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Properties

Label 16.0.45892197269...7136.5
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 3^{12}\cdot 17^{6}\cdot 97^{6}$
Root discriminant $146.68$
Ramified primes $2, 3, 17, 97$
Class number $320$ (GRH)
Class group $[2, 2, 2, 40]$ (GRH)
Galois group $C_2^3.C_2^4.C_2$ (as 16T657)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4376380753, 991021888, -541516688, -16484472, 112548612, 4311208, -1572804, -1038000, 1383262, 90376, -21320, -1672, 3933, 144, -52, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 - 52*x^14 + 144*x^13 + 3933*x^12 - 1672*x^11 - 21320*x^10 + 90376*x^9 + 1383262*x^8 - 1038000*x^7 - 1572804*x^6 + 4311208*x^5 + 112548612*x^4 - 16484472*x^3 - 541516688*x^2 + 991021888*x + 4376380753)
 
gp: K = bnfinit(x^16 - 8*x^15 - 52*x^14 + 144*x^13 + 3933*x^12 - 1672*x^11 - 21320*x^10 + 90376*x^9 + 1383262*x^8 - 1038000*x^7 - 1572804*x^6 + 4311208*x^5 + 112548612*x^4 - 16484472*x^3 - 541516688*x^2 + 991021888*x + 4376380753, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} - 52 x^{14} + 144 x^{13} + 3933 x^{12} - 1672 x^{11} - 21320 x^{10} + 90376 x^{9} + 1383262 x^{8} - 1038000 x^{7} - 1572804 x^{6} + 4311208 x^{5} + 112548612 x^{4} - 16484472 x^{3} - 541516688 x^{2} + 991021888 x + 4376380753 \)

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:  \(45892197269607041217434019101147136=2^{32}\cdot 3^{12}\cdot 17^{6}\cdot 97^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $146.68$
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 not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{5} a^{14} - \frac{2}{5} a^{13} + \frac{1}{5} a^{11} - \frac{1}{5} a^{10} + \frac{1}{5} a^{9} + \frac{2}{5} a^{8} + \frac{2}{5} a^{7} + \frac{2}{5} a^{6} - \frac{1}{5} a^{4} + \frac{2}{5} a^{3} + \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{202044034211035386419951052847517437283045370794110105651305387525653165} a^{15} + \frac{10111714939601051222028479033859201581250726182650090191197302214732226}{202044034211035386419951052847517437283045370794110105651305387525653165} a^{14} + \frac{81704417645273133306038061883170891134761993965357744030293376490345564}{202044034211035386419951052847517437283045370794110105651305387525653165} a^{13} + \frac{73225332383311214816646769490206901101034919593246171671623529078978266}{202044034211035386419951052847517437283045370794110105651305387525653165} a^{12} - \frac{19060102687000728759949426746347438678545299423724825448372756728742648}{202044034211035386419951052847517437283045370794110105651305387525653165} a^{11} + \frac{57995097005776438187458404383557584218951896381792279947093860474892518}{202044034211035386419951052847517437283045370794110105651305387525653165} a^{10} + \frac{14873509680190564349252305685342213887226885212943878231613782361148963}{40408806842207077283990210569503487456609074158822021130261077505130633} a^{9} + \frac{65842212201899274280362435293764961224128519417413353014993517132628818}{202044034211035386419951052847517437283045370794110105651305387525653165} a^{8} + \frac{79360183137304680040654508318839086513708697886857793140116666410684563}{202044034211035386419951052847517437283045370794110105651305387525653165} a^{7} + \frac{72708162325189277320543748021455529404928322041038316874598767689306626}{202044034211035386419951052847517437283045370794110105651305387525653165} a^{6} + \frac{95380896335765695718010931987581627791391115376531563426699814952052854}{202044034211035386419951052847517437283045370794110105651305387525653165} a^{5} - \frac{41044339659012667670250242937836494002594296680640135512529754868643256}{202044034211035386419951052847517437283045370794110105651305387525653165} a^{4} + \frac{36701039922129146579483776893197097215872756330682430146870399425113451}{202044034211035386419951052847517437283045370794110105651305387525653165} a^{3} - \frac{8915317346393905829346066947104448571159060900012393418933487112461129}{202044034211035386419951052847517437283045370794110105651305387525653165} a^{2} + \frac{66431728129989798367062002036143606528101391191530000695766351772302376}{202044034211035386419951052847517437283045370794110105651305387525653165} a - \frac{95983268479983187672159277015706177392771067531894114326801636499514681}{202044034211035386419951052847517437283045370794110105651305387525653165}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{40}$, which has order $320$ (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:  \( 135824302.167 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_2^4.C_2$ (as 16T657):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 34 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

\(\Q(\sqrt{6}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{3}) \), 4.0.422144.1, 4.0.949824.1, \(\Q(\sqrt{2}, \sqrt{3})\), 8.0.14434650095616.3

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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
3Data not computed
$17$17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.2.2$x^{4} - 17 x^{2} + 867$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
97Data not computed