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Properties

Label 16.8.10620043549...3584.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{46}\cdot 3^{12}\cdot 73^{4}$
Root discriminant $48.88$
Ramified primes $2, 3, 73$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^4.C_2^3$ (as 16T373)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![529, 1472, -55776, 82648, 381732, -67536, -133856, 472, 22038, 5984, -3296, -840, 204, 80, 0, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 80*x^13 + 204*x^12 - 840*x^11 - 3296*x^10 + 5984*x^9 + 22038*x^8 + 472*x^7 - 133856*x^6 - 67536*x^5 + 381732*x^4 + 82648*x^3 - 55776*x^2 + 1472*x + 529)
 
gp: K = bnfinit(x^16 - 8*x^15 + 80*x^13 + 204*x^12 - 840*x^11 - 3296*x^10 + 5984*x^9 + 22038*x^8 + 472*x^7 - 133856*x^6 - 67536*x^5 + 381732*x^4 + 82648*x^3 - 55776*x^2 + 1472*x + 529, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 80 x^{13} + 204 x^{12} - 840 x^{11} - 3296 x^{10} + 5984 x^{9} + 22038 x^{8} + 472 x^{7} - 133856 x^{6} - 67536 x^{5} + 381732 x^{4} + 82648 x^{3} - 55776 x^{2} + 1472 x + 529 \)

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:  \(1062004354962295536520003584=2^{46}\cdot 3^{12}\cdot 73^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $48.88$
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:  $2, 3, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{8} - \frac{1}{4} a^{4} + \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{9} - \frac{1}{4} a^{5} + \frac{1}{4} a$, $\frac{1}{644} a^{14} - \frac{9}{322} a^{13} - \frac{25}{322} a^{12} + \frac{51}{644} a^{11} - \frac{19}{161} a^{10} - \frac{57}{644} a^{9} + \frac{103}{644} a^{8} + \frac{8}{161} a^{7} + \frac{121}{644} a^{6} + \frac{137}{322} a^{5} + \frac{29}{161} a^{4} - \frac{179}{644} a^{3} + \frac{33}{322} a^{2} - \frac{1}{92} a + \frac{5}{28}$, $\frac{1}{2542202317033793921295344122225988} a^{15} + \frac{1301004081720327305266916107525}{2542202317033793921295344122225988} a^{14} + \frac{123416905469425330878653478615051}{2542202317033793921295344122225988} a^{13} + \frac{3213112862625228918535617420839}{2542202317033793921295344122225988} a^{12} - \frac{286631114067172426276014895855263}{2542202317033793921295344122225988} a^{11} - \frac{361084201857402373572766014495025}{2542202317033793921295344122225988} a^{10} + \frac{27183126146698771741904010700597}{110530535523208431360667135748956} a^{9} + \frac{70432199643358420183802180235125}{2542202317033793921295344122225988} a^{8} - \frac{304518428186945802252417119608817}{2542202317033793921295344122225988} a^{7} - \frac{235062679196817233213327720544489}{2542202317033793921295344122225988} a^{6} + \frac{53431237257771441973871264178661}{2542202317033793921295344122225988} a^{5} + \frac{1053803507535856482179524847217641}{2542202317033793921295344122225988} a^{4} + \frac{1145204168930272741255163398776579}{2542202317033793921295344122225988} a^{3} - \frac{169940789013296439890314973388529}{363171759576256274470763446032284} a^{2} - \frac{1035194979369200991448890369587515}{2542202317033793921295344122225988} a - \frac{33235147850017862793878011285}{15790076503315490194381019392708}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 47221753.7055 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3$ (as 16T373):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 26 conjugacy class representatives for $C_2^4.C_2^3$
Character table for $C_2^4.C_2^3$ is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{6}) \), 4.4.2018304.1, \(\Q(\sqrt{2}, \sqrt{3})\), 4.4.2018304.2, 8.8.16294204145664.5, 8.4.1550057472.2, 8.4.223208275968.1

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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$3$3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
$73$73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.4.2.1$x^{4} + 1533 x^{2} + 644809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
73.4.2.1$x^{4} + 1533 x^{2} + 644809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$