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Properties

Label 16.0.34552224922...5625.4
Degree $16$
Signature $[0, 8]$
Discriminant $5^{10}\cdot 29^{12}$
Root discriminant $34.17$
Ramified primes $5, 29$
Class number $12$ (GRH)
Class group $[2, 6]$ (GRH)
Galois group $(C_2\times C_4).D_4$ (as 16T121)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1379, -5231, 14386, -27165, 40381, -46567, 43848, -33670, 21965, -12042, 5781, -2318, 811, -224, 52, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 52*x^14 - 224*x^13 + 811*x^12 - 2318*x^11 + 5781*x^10 - 12042*x^9 + 21965*x^8 - 33670*x^7 + 43848*x^6 - 46567*x^5 + 40381*x^4 - 27165*x^3 + 14386*x^2 - 5231*x + 1379)
 
gp: K = bnfinit(x^16 - 8*x^15 + 52*x^14 - 224*x^13 + 811*x^12 - 2318*x^11 + 5781*x^10 - 12042*x^9 + 21965*x^8 - 33670*x^7 + 43848*x^6 - 46567*x^5 + 40381*x^4 - 27165*x^3 + 14386*x^2 - 5231*x + 1379, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 52 x^{14} - 224 x^{13} + 811 x^{12} - 2318 x^{11} + 5781 x^{10} - 12042 x^{9} + 21965 x^{8} - 33670 x^{7} + 43848 x^{6} - 46567 x^{5} + 40381 x^{4} - 27165 x^{3} + 14386 x^{2} - 5231 x + 1379 \)

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:  \(3455222492240908603515625=5^{10}\cdot 29^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.17$
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:  $5, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{24942595223} a^{14} - \frac{1}{3563227889} a^{13} - \frac{4019059872}{24942595223} a^{12} - \frac{828235900}{24942595223} a^{11} + \frac{1315359537}{3563227889} a^{10} + \frac{7282669697}{24942595223} a^{9} - \frac{9828256514}{24942595223} a^{8} + \frac{11449008483}{24942595223} a^{7} - \frac{3957482665}{24942595223} a^{6} + \frac{6510206587}{24942595223} a^{5} + \frac{7186904750}{24942595223} a^{4} + \frac{918039842}{3563227889} a^{3} - \frac{8655570715}{24942595223} a^{2} + \frac{4168615725}{24942595223} a + \frac{179877452}{3563227889}$, $\frac{1}{145889239459327} a^{15} + \frac{2917}{145889239459327} a^{14} + \frac{57264179551668}{145889239459327} a^{13} + \frac{53696810563524}{145889239459327} a^{12} + \frac{342603952206}{843290401499} a^{11} + \frac{25358678173964}{145889239459327} a^{10} - \frac{25532553296057}{145889239459327} a^{9} - \frac{2835959196979}{145889239459327} a^{8} - \frac{33921948970919}{145889239459327} a^{7} + \frac{62813649849113}{145889239459327} a^{6} + \frac{21936372011006}{145889239459327} a^{5} - \frac{57348698422772}{145889239459327} a^{4} + \frac{66696509338724}{145889239459327} a^{3} + \frac{1852432183585}{11222249189179} a^{2} - \frac{27418549692060}{145889239459327} a - \frac{5057616432415}{20841319922761}$

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

Class group and class number

$C_{2}\times C_{6}$, which has order $12$ (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:  \( 102675.602937 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times C_4).D_4$ (as 16T121):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 28 conjugacy class representatives for $(C_2\times C_4).D_4$
Character table for $(C_2\times C_4).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{29}) \), 4.4.4205.1, 4.0.24389.1, 4.0.121945.1, 8.4.2210253125.1, 8.4.1858822878125.1, 8.0.14870583025.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{8}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$5$5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
29Data not computed