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Properties

Label 16.12.1207094277...1881.1
Degree $16$
Signature $[12, 2]$
Discriminant $43^{2}\cdot 97^{14}$
Root discriminant $87.62$
Ramified primes $43, 97$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_2^5.C_2.C_2$ (as 16T258)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![252041, -1286813, -2970765, 147768, 2696351, 677252, -1159099, -279513, 258825, 45135, -31582, -3558, 2119, 136, -73, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 73*x^14 + 136*x^13 + 2119*x^12 - 3558*x^11 - 31582*x^10 + 45135*x^9 + 258825*x^8 - 279513*x^7 - 1159099*x^6 + 677252*x^5 + 2696351*x^4 + 147768*x^3 - 2970765*x^2 - 1286813*x + 252041)
 
gp: K = bnfinit(x^16 - 2*x^15 - 73*x^14 + 136*x^13 + 2119*x^12 - 3558*x^11 - 31582*x^10 + 45135*x^9 + 258825*x^8 - 279513*x^7 - 1159099*x^6 + 677252*x^5 + 2696351*x^4 + 147768*x^3 - 2970765*x^2 - 1286813*x + 252041, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 73 x^{14} + 136 x^{13} + 2119 x^{12} - 3558 x^{11} - 31582 x^{10} + 45135 x^{9} + 258825 x^{8} - 279513 x^{7} - 1159099 x^{6} + 677252 x^{5} + 2696351 x^{4} + 147768 x^{3} - 2970765 x^{2} - 1286813 x + 252041 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(12070942770246634791690899381881=43^{2}\cdot 97^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $87.62$
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:  $43, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{12} a^{12} - \frac{1}{12} a^{11} - \frac{1}{6} a^{10} - \frac{1}{6} a^{8} - \frac{5}{12} a^{7} - \frac{5}{12} a^{6} + \frac{1}{6} a^{5} - \frac{5}{12} a^{4} + \frac{1}{4} a^{3} - \frac{1}{12} a^{2} - \frac{5}{12} a - \frac{5}{12}$, $\frac{1}{12} a^{13} - \frac{1}{4} a^{11} - \frac{1}{6} a^{10} - \frac{1}{6} a^{9} - \frac{1}{12} a^{8} + \frac{1}{6} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} + \frac{1}{3} a^{4} + \frac{1}{6} a^{3} - \frac{1}{3} a + \frac{1}{12}$, $\frac{1}{1236} a^{14} - \frac{7}{618} a^{13} - \frac{25}{618} a^{12} - \frac{5}{412} a^{11} - \frac{5}{206} a^{10} + \frac{71}{412} a^{9} - \frac{31}{309} a^{8} - \frac{55}{206} a^{7} - \frac{29}{309} a^{6} + \frac{46}{103} a^{5} - \frac{41}{1236} a^{4} - \frac{211}{1236} a^{3} - \frac{467}{1236} a^{2} + \frac{28}{309} a + \frac{5}{12}$, $\frac{1}{1376654722282460135521486512876} a^{15} - \frac{46842267513554745319479238}{344163680570615033880371628219} a^{14} - \frac{7533830944744229429421276745}{688327361141230067760743256438} a^{13} + \frac{6979232982591361299229297343}{688327361141230067760743256438} a^{12} + \frac{51134671740974503750614744715}{458884907427486711840495504292} a^{11} + \frac{44595362960218964133389515223}{1376654722282460135521486512876} a^{10} - \frac{3689015996308398453750968729}{344163680570615033880371628219} a^{9} - \frac{29828769542335837469653841177}{229442453713743355920247752146} a^{8} - \frac{258159937468127866003811838067}{1376654722282460135521486512876} a^{7} + \frac{639405232164039554772706850191}{1376654722282460135521486512876} a^{6} - \frac{244176712443052951977162304691}{1376654722282460135521486512876} a^{5} + \frac{23670287184938731222725484477}{688327361141230067760743256438} a^{4} - \frac{282723940771976850611443024513}{688327361141230067760743256438} a^{3} - \frac{234843019512816723859490163631}{1376654722282460135521486512876} a^{2} + \frac{200610342979086025516918050005}{688327361141230067760743256438} a - \frac{1135780748672738189798276159}{13365579827985049859431907892}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  $13$
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:  \( 3004757826.11 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^5.C_2.C_2$ (as 16T258):

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

Intermediate fields

\(\Q(\sqrt{97}) \), 4.4.912673.1, 8.6.3474326232558859.1, 8.8.80798284478113.1, 8.6.35817796211947.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$43$43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
97Data not computed