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Properties

Label 16.0.55086646241...0625.9
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 41^{12}$
Root discriminant $54.18$
Ramified primes $5, 41$
Class number $144$ (GRH)
Class group $[12, 12]$ (GRH)
Galois group $C_4:C_4$ (as 16T8)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16777216, -10485760, 2097152, 491520, -471040, 135680, -4608, -5400, 649, -675, -72, 265, -115, 15, 8, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 8*x^14 + 15*x^13 - 115*x^12 + 265*x^11 - 72*x^10 - 675*x^9 + 649*x^8 - 5400*x^7 - 4608*x^6 + 135680*x^5 - 471040*x^4 + 491520*x^3 + 2097152*x^2 - 10485760*x + 16777216)
 
gp: K = bnfinit(x^16 - 5*x^15 + 8*x^14 + 15*x^13 - 115*x^12 + 265*x^11 - 72*x^10 - 675*x^9 + 649*x^8 - 5400*x^7 - 4608*x^6 + 135680*x^5 - 471040*x^4 + 491520*x^3 + 2097152*x^2 - 10485760*x + 16777216, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 8 x^{14} + 15 x^{13} - 115 x^{12} + 265 x^{11} - 72 x^{10} - 675 x^{9} + 649 x^{8} - 5400 x^{7} - 4608 x^{6} + 135680 x^{5} - 471040 x^{4} + 491520 x^{3} + 2097152 x^{2} - 10485760 x + 16777216 \)

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:  \(5508664624112838398681640625=5^{12}\cdot 41^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $54.18$
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:  $5, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{8} a^{9} + \frac{3}{8} a^{8} - \frac{1}{8} a^{6} - \frac{3}{8} a^{5} + \frac{1}{8} a^{4} - \frac{3}{8} a^{2} + \frac{1}{8} a$, $\frac{1}{64} a^{10} + \frac{3}{64} a^{9} - \frac{1}{2} a^{8} + \frac{15}{64} a^{7} + \frac{5}{64} a^{6} - \frac{15}{64} a^{5} + \frac{29}{64} a^{3} - \frac{15}{64} a^{2} - \frac{1}{4} a$, $\frac{1}{512} a^{11} + \frac{3}{512} a^{10} - \frac{1}{16} a^{9} + \frac{79}{512} a^{8} + \frac{5}{512} a^{7} - \frac{143}{512} a^{6} - \frac{227}{512} a^{4} - \frac{143}{512} a^{3} - \frac{13}{32} a^{2} - \frac{3}{8} a$, $\frac{1}{749568} a^{12} - \frac{325}{749568} a^{11} + \frac{65}{93696} a^{10} + \frac{5061}{249856} a^{9} + \frac{51917}{749568} a^{8} + \frac{80643}{249856} a^{7} + \frac{44455}{93696} a^{6} - \frac{122785}{249856} a^{5} + \frac{335945}{749568} a^{4} - \frac{10193}{31232} a^{3} + \frac{4183}{11712} a^{2} + \frac{107}{366} a + \frac{64}{183}$, $\frac{1}{5996544} a^{13} + \frac{1}{1998848} a^{12} - \frac{7}{62464} a^{11} + \frac{33487}{5996544} a^{10} + \frac{253445}{5996544} a^{9} - \frac{636943}{5996544} a^{8} + \frac{9763}{46848} a^{7} + \frac{2171549}{5996544} a^{6} - \frac{2521231}{5996544} a^{5} + \frac{35435}{374784} a^{4} + \frac{12595}{93696} a^{3} - \frac{1783}{5856} a^{2} + \frac{593}{1464} a + \frac{62}{183}$, $\frac{1}{619754815488} a^{14} - \frac{10813}{619754815488} a^{13} + \frac{1901}{6455779328} a^{12} + \frac{402802063}{619754815488} a^{11} - \frac{3868454843}{619754815488} a^{10} + \frac{4145729403}{206584938496} a^{9} - \frac{242985}{1613944832} a^{8} - \frac{26740817505}{206584938496} a^{7} + \frac{73585399995}{206584938496} a^{6} - \frac{3933552503}{12911558656} a^{5} + \frac{732153683}{2420917248} a^{4} + \frac{259790419}{605229312} a^{3} + \frac{1326391}{50435776} a^{2} + \frac{958112}{2364177} a - \frac{387562}{2364177}$, $\frac{1}{3773067316690944} a^{15} + \frac{457}{1257689105563648} a^{14} - \frac{7763819}{471633414586368} a^{13} + \frac{2497350031}{3773067316690944} a^{12} + \frac{173419421741}{3773067316690944} a^{11} + \frac{17034055995241}{3773067316690944} a^{10} + \frac{2579606445201}{157211138195456} a^{9} + \frac{263061693751455}{1257689105563648} a^{8} + \frac{383353030636067}{1257689105563648} a^{7} + \frac{20262538228019}{157211138195456} a^{6} - \frac{7840802562071}{29477088411648} a^{5} + \frac{159616906335}{2456424034304} a^{4} + \frac{84078092683}{921159012864} a^{3} + \frac{221573905}{471905232} a^{2} + \frac{2929772009}{7196554788} a + \frac{475095490}{1799138697}$

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

Class group and class number

$C_{12}\times C_{12}$, which has order $144$ (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:  \( \frac{134577}{97351892992} a^{15} - \frac{7904184027}{1257689105563648} a^{14} - \frac{36963795}{78605569097728} a^{13} + \frac{69012584489}{1257689105563648} a^{12} - \frac{108884227053}{1257689105563648} a^{11} - \frac{32702352345}{1257689105563648} a^{10} + \frac{15047558289}{78605569097728} a^{9} - \frac{304117384245}{1257689105563648} a^{8} + \frac{646052179815}{1257689105563648} a^{7} - \frac{235378160517}{39302784548864} a^{6} - \frac{141416131185}{9825696137216} a^{5} + \frac{267269429889}{1228212017152} a^{4} - \frac{8751657687}{19190812768} a^{3} - \frac{34639084635}{38381625536} a^{2} + \frac{4762928019}{1199425798} a - \frac{2200355094}{599712899} \) (order $10$)
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:  \( 478806.025823 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4:C_4$ (as 16T8):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $C_4:C_4$
Character table for $C_4:C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{205}) \), \(\Q(\sqrt{41}) \), \(\Q(\sqrt{5}, \sqrt{41})\), 4.0.1723025.1 x2, 4.0.344605.1 x2, 4.0.210125.1, \(\Q(\zeta_{5})\), 8.0.2968815150625.2, 8.0.44152515625.1, 8.8.74220378765625.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$41$41.4.3.2$x^{4} - 1476$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.3.2$x^{4} - 1476$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.3.2$x^{4} - 1476$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.3.2$x^{4} - 1476$$4$$1$$3$$C_4$$[\ ]_{4}$