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Properties

Label 16.8.57573646968...1776.2
Degree $16$
Signature $[8, 4]$
Discriminant $2^{56}\cdot 3^{14}\cdot 13^{6}\cdot 29^{4}\cdot 107^{4}\cdot 139^{4}$
Root discriminant $1983.87$
Ramified primes $2, 3, 13, 29, 107, 139$
Class number Not computed
Class group Not computed
Galois group $C_2^2.C_2^5.C_2$ (as 16T493)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![52639955977261288880155041, 0, -319644470150330689099008, 0, -2323601006970084513642, 0, -1012903008929829936, 0, 835168685147736, 0, 404452104432, 0, -54761946, 0, -26256, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 26256*x^14 - 54761946*x^12 + 404452104432*x^10 + 835168685147736*x^8 - 1012903008929829936*x^6 - 2323601006970084513642*x^4 - 319644470150330689099008*x^2 + 52639955977261288880155041)
 
gp: K = bnfinit(x^16 - 26256*x^14 - 54761946*x^12 + 404452104432*x^10 + 835168685147736*x^8 - 1012903008929829936*x^6 - 2323601006970084513642*x^4 - 319644470150330689099008*x^2 + 52639955977261288880155041, 1)
 

Normalized defining polynomial

\( x^{16} - 26256 x^{14} - 54761946 x^{12} + 404452104432 x^{10} + 835168685147736 x^{8} - 1012903008929829936 x^{6} - 2323601006970084513642 x^{4} - 319644470150330689099008 x^{2} + 52639955977261288880155041 \)

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:  \(57573646968444953741950777740415027177610419976011776=2^{56}\cdot 3^{14}\cdot 13^{6}\cdot 29^{4}\cdot 107^{4}\cdot 139^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1983.87$
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:  $2, 3, 13, 29, 107, 139$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8}$, $\frac{1}{3} a^{9}$, $\frac{1}{1293951} a^{10} - \frac{8752}{431317} a^{8} - \frac{138668}{431317} a^{6} + \frac{182137}{431317} a^{4} - \frac{213400}{431317} a^{2}$, $\frac{1}{1293951} a^{11} - \frac{8752}{431317} a^{9} - \frac{138668}{431317} a^{7} + \frac{182137}{431317} a^{5} - \frac{213400}{431317} a^{3}$, $\frac{1}{1674309190401} a^{12} - \frac{8752}{558103063467} a^{10} - \frac{18253982}{558103063467} a^{8} - \frac{51216986345}{558103063467} a^{6} + \frac{268201754857}{558103063467} a^{4} + \frac{121944}{431317} a^{2}$, $\frac{1}{1674309190401} a^{13} - \frac{8752}{558103063467} a^{11} - \frac{18253982}{558103063467} a^{9} - \frac{51216986345}{558103063467} a^{7} + \frac{268201754857}{558103063467} a^{5} + \frac{121944}{431317} a^{3}$, $\frac{1}{61589251976113128701273366145209266683439230869627028292079583566085648603513} a^{14} + \frac{8054948598863476486725437892272897983041967496132816661635765623}{61589251976113128701273366145209266683439230869627028292079583566085648603513} a^{12} - \frac{553556335878103861125559498455523834172886397767621715890214480014518}{1866340968973125112159798974097250505558764571806879645214532835335928745561} a^{10} - \frac{248302352254484150943329342197988122372511478226725603272340048611194367824}{20529750658704376233757788715069755561146410289875676097359861188695216201171} a^{8} - \frac{6803475824068285276050514864620707993435876515925693352442916203421625303250}{20529750658704376233757788715069755561146410289875676097359861188695216201171} a^{6} - \frac{5686042052015538431683885876443428023961811694830311312421662376930950}{47597824010424760057585925699821142132460372046257569484531936345414663} a^{4} + \frac{2187359657247429389620557911696719784935159871862085961421278197}{36784873623827146512955997328972381591312477865280500949828808313} a^{2} - \frac{3012838793691740697105750322521553933568459204128612347022}{6560385200145876379866957985920471769971163073755765382953}$, $\frac{1}{61589251976113128701273366145209266683439230869627028292079583566085648603513} a^{15} + \frac{8054948598863476486725437892272897983041967496132816661635765623}{61589251976113128701273366145209266683439230869627028292079583566085648603513} a^{13} - \frac{553556335878103861125559498455523834172886397767621715890214480014518}{1866340968973125112159798974097250505558764571806879645214532835335928745561} a^{11} - \frac{248302352254484150943329342197988122372511478226725603272340048611194367824}{20529750658704376233757788715069755561146410289875676097359861188695216201171} a^{9} - \frac{6803475824068285276050514864620707993435876515925693352442916203421625303250}{20529750658704376233757788715069755561146410289875676097359861188695216201171} a^{7} - \frac{5686042052015538431683885876443428023961811694830311312421662376930950}{47597824010424760057585925699821142132460372046257569484531936345414663} a^{5} + \frac{2187359657247429389620557911696719784935159871862085961421278197}{36784873623827146512955997328972381591312477865280500949828808313} a^{3} - \frac{3012838793691740697105750322521553933568459204128612347022}{6560385200145876379866957985920471769971163073755765382953} a$

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

Class group and class number

Not computed

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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2.C_2^5.C_2$ (as 16T493):

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 $C_2^2.C_2^5.C_2$
Character table for $C_2^2.C_2^5.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), 4.4.7488.1, 4.4.179712.2, 4.4.13824.1, 8.8.129185611776.1

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 ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
2Data not computed
3Data not computed
$13$13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$29$29.4.2.2$x^{4} - 29 x^{2} + 2523$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
29.4.2.2$x^{4} - 29 x^{2} + 2523$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
107Data not computed
$139$139.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
139.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
139.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
139.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
139.4.2.1$x^{4} + 417 x^{2} + 77284$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
139.4.2.1$x^{4} + 417 x^{2} + 77284$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$