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Properties

Label 16.0.12003260674...9641.6
Degree $16$
Signature $[0, 8]$
Discriminant $13^{12}\cdot 61^{6}$
Root discriminant $31.99$
Ramified primes $13, 61$
Class number $2$ (GRH)
Class group $[2]$ (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![211, -728, 30, 2618, -3621, -125, 7564, -12586, 12611, -9309, 5252, -2364, 849, -238, 54, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 54*x^14 - 238*x^13 + 849*x^12 - 2364*x^11 + 5252*x^10 - 9309*x^9 + 12611*x^8 - 12586*x^7 + 7564*x^6 - 125*x^5 - 3621*x^4 + 2618*x^3 + 30*x^2 - 728*x + 211)
 
gp: K = bnfinit(x^16 - 8*x^15 + 54*x^14 - 238*x^13 + 849*x^12 - 2364*x^11 + 5252*x^10 - 9309*x^9 + 12611*x^8 - 12586*x^7 + 7564*x^6 - 125*x^5 - 3621*x^4 + 2618*x^3 + 30*x^2 - 728*x + 211, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 54 x^{14} - 238 x^{13} + 849 x^{12} - 2364 x^{11} + 5252 x^{10} - 9309 x^{9} + 12611 x^{8} - 12586 x^{7} + 7564 x^{6} - 125 x^{5} - 3621 x^{4} + 2618 x^{3} + 30 x^{2} - 728 x + 211 \)

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:  \(1200326067404665657109641=13^{12}\cdot 61^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.99$
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:  $13, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8} + \frac{1}{5} a^{7} - \frac{1}{5} a^{6} + \frac{2}{5} a^{5} + \frac{2}{5} a^{4} - \frac{2}{5} a^{3} - \frac{2}{5} a^{2} - \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{9} - \frac{2}{5} a^{7} - \frac{2}{5} a^{6} + \frac{1}{5} a^{4} + \frac{1}{5} a^{2} + \frac{1}{5}$, $\frac{1}{5} a^{10} - \frac{2}{5} a^{6} - \frac{1}{5} a^{4} + \frac{2}{5} a^{3} + \frac{1}{5} a^{2} - \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{5} a^{11} - \frac{2}{5} a^{7} - \frac{1}{5} a^{5} + \frac{2}{5} a^{4} + \frac{1}{5} a^{3} - \frac{1}{5} a^{2} - \frac{2}{5} a$, $\frac{1}{5} a^{12} + \frac{2}{5} a^{7} + \frac{2}{5} a^{6} + \frac{1}{5} a^{5} - \frac{1}{5} a^{2} - \frac{2}{5} a - \frac{2}{5}$, $\frac{1}{5} a^{13} - \frac{2}{5} a^{6} + \frac{1}{5} a^{5} + \frac{1}{5} a^{4} - \frac{2}{5} a^{3} + \frac{2}{5} a^{2} + \frac{2}{5}$, $\frac{1}{171123635} a^{14} - \frac{7}{171123635} a^{13} - \frac{1416159}{34224727} a^{12} + \frac{8260134}{171123635} a^{11} + \frac{1185109}{34224727} a^{10} - \frac{8375727}{171123635} a^{9} + \frac{16235739}{171123635} a^{8} - \frac{60725742}{171123635} a^{7} + \frac{9095414}{34224727} a^{6} + \frac{56165713}{171123635} a^{5} + \frac{16356820}{34224727} a^{4} + \frac{76405192}{171123635} a^{3} + \frac{66561592}{171123635} a^{2} + \frac{12322891}{34224727} a - \frac{3858092}{34224727}$, $\frac{1}{112770475465} a^{15} + \frac{322}{112770475465} a^{14} + \frac{4134108869}{112770475465} a^{13} + \frac{925259632}{22554095093} a^{12} + \frac{3681801987}{112770475465} a^{11} - \frac{1316165689}{22554095093} a^{10} + \frac{6022151668}{112770475465} a^{9} - \frac{1940585008}{112770475465} a^{8} + \frac{3921342671}{112770475465} a^{7} - \frac{9486362}{777727417} a^{6} - \frac{59504180}{22554095093} a^{5} - \frac{28460683648}{112770475465} a^{4} - \frac{1017002727}{22554095093} a^{3} - \frac{3365919757}{112770475465} a^{2} + \frac{29252968436}{112770475465} a + \frac{47762732047}{112770475465}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 120660.872306 \) (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

\(\Q(\sqrt{13}) \), 4.4.10309.1, 4.0.2197.1, 4.0.134017.1, 8.4.1095593933629.1, 8.4.6482804341.1, 8.0.17960556289.2

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$13$13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
$61$61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.4.3.1$x^{4} - 61$$4$$1$$3$$C_4$$[\ ]_{4}$
61.4.3.1$x^{4} - 61$$4$$1$$3$$C_4$$[\ ]_{4}$