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Properties

Label 16.0.35664401793...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 3^{12}\cdot 5^{6}$
Root discriminant $16.67$
Ramified primes $2, 3, 5$
Class number $2$
Class group $[2]$
Galois group $C_8:C_2^2$ (as 16T38)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 4, 2, 120, 284, -796, 604, -444, 475, -276, 124, -104, 44, 0, 2, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 2*x^14 + 44*x^12 - 104*x^11 + 124*x^10 - 276*x^9 + 475*x^8 - 444*x^7 + 604*x^6 - 796*x^5 + 284*x^4 + 120*x^3 + 2*x^2 + 4*x + 1)
 
gp: K = bnfinit(x^16 - 4*x^15 + 2*x^14 + 44*x^12 - 104*x^11 + 124*x^10 - 276*x^9 + 475*x^8 - 444*x^7 + 604*x^6 - 796*x^5 + 284*x^4 + 120*x^3 + 2*x^2 + 4*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 2 x^{14} + 44 x^{12} - 104 x^{11} + 124 x^{10} - 276 x^{9} + 475 x^{8} - 444 x^{7} + 604 x^{6} - 796 x^{5} + 284 x^{4} + 120 x^{3} + 2 x^{2} + 4 x + 1 \)

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:  \(35664401793024000000=2^{32}\cdot 3^{12}\cdot 5^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.67$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{9} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{6} + \frac{1}{3} a^{3} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{115659} a^{14} - \frac{15314}{115659} a^{13} - \frac{6634}{115659} a^{12} + \frac{7823}{115659} a^{11} - \frac{16363}{115659} a^{10} - \frac{7265}{115659} a^{9} - \frac{16222}{115659} a^{8} - \frac{51997}{115659} a^{7} - \frac{54794}{115659} a^{6} - \frac{46337}{115659} a^{5} + \frac{12826}{115659} a^{4} + \frac{12005}{115659} a^{3} + \frac{38569}{115659} a^{2} - \frac{49883}{115659} a + \frac{26957}{115659}$, $\frac{1}{9153831555} a^{15} + \frac{25634}{9153831555} a^{14} + \frac{297237043}{3051277185} a^{13} - \frac{1411383548}{9153831555} a^{12} + \frac{95816219}{1830766311} a^{11} - \frac{344338628}{3051277185} a^{10} - \frac{34906516}{277388835} a^{9} - \frac{112729714}{1830766311} a^{8} - \frac{149903299}{610255437} a^{7} - \frac{1269709504}{9153831555} a^{6} + \frac{3686968847}{9153831555} a^{5} + \frac{17032951}{610255437} a^{4} + \frac{514641148}{3051277185} a^{3} + \frac{3916202177}{9153831555} a^{2} + \frac{61105166}{277388835} a + \frac{1183023098}{9153831555}$

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:  \( -\frac{139402744}{3051277185} a^{15} + \frac{631493699}{3051277185} a^{14} - \frac{677908096}{3051277185} a^{13} + \frac{161787084}{1017092395} a^{12} - \frac{1226688550}{610255437} a^{11} + \frac{17878679096}{3051277185} a^{10} - \frac{2672599228}{277388835} a^{9} + \frac{3680210041}{203418479} a^{8} - \frac{6382039344}{203418479} a^{7} + \frac{40489685272}{1017092395} a^{6} - \frac{150998772548}{3051277185} a^{5} + \frac{37311072940}{610255437} a^{4} - \frac{50323505752}{1017092395} a^{3} + \frac{19181660319}{1017092395} a^{2} - \frac{212279812}{277388835} a - \frac{1884165392}{3051277185} \) (order $12$)
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:  \( 5042.1967116 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_8:C_2^2$ (as 16T38):

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 $C_8:C_2^2$
Character table for $C_8:C_2^2$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), 4.0.2880.1, 4.0.320.1, \(\Q(\zeta_{12})\), 8.0.5971968000.1, 8.0.5971968000.2, 8.0.8294400.2

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 R R R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
2Data not computed
3Data not computed
$5$5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$