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Properties

Label 16.0.13066428858...0625.2
Degree $16$
Signature $[0, 8]$
Discriminant $3^{8}\cdot 5^{12}\cdot 13^{8}$
Root discriminant $20.88$
Ramified primes $3, 5, 13$
Class number $4$
Class group $[4]$
Galois group $D_{8}$ (as 16T13)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -8, 58, 0, -175, 163, 474, -445, 484, -245, 111, -4, 20, 0, 7, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 7*x^14 + 20*x^12 - 4*x^11 + 111*x^10 - 245*x^9 + 484*x^8 - 445*x^7 + 474*x^6 + 163*x^5 - 175*x^4 + 58*x^2 - 8*x + 1)
 
gp: K = bnfinit(x^16 - x^15 + 7*x^14 + 20*x^12 - 4*x^11 + 111*x^10 - 245*x^9 + 484*x^8 - 445*x^7 + 474*x^6 + 163*x^5 - 175*x^4 + 58*x^2 - 8*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 7 x^{14} + 20 x^{12} - 4 x^{11} + 111 x^{10} - 245 x^{9} + 484 x^{8} - 445 x^{7} + 474 x^{6} + 163 x^{5} - 175 x^{4} + 58 x^{2} - 8 x + 1 \)

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:  \(1306642885859619140625=3^{8}\cdot 5^{12}\cdot 13^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.88$
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:  $3, 5, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{4} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{7} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{273} a^{12} + \frac{2}{21} a^{11} + \frac{1}{7} a^{10} + \frac{6}{91} a^{9} - \frac{2}{21} a^{8} + \frac{1}{7} a^{7} - \frac{85}{273} a^{6} - \frac{2}{7} a^{5} - \frac{10}{21} a^{4} + \frac{92}{273} a^{3} + \frac{4}{21} a^{2} - \frac{1}{3} a + \frac{38}{273}$, $\frac{1}{273} a^{13} + \frac{5}{273} a^{10} - \frac{1}{7} a^{9} - \frac{1}{21} a^{8} + \frac{4}{13} a^{7} + \frac{1}{7} a^{6} + \frac{2}{7} a^{5} + \frac{5}{13} a^{4} + \frac{3}{7} a^{3} - \frac{2}{7} a^{2} + \frac{38}{273} a + \frac{8}{21}$, $\frac{1}{10101} a^{14} - \frac{5}{3367} a^{13} - \frac{16}{10101} a^{12} - \frac{228}{3367} a^{11} + \frac{263}{10101} a^{10} + \frac{1285}{10101} a^{9} - \frac{375}{3367} a^{8} + \frac{2887}{10101} a^{7} - \frac{1293}{3367} a^{6} - \frac{1031}{3367} a^{5} - \frac{2836}{10101} a^{4} - \frac{1849}{10101} a^{3} - \frac{1270}{3367} a^{2} + \frac{3083}{10101} a - \frac{2896}{10101}$, $\frac{1}{8016385923} a^{15} + \frac{188410}{8016385923} a^{14} - \frac{1138255}{2672128641} a^{13} - \frac{342220}{2672128641} a^{12} + \frac{263941124}{8016385923} a^{11} - \frac{1953312}{24073231} a^{10} - \frac{421515175}{2672128641} a^{9} + \frac{815134846}{8016385923} a^{8} - \frac{285425354}{2672128641} a^{7} + \frac{1871571659}{8016385923} a^{6} + \frac{1954585231}{8016385923} a^{5} + \frac{1274394224}{2672128641} a^{4} + \frac{2151135437}{8016385923} a^{3} + \frac{3317960788}{8016385923} a^{2} - \frac{1023497903}{2672128641} a - \frac{6059219}{8016385923}$

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

Class group and class number

$C_{4}$, which has order $4$

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:  \( -\frac{3409347}{24073231} a^{15} + \frac{37230}{264541} a^{14} - \frac{23733985}{24073231} a^{13} - \frac{303155}{72219693} a^{12} - \frac{5194995}{1851787} a^{11} + \frac{14277097}{24073231} a^{10} - \frac{1127702560}{72219693} a^{9} + \frac{64284160}{1851787} a^{8} - \frac{1632819785}{24073231} a^{7} + \frac{4491594295}{72219693} a^{6} - \frac{122175216}{1851787} a^{5} - \frac{548784290}{24073231} a^{4} + \frac{1768060030}{72219693} a^{3} + \frac{6519810}{1851787} a^{2} - \frac{195431130}{24073231} a + \frac{11552338}{10317099} \) (order $6$)
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:  \( 10354.9857003 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_8$ (as 16T13):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 7 conjugacy class representatives for $D_{8}$
Character table for $D_{8}$

Intermediate fields

\(\Q(\sqrt{-39}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-3}, \sqrt{13})\), 4.2.12675.1 x2, 4.0.2925.1 x2, 8.0.1445900625.2, 8.2.12049171875.2 x4, 8.0.2780578125.2 x4

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 8 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ R R ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$5$5.8.6.3$x^{8} + 25 x^{4} + 200$$4$$2$$6$$C_8$$[\ ]_{4}^{2}$
5.8.6.3$x^{8} + 25 x^{4} + 200$$4$$2$$6$$C_8$$[\ ]_{4}^{2}$
$13$13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$