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Properties

Label 16.0.10620043549...3584.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{46}\cdot 3^{12}\cdot 73^{4}$
Root discriminant $48.88$
Ramified primes $2, 3, 73$
Class number $128$ (GRH)
Class group $[2, 8, 8]$ (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![160632, -38016, 326736, -144000, 310344, -107184, 171600, -60864, 17656, -6688, 2840, -1336, 316, -32, 20, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 20*x^14 - 32*x^13 + 316*x^12 - 1336*x^11 + 2840*x^10 - 6688*x^9 + 17656*x^8 - 60864*x^7 + 171600*x^6 - 107184*x^5 + 310344*x^4 - 144000*x^3 + 326736*x^2 - 38016*x + 160632)
 
gp: K = bnfinit(x^16 - 8*x^15 + 20*x^14 - 32*x^13 + 316*x^12 - 1336*x^11 + 2840*x^10 - 6688*x^9 + 17656*x^8 - 60864*x^7 + 171600*x^6 - 107184*x^5 + 310344*x^4 - 144000*x^3 + 326736*x^2 - 38016*x + 160632, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 20 x^{14} - 32 x^{13} + 316 x^{12} - 1336 x^{11} + 2840 x^{10} - 6688 x^{9} + 17656 x^{8} - 60864 x^{7} + 171600 x^{6} - 107184 x^{5} + 310344 x^{4} - 144000 x^{3} + 326736 x^{2} - 38016 x + 160632 \)

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:  \(1062004354962295536520003584=2^{46}\cdot 3^{12}\cdot 73^{4}\)
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{4} a^{11}$, $\frac{1}{36} a^{12} + \frac{1}{36} a^{11} + \frac{2}{9} a^{10} + \frac{1}{9} a^{9} + \frac{1}{9} a^{8} - \frac{1}{9} a^{7} + \frac{2}{9} a^{6} - \frac{4}{9} a^{5} + \frac{1}{9} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{36} a^{13} - \frac{1}{18} a^{11} - \frac{1}{9} a^{10} - \frac{2}{9} a^{8} - \frac{1}{6} a^{7} - \frac{1}{6} a^{6} - \frac{4}{9} a^{5} - \frac{4}{9} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3}$, $\frac{1}{36} a^{14} - \frac{1}{18} a^{11} - \frac{1}{18} a^{10} + \frac{1}{18} a^{8} + \frac{1}{9} a^{7} - \frac{1}{3} a^{5} - \frac{1}{9} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{983495575499646013816833375786219569652} a^{15} - \frac{12622486814205186403358005327989637289}{983495575499646013816833375786219569652} a^{14} - \frac{2010314743339574272750212465870031135}{163915929249941002302805562631036594942} a^{13} - \frac{12227356938571028974563959598539730985}{983495575499646013816833375786219569652} a^{12} + \frac{18595141955145929964718529369556563237}{163915929249941002302805562631036594942} a^{11} + \frac{6707852367138326780543377981012531039}{163915929249941002302805562631036594942} a^{10} - \frac{38082772561497470670351248002685227595}{163915929249941002302805562631036594942} a^{9} - \frac{3309668998637313991142331069221767759}{14463170227935970791424020232150287789} a^{8} - \frac{15136234596899663112677274701172205466}{81957964624970501151402781315518297471} a^{7} + \frac{38270277587340290568319819709203233062}{245873893874911503454208343946554892413} a^{6} - \frac{102587872720144355372369215558778204756}{245873893874911503454208343946554892413} a^{5} + \frac{2551053416563767256582394953031740493}{81957964624970501151402781315518297471} a^{4} - \frac{16870350513779791012112205691653576121}{81957964624970501151402781315518297471} a^{3} - \frac{93713042637675643366313014087055450}{11708280660710071593057540187931185353} a^{2} + \frac{21518543395583122980433606069404171781}{81957964624970501151402781315518297471} a - \frac{104122358916341577634119867866630739}{281642490120173543475610932355732981}$

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

Class group and class number

$C_{2}\times C_{8}\times C_{8}$, which has order $128$ (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:  \( 120190.797574 \) (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

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{2}, \sqrt{3})\), 4.0.1009152.1, 4.0.1009152.2, 8.0.16294204145664.2, 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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$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}$