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Properties

Label 16.0.48794117818...0625.2
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 29^{4}\cdot 41^{4}$
Root discriminant $19.63$
Ramified primes $5, 29, 41$
Class number $2$
Class group $[2]$
Galois group $C_2^5.C_2$ (as 16T79)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6241, -1343, -192, -332, -362, -1583, 929, 19, 158, -123, 141, -70, -2, -1, 2, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 2*x^14 - x^13 - 2*x^12 - 70*x^11 + 141*x^10 - 123*x^9 + 158*x^8 + 19*x^7 + 929*x^6 - 1583*x^5 - 362*x^4 - 332*x^3 - 192*x^2 - 1343*x + 6241)
 
gp: K = bnfinit(x^16 - 2*x^15 + 2*x^14 - x^13 - 2*x^12 - 70*x^11 + 141*x^10 - 123*x^9 + 158*x^8 + 19*x^7 + 929*x^6 - 1583*x^5 - 362*x^4 - 332*x^3 - 192*x^2 - 1343*x + 6241, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 2 x^{14} - x^{13} - 2 x^{12} - 70 x^{11} + 141 x^{10} - 123 x^{9} + 158 x^{8} + 19 x^{7} + 929 x^{6} - 1583 x^{5} - 362 x^{4} - 332 x^{3} - 192 x^{2} - 1343 x + 6241 \)

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:  \(487941178183837890625=5^{12}\cdot 29^{4}\cdot 41^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.63$
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:  $5, 29, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{8618} a^{14} - \frac{1771}{8618} a^{13} + \frac{1015}{4309} a^{12} + \frac{1375}{8618} a^{11} + \frac{1353}{8618} a^{10} - \frac{117}{8618} a^{9} + \frac{997}{4309} a^{8} + \frac{535}{4309} a^{7} + \frac{1638}{4309} a^{6} + \frac{230}{4309} a^{5} + \frac{2439}{8618} a^{4} - \frac{1073}{4309} a^{3} + \frac{3973}{8618} a^{2} - \frac{1382}{4309} a + \frac{1084}{4309}$, $\frac{1}{42441760526863469710378} a^{15} + \frac{1662499782646896353}{42441760526863469710378} a^{14} - \frac{3356150645194546060166}{21220880263431734855189} a^{13} - \frac{80290074595074018017}{719351873336668978142} a^{12} - \frac{6417945313320881197}{62231320420620923329} a^{11} + \frac{4286956418918364682335}{21220880263431734855189} a^{10} + \frac{17598692171728520624199}{42441760526863469710378} a^{9} - \frac{11440288695000935981727}{42441760526863469710378} a^{8} - \frac{76501077256234540764}{268618737511794112091} a^{7} - \frac{3880276553662596621947}{42441760526863469710378} a^{6} + \frac{19892752527296221238601}{42441760526863469710378} a^{5} + \frac{158299097282834186255}{719351873336668978142} a^{4} - \frac{10161153287901097444217}{21220880263431734855189} a^{3} + \frac{323849106056873392966}{684544524626830156619} a^{2} + \frac{5413928596809059549639}{21220880263431734855189} a + \frac{3676290144692699091}{537237475023588224182}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( \frac{92264872579604276}{152668203334041257951} a^{15} - \frac{4406957200476015}{305336406668082515902} a^{14} + \frac{234005090517368381}{305336406668082515902} a^{13} + \frac{13195175322381869}{5175193333357330778} a^{12} + \frac{51025277566374707}{27757855151643865082} a^{11} - \frac{5848258757683258584}{152668203334041257951} a^{10} + \frac{2761569217922381159}{305336406668082515902} a^{9} - \frac{4519825600479675209}{152668203334041257951} a^{8} - \frac{289206055611859483}{3865017805925095138} a^{7} - \frac{247521067192834129}{305336406668082515902} a^{6} + \frac{76034068571275579932}{152668203334041257951} a^{5} + \frac{306521850217687950}{2587596666678665389} a^{4} - \frac{70561521861773684263}{305336406668082515902} a^{3} + \frac{56875579722137415193}{152668203334041257951} a^{2} - \frac{41288391908173626561}{152668203334041257951} a - \frac{3583318938837835874}{1932508902962547569} \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 19099.2258509 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^5.C_2$ (as 16T79):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 28 conjugacy class representatives for $C_2^5.C_2$
Character table for $C_2^5.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.29725.2, 4.0.3625.1, 4.0.1025.1, \(\Q(\zeta_{5})\), 4.4.5125.1, 4.4.725.1, 4.4.148625.2, 8.8.22089390625.1, 8.0.22089390625.5, 8.0.26265625.1, 8.0.883575625.2, 8.0.13140625.1, 8.0.22089390625.3, 8.0.22089390625.7

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ 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}$
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$41$41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$