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Properties

Label 16.4.47554978486...0000.2
Degree $16$
Signature $[4, 6]$
Discriminant $2^{40}\cdot 5^{12}\cdot 11^{6}$
Root discriminant $46.49$
Ramified primes $2, 5, 11$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group 16T790

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9801, 39204, 74052, 95832, 87604, 52800, 21252, 8976, 6147, 3144, 600, -192, -130, -36, -4, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^14 - 36*x^13 - 130*x^12 - 192*x^11 + 600*x^10 + 3144*x^9 + 6147*x^8 + 8976*x^7 + 21252*x^6 + 52800*x^5 + 87604*x^4 + 95832*x^3 + 74052*x^2 + 39204*x + 9801)
 
gp: K = bnfinit(x^16 - 4*x^14 - 36*x^13 - 130*x^12 - 192*x^11 + 600*x^10 + 3144*x^9 + 6147*x^8 + 8976*x^7 + 21252*x^6 + 52800*x^5 + 87604*x^4 + 95832*x^3 + 74052*x^2 + 39204*x + 9801, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{14} - 36 x^{13} - 130 x^{12} - 192 x^{11} + 600 x^{10} + 3144 x^{9} + 6147 x^{8} + 8976 x^{7} + 21252 x^{6} + 52800 x^{5} + 87604 x^{4} + 95832 x^{3} + 74052 x^{2} + 39204 x + 9801 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(475549784866816000000000000=2^{40}\cdot 5^{12}\cdot 11^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $46.49$
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:  $2, 5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{33} a^{12} - \frac{5}{11} a^{10} - \frac{1}{11} a^{9} + \frac{2}{33} a^{8} + \frac{2}{11} a^{7} - \frac{16}{33} a^{6} + \frac{3}{11} a^{5} - \frac{13}{33} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{99} a^{13} - \frac{1}{99} a^{12} - \frac{5}{33} a^{11} + \frac{4}{33} a^{10} + \frac{38}{99} a^{9} + \frac{4}{99} a^{8} - \frac{2}{9} a^{7} - \frac{8}{99} a^{6} + \frac{1}{9} a^{5} + \frac{13}{99} a^{4} + \frac{1}{9} a^{3} - \frac{1}{9} a^{2} - \frac{1}{3} a$, $\frac{1}{297} a^{14} - \frac{4}{297} a^{12} - \frac{4}{33} a^{11} + \frac{68}{297} a^{10} + \frac{35}{99} a^{9} + \frac{2}{99} a^{8} - \frac{41}{99} a^{7} + \frac{1}{33} a^{6} + \frac{2}{9} a^{5} - \frac{1}{9} a^{4} - \frac{2}{9} a^{3} + \frac{8}{27} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{520994577864667922522991693} a^{15} - \frac{13584835373251805042051}{57888286429407546946999077} a^{14} - \frac{1004842088855857063173784}{520994577864667922522991693} a^{13} - \frac{704029997857456422155218}{57888286429407546946999077} a^{12} - \frac{62497973115556679005777993}{520994577864667922522991693} a^{11} + \frac{84585317464214805563992625}{173664859288222640840997231} a^{10} - \frac{25535989959812655440974885}{173664859288222640840997231} a^{9} + \frac{55248630244418721406743649}{173664859288222640840997231} a^{8} + \frac{185947342579084030576453}{1754190497860834755969669} a^{7} + \frac{43038668206362208695616441}{173664859288222640840997231} a^{6} - \frac{20630884218831189499284140}{173664859288222640840997231} a^{5} - \frac{312313942337922418305112}{173664859288222640840997231} a^{4} - \frac{17223345674398960784436628}{47363143442242538411181063} a^{3} - \frac{175403365826673797837869}{5262571493582504267909007} a^{2} - \frac{2211409560734464249546567}{5262571493582504267909007} a - \frac{192124689512163451326520}{584730165953611585323223}$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  $9$
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:  \( 7392645.01607 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T790:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 62 conjugacy class representatives for t16n790 are not computed
Character table for t16n790 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.2.17600.1, 4.4.88000.1, 4.2.2000.1, 8.4.123904000000.20

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$5$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$11$11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.8.6.1$x^{8} + 143 x^{4} + 5929$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$