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Properties

Label 16.0.43598484006...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{8}\cdot 5^{12}\cdot 17^{8}$
Root discriminant $19.50$
Ramified primes $2, 5, 17$
Class number $2$
Class group $[2]$
Galois group $A_4:C_4$ (as 16T62)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 6, 18, 24, 3, -54, 10, 90, 291, -90, 10, 54, 3, -24, 18, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 18*x^14 - 24*x^13 + 3*x^12 + 54*x^11 + 10*x^10 - 90*x^9 + 291*x^8 + 90*x^7 + 10*x^6 - 54*x^5 + 3*x^4 + 24*x^3 + 18*x^2 + 6*x + 1)
 
gp: K = bnfinit(x^16 - 6*x^15 + 18*x^14 - 24*x^13 + 3*x^12 + 54*x^11 + 10*x^10 - 90*x^9 + 291*x^8 + 90*x^7 + 10*x^6 - 54*x^5 + 3*x^4 + 24*x^3 + 18*x^2 + 6*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 18 x^{14} - 24 x^{13} + 3 x^{12} + 54 x^{11} + 10 x^{10} - 90 x^{9} + 291 x^{8} + 90 x^{7} + 10 x^{6} - 54 x^{5} + 3 x^{4} + 24 x^{3} + 18 x^{2} + 6 x + 1 \)

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:  \(435984840062500000000=2^{8}\cdot 5^{12}\cdot 17^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.50$
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:  $2, 5, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} + \frac{1}{3}$, $\frac{1}{3} a^{11} + \frac{1}{3} a$, $\frac{1}{270} a^{12} - \frac{11}{90} a^{11} - \frac{2}{27} a^{10} + \frac{2}{9} a^{9} - \frac{43}{90} a^{8} + \frac{4}{9} a^{7} - \frac{1}{30} a^{6} - \frac{4}{9} a^{5} - \frac{43}{90} a^{4} - \frac{2}{9} a^{3} - \frac{2}{27} a^{2} - \frac{19}{90} a + \frac{1}{270}$, $\frac{1}{270} a^{13} - \frac{29}{270} a^{11} + \frac{1}{9} a^{10} - \frac{13}{90} a^{9} - \frac{29}{90} a^{8} - \frac{11}{30} a^{7} + \frac{41}{90} a^{6} - \frac{13}{90} a^{5} + \frac{1}{90} a^{4} - \frac{11}{27} a^{3} + \frac{31}{90} a^{2} + \frac{1}{27} a + \frac{41}{90}$, $\frac{1}{8910} a^{14} + \frac{1}{990} a^{13} + \frac{1}{810} a^{12} + \frac{83}{2970} a^{11} - \frac{299}{8910} a^{10} + \frac{94}{495} a^{9} - \frac{22}{135} a^{8} - \frac{76}{495} a^{7} + \frac{23}{135} a^{6} - \frac{181}{495} a^{5} - \frac{383}{8910} a^{4} - \frac{1369}{2970} a^{3} + \frac{397}{810} a^{2} + \frac{1081}{2970} a - \frac{1013}{8910}$, $\frac{1}{44550} a^{15} + \frac{1}{44550} a^{14} + \frac{1}{8910} a^{13} - \frac{2}{22275} a^{12} - \frac{94}{891} a^{11} + \frac{227}{22275} a^{10} - \frac{122}{7425} a^{9} - \frac{313}{14850} a^{8} + \frac{658}{7425} a^{7} - \frac{1933}{14850} a^{6} - \frac{10553}{44550} a^{5} - \frac{304}{891} a^{4} - \frac{13597}{44550} a^{3} + \frac{2677}{8910} a^{2} + \frac{10859}{22275} a + \frac{16057}{44550}$

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:  \( \frac{763}{810} a^{15} - \frac{9292}{1485} a^{14} + \frac{183829}{8910} a^{13} - \frac{1514}{45} a^{12} + \frac{154379}{8910} a^{11} + \frac{150589}{2970} a^{10} - \frac{81929}{2970} a^{9} - \frac{1549}{18} a^{8} + \frac{996301}{2970} a^{7} - \frac{593}{6} a^{6} - \frac{38563}{891} a^{5} - \frac{27301}{2970} a^{4} + \frac{100757}{4455} a^{3} + \frac{3929}{270} a^{2} + \frac{1357}{4455} a - \frac{487}{1485} \) (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:  \( 26185.7676977 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$A_4:C_4$ (as 16T62):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 48
The 10 conjugacy class representatives for $A_4:C_4$
Character table for $A_4:C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.144500.1, \(\Q(\zeta_{5})\), 8.0.20880250000.1

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

Sibling fields

Degree 12 siblings: data not computed
Degree 24 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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$2$2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.12.8.1$x^{12} - 6 x^{9} + 12 x^{6} - 8 x^{3} + 16$$3$$4$$8$$C_3 : C_4$$[\ ]_{3}^{4}$
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
$17$17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.12.8.1$x^{12} - 51 x^{9} + 867 x^{6} - 4913 x^{3} + 111166451$$3$$4$$8$$C_3 : C_4$$[\ ]_{3}^{4}$