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Properties

Label 16.16.7803965950...8064.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{36}\cdot 3^{12}\cdot 17^{6}\cdot 97^{4}$
Root discriminant $98.46$
Ramified primes $2, 3, 17, 97$
Class number $2$ (GRH)
Class group $[2]$ (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![81973873, 699011236, -435859146, -423594144, 213981992, 99037596, -43341016, -11767196, 4565499, 776384, -269044, -28560, 8852, 540, -150, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 150*x^14 + 540*x^13 + 8852*x^12 - 28560*x^11 - 269044*x^10 + 776384*x^9 + 4565499*x^8 - 11767196*x^7 - 43341016*x^6 + 99037596*x^5 + 213981992*x^4 - 423594144*x^3 - 435859146*x^2 + 699011236*x + 81973873)
 
gp: K = bnfinit(x^16 - 4*x^15 - 150*x^14 + 540*x^13 + 8852*x^12 - 28560*x^11 - 269044*x^10 + 776384*x^9 + 4565499*x^8 - 11767196*x^7 - 43341016*x^6 + 99037596*x^5 + 213981992*x^4 - 423594144*x^3 - 435859146*x^2 + 699011236*x + 81973873, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 150 x^{14} + 540 x^{13} + 8852 x^{12} - 28560 x^{11} - 269044 x^{10} + 776384 x^{9} + 4565499 x^{8} - 11767196 x^{7} - 43341016 x^{6} + 99037596 x^{5} + 213981992 x^{4} - 423594144 x^{3} - 435859146 x^{2} + 699011236 x + 81973873 \)

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:  \(78039659508312536877345552728064=2^{36}\cdot 3^{12}\cdot 17^{6}\cdot 97^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $98.46$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{7} a^{14} + \frac{2}{7} a^{13} + \frac{1}{7} a^{12} - \frac{2}{7} a^{11} - \frac{2}{7} a^{10} - \frac{3}{7} a^{9} - \frac{1}{7} a^{8} - \frac{3}{7} a^{7} - \frac{2}{7} a^{6} - \frac{2}{7} a^{5} - \frac{1}{7} a^{4} + \frac{3}{7} a^{3} - \frac{2}{7} a^{2} - \frac{2}{7} a + \frac{2}{7}$, $\frac{1}{34344928357142352789736098093421774902546541768668917} a^{15} + \frac{22990717299056233863980659061108126240506438702207}{2641917565934027137672007545647828838657426289897609} a^{14} + \frac{8010872483073463667348237527574383185080052688304503}{34344928357142352789736098093421774902546541768668917} a^{13} + \frac{15210666701412559995491406607340377115356260578865876}{34344928357142352789736098093421774902546541768668917} a^{12} - \frac{12200032452364783533592257583859373160850790346991151}{34344928357142352789736098093421774902546541768668917} a^{11} + \frac{11428810510376702625641115185482619112817810348809211}{34344928357142352789736098093421774902546541768668917} a^{10} - \frac{409869661520049011825840671490233023296145009801869}{2641917565934027137672007545647828838657426289897609} a^{9} - \frac{297542861080037819202379396424931054160568200043908}{4906418336734621827105156870488824986078077395524131} a^{8} - \frac{1500687108847435427018624013279569138102484753326519}{4906418336734621827105156870488824986078077395524131} a^{7} + \frac{4143771683838982552868806828435461081869417430527942}{34344928357142352789736098093421774902546541768668917} a^{6} + \frac{16618877353724805534444063610000047024694426354121041}{34344928357142352789736098093421774902546541768668917} a^{5} - \frac{1185998128392283919047056583840967807385100142566040}{2641917565934027137672007545647828838657426289897609} a^{4} + \frac{9275456964231882186054087647473709414223607167867140}{34344928357142352789736098093421774902546541768668917} a^{3} - \frac{152887260756871948491836457556522623264270829953972}{2641917565934027137672007545647828838657426289897609} a^{2} - \frac{5959976292844407478067006794847242251739356935605604}{34344928357142352789736098093421774902546541768668917} a - \frac{15496321680620006022715166910582322178186648283261481}{34344928357142352789736098093421774902546541768668917}$

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:  \( 28790024467.5 \) (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

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{3}) \), 4.4.9792.1, 4.4.4352.1, \(\Q(\sqrt{2}, \sqrt{3})\), 8.8.1534132224.1

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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
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