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Properties

Label 16.0.46968797499...3969.7
Degree $16$
Signature $[0, 8]$
Discriminant $13^{12}\cdot 17^{10}$
Root discriminant $40.22$
Ramified primes $13, 17$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^2.D_4$ (as 16T33)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8128, -13280, 11504, 4240, -1564, -8502, 20601, -25692, 21952, -14408, 7385, -3044, 1023, -266, 58, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 58*x^14 - 266*x^13 + 1023*x^12 - 3044*x^11 + 7385*x^10 - 14408*x^9 + 21952*x^8 - 25692*x^7 + 20601*x^6 - 8502*x^5 - 1564*x^4 + 4240*x^3 + 11504*x^2 - 13280*x + 8128)
 
gp: K = bnfinit(x^16 - 8*x^15 + 58*x^14 - 266*x^13 + 1023*x^12 - 3044*x^11 + 7385*x^10 - 14408*x^9 + 21952*x^8 - 25692*x^7 + 20601*x^6 - 8502*x^5 - 1564*x^4 + 4240*x^3 + 11504*x^2 - 13280*x + 8128, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 58 x^{14} - 266 x^{13} + 1023 x^{12} - 3044 x^{11} + 7385 x^{10} - 14408 x^{9} + 21952 x^{8} - 25692 x^{7} + 20601 x^{6} - 8502 x^{5} - 1564 x^{4} + 4240 x^{3} + 11504 x^{2} - 13280 x + 8128 \)

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:  \(46968797499063289085893969=13^{12}\cdot 17^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $40.22$
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:  $13, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{16} a^{10} - \frac{1}{16} a^{9} - \frac{1}{8} a^{8} - \frac{1}{8} a^{7} - \frac{1}{4} a^{6} + \frac{1}{8} a^{5} + \frac{1}{16} a^{4} - \frac{3}{16} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{16} a^{11} + \frac{1}{16} a^{9} + \frac{1}{8} a^{7} - \frac{1}{8} a^{6} + \frac{3}{16} a^{5} - \frac{1}{8} a^{4} + \frac{1}{16} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{384} a^{12} - \frac{1}{64} a^{11} - \frac{11}{384} a^{10} + \frac{7}{192} a^{9} + \frac{1}{192} a^{8} + \frac{17}{192} a^{7} + \frac{5}{128} a^{6} - \frac{5}{96} a^{5} - \frac{71}{384} a^{4} + \frac{31}{64} a^{3} - \frac{5}{48} a^{2} - \frac{13}{48} a - \frac{7}{24}$, $\frac{1}{384} a^{13} + \frac{1}{384} a^{11} - \frac{1}{96} a^{10} - \frac{5}{192} a^{9} + \frac{23}{192} a^{8} + \frac{9}{128} a^{7} - \frac{13}{192} a^{6} + \frac{49}{384} a^{5} - \frac{1}{4} a^{4} + \frac{29}{96} a^{3} + \frac{17}{48} a^{2} + \frac{1}{12} a + \frac{1}{4}$, $\frac{1}{66878976} a^{14} - \frac{7}{66878976} a^{13} - \frac{3893}{22292992} a^{12} + \frac{70165}{66878976} a^{11} - \frac{784343}{33439488} a^{10} + \frac{600007}{5573248} a^{9} + \frac{979795}{22292992} a^{8} + \frac{4231011}{22292992} a^{7} + \frac{29293}{22292992} a^{6} - \frac{15221015}{66878976} a^{5} - \frac{3916715}{16719744} a^{4} - \frac{2345617}{16719744} a^{3} + \frac{3092545}{8359872} a^{2} - \frac{15313}{174164} a + \frac{170363}{2089968}$, $\frac{1}{1175665519104} a^{15} + \frac{4391}{587832759552} a^{14} + \frac{126667709}{587832759552} a^{13} - \frac{36486353}{195944253184} a^{12} - \frac{1330126217}{1175665519104} a^{11} - \frac{2596457827}{195944253184} a^{10} - \frac{50572450715}{1175665519104} a^{9} - \frac{26716774993}{587832759552} a^{8} + \frac{9880189621}{146958189888} a^{7} + \frac{46051805917}{293916379776} a^{6} + \frac{197777294717}{1175665519104} a^{5} - \frac{2516340363}{24493031648} a^{4} + \frac{43927728819}{97972126592} a^{3} - \frac{31752055535}{146958189888} a^{2} + \frac{4192819965}{12246515824} a - \frac{2886583303}{12246515824}$

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:  \( 7162693.22826 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2.D_4$ (as 16T33):

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 $C_2^2.D_4$
Character table for $C_2^2.D_4$

Intermediate fields

\(\Q(\sqrt{221}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{17}) \), 4.0.3757.1 x2, 4.0.2873.1 x2, \(\Q(\sqrt{13}, \sqrt{17})\), 8.0.2385443281.2, 8.0.403139914489.2 x2, 8.4.6853378546313.1 x2

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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R R ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$13$13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$17$17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.3.2$x^{4} - 153$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.2$x^{4} - 153$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$