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Properties

Label 16.16.3603117672...0000.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{32}\cdot 5^{4}\cdot 41^{10}$
Root discriminant $60.92$
Ramified primes $2, 5, 41$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $OD_{16}:C_2^2$ (as 16T106)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1471, -66092, -161348, 146816, 448317, -131408, -337526, 40592, 107128, -5040, -16978, 256, 1421, -4, -60, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 60*x^14 - 4*x^13 + 1421*x^12 + 256*x^11 - 16978*x^10 - 5040*x^9 + 107128*x^8 + 40592*x^7 - 337526*x^6 - 131408*x^5 + 448317*x^4 + 146816*x^3 - 161348*x^2 - 66092*x - 1471)
 
gp: K = bnfinit(x^16 - 60*x^14 - 4*x^13 + 1421*x^12 + 256*x^11 - 16978*x^10 - 5040*x^9 + 107128*x^8 + 40592*x^7 - 337526*x^6 - 131408*x^5 + 448317*x^4 + 146816*x^3 - 161348*x^2 - 66092*x - 1471, 1)
 

Normalized defining polynomial

\( x^{16} - 60 x^{14} - 4 x^{13} + 1421 x^{12} + 256 x^{11} - 16978 x^{10} - 5040 x^{9} + 107128 x^{8} + 40592 x^{7} - 337526 x^{6} - 131408 x^{5} + 448317 x^{4} + 146816 x^{3} - 161348 x^{2} - 66092 x - 1471 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(36031176726534051919298560000=2^{32}\cdot 5^{4}\cdot 41^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $60.92$
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:  $2, 5, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2}$, $\frac{1}{34} a^{13} - \frac{4}{17} a^{12} - \frac{1}{17} a^{11} + \frac{3}{34} a^{10} - \frac{5}{34} a^{9} + \frac{1}{34} a^{8} + \frac{9}{34} a^{7} + \frac{6}{17} a^{6} + \frac{4}{17} a^{5} - \frac{7}{34} a^{4} + \frac{13}{34} a^{3} + \frac{11}{34} a^{2} - \frac{6}{17} a - \frac{5}{17}$, $\frac{1}{68} a^{14} - \frac{1}{68} a^{13} + \frac{5}{34} a^{12} - \frac{11}{68} a^{11} + \frac{4}{17} a^{10} - \frac{1}{4} a^{9} - \frac{1}{68} a^{8} + \frac{6}{17} a^{7} + \frac{7}{68} a^{6} + \frac{15}{68} a^{5} - \frac{1}{34} a^{4} - \frac{1}{4} a^{3} - \frac{5}{17} a^{2} - \frac{9}{68} a - \frac{19}{68}$, $\frac{1}{29111674794356389867516} a^{15} + \frac{114801349513463095663}{29111674794356389867516} a^{14} + \frac{14789494738369739634}{7277918698589097466879} a^{13} + \frac{2412484698744464858157}{29111674794356389867516} a^{12} + \frac{169991795848410927003}{14555837397178194933758} a^{11} - \frac{2676469024488991125977}{29111674794356389867516} a^{10} + \frac{1382948726136368532017}{29111674794356389867516} a^{9} - \frac{13021318310959323680}{68017931762514929597} a^{8} - \frac{194264796352501281361}{29111674794356389867516} a^{7} - \frac{13185516841348559495129}{29111674794356389867516} a^{6} + \frac{192535349010056118751}{428112864622888086287} a^{5} + \frac{14274490438642096751803}{29111674794356389867516} a^{4} + \frac{1241463407742742231}{141318809681341698386} a^{3} + \frac{13450126688229141231}{1712451458491552345148} a^{2} + \frac{8931584938309672624079}{29111674794356389867516} a + \frac{2958430870677535315491}{7277918698589097466879}$

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:  $15$
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:  \( 839195284.563 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$OD_{16}:C_2^2$ (as 16T106):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 22 conjugacy class representatives for $OD_{16}:C_2^2$
Character table for $OD_{16}:C_2^2$ is not computed

Intermediate fields

\(\Q(\sqrt{41}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{82}) \), 4.4.13448.1 x2, 4.4.2624.1 x2, \(\Q(\sqrt{2}, \sqrt{41})\), 8.8.11574317056.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$2.8.16.3$x^{8} + 2 x^{6} + 6 x^{4} + 4 x^{2} + 8 x + 28$$4$$2$$16$$C_4\times C_2$$[2, 3]^{2}$
2.8.16.3$x^{8} + 2 x^{6} + 6 x^{4} + 4 x^{2} + 8 x + 28$$4$$2$$16$$C_4\times C_2$$[2, 3]^{2}$
$5$5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$41$41.8.4.1$x^{8} + 57154 x^{4} - 68921 x^{2} + 816644929$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
41.8.6.2$x^{8} + 943 x^{4} + 242064$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$