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Properties

Label 16.0.61552138713...0625.2
Degree $16$
Signature $[0, 8]$
Discriminant $5^{14}\cdot 11^{8}\cdot 19^{6}$
Root discriminant $40.91$
Ramified primes $5, 11, 19$
Class number $32$ (GRH)
Class group $[2, 4, 4]$ (GRH)
Galois group $C_2^3.C_4$ (as 16T41)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![30761, -107825, 157306, -142525, 100442, -47550, 14128, -1685, -10, 90, 212, -85, 152, -65, 29, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 29*x^14 - 65*x^13 + 152*x^12 - 85*x^11 + 212*x^10 + 90*x^9 - 10*x^8 - 1685*x^7 + 14128*x^6 - 47550*x^5 + 100442*x^4 - 142525*x^3 + 157306*x^2 - 107825*x + 30761)
 
gp: K = bnfinit(x^16 - 5*x^15 + 29*x^14 - 65*x^13 + 152*x^12 - 85*x^11 + 212*x^10 + 90*x^9 - 10*x^8 - 1685*x^7 + 14128*x^6 - 47550*x^5 + 100442*x^4 - 142525*x^3 + 157306*x^2 - 107825*x + 30761, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 29 x^{14} - 65 x^{13} + 152 x^{12} - 85 x^{11} + 212 x^{10} + 90 x^{9} - 10 x^{8} - 1685 x^{7} + 14128 x^{6} - 47550 x^{5} + 100442 x^{4} - 142525 x^{3} + 157306 x^{2} - 107825 x + 30761 \)

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:  \(61552138713495855712890625=5^{14}\cdot 11^{8}\cdot 19^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $40.91$
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:  $5, 11, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{11} a^{8} - \frac{2}{11} a^{7} + \frac{5}{11} a^{6} + \frac{2}{11} a^{5} + \frac{1}{11} a^{4} - \frac{2}{11} a^{3} + \frac{4}{11} a^{2} - \frac{2}{11} a + \frac{5}{11}$, $\frac{1}{11} a^{9} + \frac{1}{11} a^{7} + \frac{1}{11} a^{6} + \frac{5}{11} a^{5} - \frac{5}{11} a^{2} + \frac{1}{11} a - \frac{1}{11}$, $\frac{1}{11} a^{10} + \frac{3}{11} a^{7} - \frac{2}{11} a^{5} - \frac{1}{11} a^{4} - \frac{3}{11} a^{3} - \frac{3}{11} a^{2} + \frac{1}{11} a - \frac{5}{11}$, $\frac{1}{11} a^{11} - \frac{5}{11} a^{7} + \frac{5}{11} a^{6} + \frac{4}{11} a^{5} + \frac{5}{11} a^{4} + \frac{3}{11} a^{3} + \frac{1}{11} a - \frac{4}{11}$, $\frac{1}{11} a^{12} - \frac{5}{11} a^{7} - \frac{4}{11} a^{6} + \frac{4}{11} a^{5} - \frac{3}{11} a^{4} + \frac{1}{11} a^{3} - \frac{1}{11} a^{2} - \frac{3}{11} a + \frac{3}{11}$, $\frac{1}{11} a^{13} - \frac{3}{11} a^{7} - \frac{4}{11} a^{6} - \frac{4}{11} a^{5} - \frac{5}{11} a^{4} - \frac{5}{11} a^{2} + \frac{4}{11} a + \frac{3}{11}$, $\frac{1}{121} a^{14} + \frac{2}{121} a^{13} - \frac{5}{121} a^{12} + \frac{2}{121} a^{11} - \frac{1}{121} a^{10} - \frac{1}{121} a^{9} + \frac{50}{121} a^{7} - \frac{56}{121} a^{6} - \frac{3}{11} a^{5} - \frac{47}{121} a^{4} - \frac{51}{121} a^{3} + \frac{30}{121} a^{2} + \frac{20}{121} a + \frac{37}{121}$, $\frac{1}{96636617856352349354898913891} a^{15} - \frac{152026738024127220786599964}{96636617856352349354898913891} a^{14} - \frac{2853938759777725595824070760}{96636617856352349354898913891} a^{13} + \frac{1421486758306691571207148511}{96636617856352349354898913891} a^{12} + \frac{1995418391125956460418488200}{96636617856352349354898913891} a^{11} - \frac{957483736762412772313781472}{96636617856352349354898913891} a^{10} + \frac{415636292641702215645242865}{96636617856352349354898913891} a^{9} - \frac{1979050027658756182468298647}{96636617856352349354898913891} a^{8} + \frac{41373064801970765068262770873}{96636617856352349354898913891} a^{7} + \frac{1333567490803116287562959443}{3332297167460425839824100479} a^{6} - \frac{28757132095313893420751568783}{96636617856352349354898913891} a^{5} + \frac{6176867193539692976699566573}{96636617856352349354898913891} a^{4} + \frac{39075730220636590698789740573}{96636617856352349354898913891} a^{3} + \frac{2704199228695389748490265880}{8785147077850213577718083081} a^{2} - \frac{20080486919451886244957023848}{96636617856352349354898913891} a - \frac{19437689283496719757200611349}{96636617856352349354898913891}$

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

Class group and class number

$C_{2}\times C_{4}\times C_{4}$, which has order $32$ (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:  $7$
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:  \( 83143.0598167 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_4$ (as 16T41):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $C_2^3.C_4$
Character table for $C_2^3.C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.15125.1, 4.4.5225.1, 4.4.26125.1, 8.0.412921953125.1 x2, 8.8.82584390625.1

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

Sibling fields

Galois closure: data not computed
Degree 8 siblings: data not computed
Degree 16 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
5Data not computed
$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}$
$19$19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$