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Properties

Label 16.0.10807201753...8125.2
Degree $16$
Signature $[0, 8]$
Discriminant $5^{11}\cdot 19^{12}$
Root discriminant $27.52$
Ramified primes $5, 19$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $C_4:D_4.D_4$ (as 16T681)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1255, 2910, 3224, -1847, -4339, -318, 3284, 1814, -517, -804, -108, 154, 76, -3, -11, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 11*x^14 - 3*x^13 + 76*x^12 + 154*x^11 - 108*x^10 - 804*x^9 - 517*x^8 + 1814*x^7 + 3284*x^6 - 318*x^5 - 4339*x^4 - 1847*x^3 + 3224*x^2 + 2910*x + 1255)
 
gp: K = bnfinit(x^16 - 2*x^15 - 11*x^14 - 3*x^13 + 76*x^12 + 154*x^11 - 108*x^10 - 804*x^9 - 517*x^8 + 1814*x^7 + 3284*x^6 - 318*x^5 - 4339*x^4 - 1847*x^3 + 3224*x^2 + 2910*x + 1255, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 11 x^{14} - 3 x^{13} + 76 x^{12} + 154 x^{11} - 108 x^{10} - 804 x^{9} - 517 x^{8} + 1814 x^{7} + 3284 x^{6} - 318 x^{5} - 4339 x^{4} - 1847 x^{3} + 3224 x^{2} + 2910 x + 1255 \)

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:  \(108072017532527392578125=5^{11}\cdot 19^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.52$
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:  $5, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{5} a^{6} - \frac{2}{5} a^{5} - \frac{1}{5} a^{4} + \frac{2}{5} a^{3} + \frac{1}{5} a^{2}$, $\frac{1}{5} a^{7} + \frac{2}{5} a^{2}$, $\frac{1}{5} a^{8} + \frac{2}{5} a^{3}$, $\frac{1}{25} a^{9} - \frac{1}{25} a^{8} - \frac{1}{25} a^{7} - \frac{2}{5} a^{5} + \frac{12}{25} a^{4} + \frac{8}{25} a^{3} - \frac{12}{25} a^{2} + \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{25} a^{10} - \frac{2}{25} a^{8} - \frac{1}{25} a^{7} + \frac{7}{25} a^{5} + \frac{2}{5} a^{4} - \frac{9}{25} a^{3} + \frac{8}{25} a^{2} - \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{25} a^{11} + \frac{2}{25} a^{8} - \frac{2}{25} a^{7} + \frac{2}{25} a^{6} - \frac{1}{5} a^{4} - \frac{1}{25} a^{3} - \frac{9}{25} a^{2} + \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{125} a^{12} - \frac{2}{25} a^{8} + \frac{9}{125} a^{7} - \frac{7}{25} a^{5} - \frac{2}{5} a^{4} + \frac{6}{25} a^{3} - \frac{61}{125} a^{2} + \frac{1}{5} a - \frac{9}{25}$, $\frac{1}{125} a^{13} - \frac{1}{125} a^{8} - \frac{2}{25} a^{7} - \frac{2}{25} a^{6} + \frac{2}{5} a^{5} - \frac{56}{125} a^{3} + \frac{11}{25} a^{2} + \frac{11}{25} a - \frac{1}{5}$, $\frac{1}{26125} a^{14} + \frac{3}{26125} a^{13} + \frac{12}{26125} a^{12} + \frac{58}{5225} a^{11} - \frac{2}{1045} a^{10} + \frac{134}{26125} a^{9} + \frac{162}{26125} a^{8} + \frac{98}{2375} a^{7} - \frac{96}{1045} a^{6} + \frac{21}{475} a^{5} - \frac{736}{26125} a^{4} - \frac{9388}{26125} a^{3} - \frac{447}{2375} a^{2} + \frac{117}{275} a - \frac{2043}{5225}$, $\frac{1}{4819874687375} a^{15} + \frac{13719037}{963974937475} a^{14} + \frac{290255189}{438170426125} a^{13} + \frac{673915688}{438170426125} a^{12} + \frac{59170628}{3505363409} a^{11} + \frac{28594521014}{4819874687375} a^{10} + \frac{738662589}{87634085225} a^{9} - \frac{371658134814}{4819874687375} a^{8} + \frac{24619897618}{253677615125} a^{7} + \frac{81608030642}{963974937475} a^{6} - \frac{1324159588531}{4819874687375} a^{5} - \frac{14497282924}{192794987495} a^{4} - \frac{743854769594}{4819874687375} a^{3} - \frac{1933727306088}{4819874687375} a^{2} - \frac{19762993251}{963974937475} a - \frac{71126703022}{963974937475}$

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

Class group and class number

$C_{2}\times C_{2}$, 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:  $7$
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:  \( 49539.9976127 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4:D_4.D_4$ (as 16T681):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 19 conjugacy class representatives for $C_4:D_4.D_4$
Character table for $C_4:D_4.D_4$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.2.475.1, 8.0.407253125.1

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 $16$ $16$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ $16$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ $16$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
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$$[\ ]_{4}$
$19$19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.4.3.2$x^{4} - 19$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
19.8.7.1$x^{8} + 76$$8$$1$$7$$QD_{16}$$[\ ]_{8}^{2}$