - \frac{10417143205982208483366719499}{47889822606525982073893463548} a^{9} - \frac{1066797896770545492514342380}{11972455651631495518473365887} a^{8} + \frac{372718436490095462501977545}{11972455651631495518473365887} a^{7} - \frac{576359493626208953787105121}{11972455651631495518473365887} a^{6} - \frac{31611710666022977603603132}{1710350807375927931210480841} a^{5} + \frac{119931323153418251860609129}{23944911303262991036946731774} a^{4} + \frac{1228499109613289991420578294}{11972455651631495518473365887} a^{3} - \frac{256538916011891302335038545}{1710350807375927931210480841} a^{2} + \frac{767392816503163909738854707}{1710350807375927931210480841} a - \frac{76067186089393967616882516}{244335829625132561601497263}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $11$
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:  \( 43586613.7889 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$\GL(2,Z/4)$ (as 16T193):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 96
The 14 conjugacy class representatives for $\GL(2,Z/4)$
Character table for $\GL(2,Z/4)$

Intermediate fields

\(\Q(\sqrt{7}) \), 4.4.32368.1, 4.2.9408.2, 8.8.821386940416.1

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

Sibling fields

Degree 12 siblings: data not computed
Degree 16 sibling: data not computed
Degree 24 siblings: data not computed
Degree 32 sibling: 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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.8.12.13$x^{8} + 12 x^{4} + 16$$4$$2$$12$$D_4$$[2, 2]^{2}$
2.8.12.13$x^{8} + 12 x^{4} + 16$$4$$2$$12$$D_4$$[2, 2]^{2}$
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
3.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$17$17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.12.8.1$x^{12} - 51 x^{9} + 867 x^{6} - 4913 x^{3} + 111166451$$3$$4$$8$$C_3 : C_4$$[\ ]_{3}^{4}$