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Properties

Label 16.8.66688975910...2433.3
Degree $16$
Signature $[8, 4]$
Discriminant $13^{12}\cdot 17^{15}$
Root discriminant $97.50$
Ramified primes $13, 17$
Class number $8$ (GRH)
Class group $[2, 4]$ (GRH)
Galois group $(C_2\times C_8).D_4$ (as 16T306)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![21153679, 8743206, -5505995, -5407891, -2204236, 778026, 99993, -67715, 73917, -1016, -1667, 999, -424, 21, -1, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - x^14 + 21*x^13 - 424*x^12 + 999*x^11 - 1667*x^10 - 1016*x^9 + 73917*x^8 - 67715*x^7 + 99993*x^6 + 778026*x^5 - 2204236*x^4 - 5407891*x^3 - 5505995*x^2 + 8743206*x + 21153679)
 
gp: K = bnfinit(x^16 - 4*x^15 - x^14 + 21*x^13 - 424*x^12 + 999*x^11 - 1667*x^10 - 1016*x^9 + 73917*x^8 - 67715*x^7 + 99993*x^6 + 778026*x^5 - 2204236*x^4 - 5407891*x^3 - 5505995*x^2 + 8743206*x + 21153679, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - x^{14} + 21 x^{13} - 424 x^{12} + 999 x^{11} - 1667 x^{10} - 1016 x^{9} + 73917 x^{8} - 67715 x^{7} + 99993 x^{6} + 778026 x^{5} - 2204236 x^{4} - 5407891 x^{3} - 5505995 x^{2} + 8743206 x + 21153679 \)

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:  \(66688975910627504451630153142433=13^{12}\cdot 17^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $97.50$
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:  $13, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{13} a^{8} - \frac{2}{13} a^{7} + \frac{4}{13} a^{6} - \frac{1}{13} a^{5} - \frac{1}{13} a^{4} + \frac{1}{13} a^{3} + \frac{4}{13} a^{2} + \frac{2}{13} a + \frac{1}{13}$, $\frac{1}{13} a^{9} - \frac{6}{13} a^{6} - \frac{3}{13} a^{5} - \frac{1}{13} a^{4} + \frac{6}{13} a^{3} - \frac{3}{13} a^{2} + \frac{5}{13} a + \frac{2}{13}$, $\frac{1}{13} a^{10} - \frac{6}{13} a^{7} - \frac{3}{13} a^{6} - \frac{1}{13} a^{5} + \frac{6}{13} a^{4} - \frac{3}{13} a^{3} + \frac{5}{13} a^{2} + \frac{2}{13} a$, $\frac{1}{13} a^{11} - \frac{2}{13} a^{7} - \frac{3}{13} a^{6} + \frac{4}{13} a^{4} - \frac{2}{13} a^{3} - \frac{1}{13} a + \frac{6}{13}$, $\frac{1}{26} a^{12} - \frac{1}{26} a^{10} - \frac{1}{26} a^{8} - \frac{6}{13} a^{7} - \frac{3}{13} a^{6} - \frac{9}{26} a^{5} + \frac{2}{13} a^{4} - \frac{9}{26} a^{3} - \frac{1}{13} a^{2} - \frac{7}{26} a + \frac{1}{26}$, $\frac{1}{26} a^{13} - \frac{1}{26} a^{11} - \frac{1}{26} a^{9} - \frac{2}{13} a^{7} - \frac{1}{2} a^{6} - \frac{4}{13} a^{5} + \frac{5}{26} a^{4} + \frac{5}{13} a^{3} - \frac{11}{26} a^{2} - \frac{1}{26} a + \frac{6}{13}$, $\frac{1}{503854} a^{14} + \frac{2533}{503854} a^{13} - \frac{8877}{503854} a^{12} + \frac{7}{503854} a^{11} - \frac{15709}{503854} a^{10} - \frac{18393}{503854} a^{9} - \frac{3173}{251927} a^{8} + \frac{30911}{503854} a^{7} - \frac{172827}{503854} a^{6} - \frac{102281}{503854} a^{5} - \frac{32427}{503854} a^{4} + \frac{70985}{503854} a^{3} + \frac{84073}{251927} a^{2} - \frac{107943}{503854} a - \frac{21482}{251927}$, $\frac{1}{26269211257704686895231630900345056129918} a^{15} - \frac{2061056797578473093361582089809909}{13134605628852343447615815450172528064959} a^{14} + \frac{84037611755570374419469636840836608841}{26269211257704686895231630900345056129918} a^{13} - \frac{79546471701104220074633347519286021925}{26269211257704686895231630900345056129918} a^{12} + \frac{482702550446788168438447501473543947197}{26269211257704686895231630900345056129918} a^{11} - \frac{286806461586915074593050476526166695485}{26269211257704686895231630900345056129918} a^{10} - \frac{417898943440291370887632198161543724112}{13134605628852343447615815450172528064959} a^{9} - \frac{250107163842048837391022001309297752931}{13134605628852343447615815450172528064959} a^{8} - \frac{3502493199978022128794801919751636455014}{13134605628852343447615815450172528064959} a^{7} + \frac{10464999477133287292350152316549431552593}{26269211257704686895231630900345056129918} a^{6} - \frac{5434408267611834428262255516697439672471}{26269211257704686895231630900345056129918} a^{5} - \frac{3048889587333034280328988638103094132029}{26269211257704686895231630900345056129918} a^{4} + \frac{3801348931703655595844302943755894043549}{13134605628852343447615815450172528064959} a^{3} + \frac{207408265188569518942580508057219838609}{1010354279142487957508908880782502158843} a^{2} - \frac{153560266417488030770208604292683352939}{2020708558284975915017817761565004317686} a + \frac{7733463200826619994563341053543287151741}{26269211257704686895231630900345056129918}$

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

Class group and class number

$C_{2}\times C_{4}$, which has order $8$ (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:  $11$
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:  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:  \( 1020867179.1 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times C_8).D_4$ (as 16T306):

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 $(C_2\times C_8).D_4$
Character table for $(C_2\times C_8).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, 8.8.11719682839553.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 16 sibling: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ R R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$13$13.4.3.4$x^{4} + 104$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.3$x^{4} + 26$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.4$x^{4} + 104$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.4$x^{4} + 104$$4$$1$$3$$C_4$$[\ ]_{4}$
17Data not computed