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Properties

Label 16.0.34336147906...0625.5
Degree $16$
Signature $[0, 8]$
Discriminant $3^{8}\cdot 5^{12}\cdot 11^{8}$
Root discriminant $19.21$
Ramified primes $3, 5, 11$
Class number $4$
Class group $[4]$
Galois group $C_2^2 : C_4$ (as 16T10)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![81, -162, 216, -243, 225, -342, 393, -324, 163, 89, -12, -41, 14, 1, -3, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 3*x^14 + x^13 + 14*x^12 - 41*x^11 - 12*x^10 + 89*x^9 + 163*x^8 - 324*x^7 + 393*x^6 - 342*x^5 + 225*x^4 - 243*x^3 + 216*x^2 - 162*x + 81)
 
gp: K = bnfinit(x^16 - x^15 - 3*x^14 + x^13 + 14*x^12 - 41*x^11 - 12*x^10 + 89*x^9 + 163*x^8 - 324*x^7 + 393*x^6 - 342*x^5 + 225*x^4 - 243*x^3 + 216*x^2 - 162*x + 81, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 3 x^{14} + x^{13} + 14 x^{12} - 41 x^{11} - 12 x^{10} + 89 x^{9} + 163 x^{8} - 324 x^{7} + 393 x^{6} - 342 x^{5} + 225 x^{4} - 243 x^{3} + 216 x^{2} - 162 x + 81 \)

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:  \(343361479062744140625=3^{8}\cdot 5^{12}\cdot 11^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.21$
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:  $3, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{6} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{6} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{2}$, $\frac{1}{6} a^{11} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{2} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{2} - \frac{1}{2} a$, $\frac{1}{90} a^{12} - \frac{2}{45} a^{11} + \frac{1}{9} a^{9} - \frac{17}{45} a^{8} + \frac{5}{18} a^{7} - \frac{1}{15} a^{6} - \frac{1}{9} a^{5} - \frac{7}{45} a^{4} - \frac{1}{3} a^{3} + \frac{1}{6} a^{2} + \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{388890} a^{13} + \frac{169}{38889} a^{12} + \frac{13922}{194445} a^{11} + \frac{6041}{77778} a^{10} - \frac{75862}{194445} a^{9} + \frac{131219}{388890} a^{8} + \frac{43762}{194445} a^{7} - \frac{39107}{194445} a^{6} - \frac{150319}{388890} a^{5} - \frac{33368}{194445} a^{4} + \frac{1809}{8642} a^{3} - \frac{3633}{21605} a^{2} - \frac{1033}{4321} a + \frac{14571}{43210}$, $\frac{1}{1166670} a^{14} - \frac{1}{1166670} a^{13} + \frac{103}{388890} a^{12} + \frac{11648}{583335} a^{11} - \frac{24529}{1166670} a^{10} - \frac{186947}{1166670} a^{9} - \frac{9755}{25926} a^{8} - \frac{65443}{1166670} a^{7} - \frac{4823}{116667} a^{6} + \frac{1561}{388890} a^{5} + \frac{108907}{388890} a^{4} - \frac{49231}{129630} a^{3} + \frac{26441}{129630} a^{2} - \frac{6509}{21605} a - \frac{3287}{43210}$, $\frac{1}{5833350} a^{15} + \frac{1}{2916675} a^{14} + \frac{1}{1944450} a^{13} + \frac{1771}{583335} a^{12} - \frac{302461}{5833350} a^{11} - \frac{468179}{5833350} a^{10} - \frac{87929}{972225} a^{9} + \frac{780617}{5833350} a^{8} + \frac{909542}{2916675} a^{7} - \frac{417449}{1944450} a^{6} - \frac{216227}{648150} a^{5} - \frac{377366}{972225} a^{4} - \frac{75359}{216050} a^{3} - \frac{121354}{324075} a^{2} - \frac{405}{8642} a + \frac{49977}{108025}$

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

Class group and class number

$C_{4}$, which has order $4$

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:  \( -\frac{14557}{1166670} a^{15} + \frac{1369}{116667} a^{14} + \frac{14557}{388890} a^{13} - \frac{14557}{1166670} a^{12} - \frac{101899}{583335} a^{11} + \frac{596837}{1166670} a^{10} + \frac{11782}{64815} a^{9} - \frac{1295573}{1166670} a^{8} - \frac{2372791}{1166670} a^{7} + \frac{87342}{21605} a^{6} - \frac{1906967}{388890} a^{5} + \frac{644572}{194445} a^{4} - \frac{72785}{25926} a^{3} + \frac{131013}{43210} a^{2} - \frac{58228}{21605} a + \frac{43671}{21605} \) (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:  \( 14720.183108 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:C_4$ (as 16T10):

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 $C_2^2 : C_4$
Character table for $C_2^2 : C_4$

Intermediate fields

\(\Q(\sqrt{-55}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{5}, \sqrt{-11})\), 4.2.2475.1 x2, 4.0.5445.1 x2, 4.0.136125.1 x2, 4.2.12375.1 x2, \(\Q(\zeta_{5})\), 4.4.15125.1, 8.0.741200625.3, 8.0.18530015625.6, 8.0.228765625.1, 8.0.153140625.2 x2, 8.4.18530015625.1 x2

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

Sibling fields

Degree 8 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{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}$
$11$11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$