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Properties

Label 16.8.26428481046...2569.5
Degree $16$
Signature $[8, 4]$
Discriminant $13^{10}\cdot 61^{8}$
Root discriminant $38.80$
Ramified primes $13, 61$
Class number $1$ (GRH)
Class group Trivial (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![9763, 17394, -9984, -3822, -19717, 10916, 17454, -10748, -5172, 3362, 938, -556, -150, 84, 8, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 8*x^14 + 84*x^13 - 150*x^12 - 556*x^11 + 938*x^10 + 3362*x^9 - 5172*x^8 - 10748*x^7 + 17454*x^6 + 10916*x^5 - 19717*x^4 - 3822*x^3 - 9984*x^2 + 17394*x + 9763)
 
gp: K = bnfinit(x^16 - 8*x^15 + 8*x^14 + 84*x^13 - 150*x^12 - 556*x^11 + 938*x^10 + 3362*x^9 - 5172*x^8 - 10748*x^7 + 17454*x^6 + 10916*x^5 - 19717*x^4 - 3822*x^3 - 9984*x^2 + 17394*x + 9763, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 8 x^{14} + 84 x^{13} - 150 x^{12} - 556 x^{11} + 938 x^{10} + 3362 x^{9} - 5172 x^{8} - 10748 x^{7} + 17454 x^{6} + 10916 x^{5} - 19717 x^{4} - 3822 x^{3} - 9984 x^{2} + 17394 x + 9763 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(26428481046229354497662569=13^{10}\cdot 61^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.80$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{6} a^{7} + \frac{1}{6} a^{6} + \frac{1}{6} a^{5} + \frac{1}{6} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{6} a - \frac{1}{3}$, $\frac{1}{12} a^{8} + \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{12}$, $\frac{1}{36} a^{9} + \frac{1}{18} a^{7} - \frac{1}{9} a^{6} + \frac{1}{18} a^{5} + \frac{2}{9} a^{4} - \frac{7}{36} a^{3} + \frac{2}{9} a^{2} + \frac{7}{36} a + \frac{7}{18}$, $\frac{1}{36} a^{10} - \frac{1}{36} a^{8} + \frac{1}{18} a^{7} + \frac{2}{9} a^{6} - \frac{1}{9} a^{5} - \frac{1}{36} a^{4} - \frac{1}{9} a^{3} + \frac{1}{9} a^{2} - \frac{4}{9} a - \frac{1}{4}$, $\frac{1}{36} a^{11} - \frac{1}{36} a^{8} - \frac{1}{18} a^{7} - \frac{1}{18} a^{6} + \frac{7}{36} a^{5} - \frac{2}{9} a^{4} + \frac{1}{12} a^{3} + \frac{7}{36} a^{2} - \frac{7}{18} a + \frac{5}{36}$, $\frac{1}{864} a^{12} - \frac{1}{144} a^{11} - \frac{1}{72} a^{10} - \frac{5}{864} a^{9} + \frac{7}{864} a^{8} - \frac{1}{54} a^{7} + \frac{107}{864} a^{6} - \frac{1}{216} a^{5} + \frac{163}{864} a^{4} - \frac{289}{864} a^{3} + \frac{419}{864} a^{2} - \frac{101}{864} a - \frac{323}{864}$, $\frac{1}{864} a^{13} - \frac{5}{864} a^{10} + \frac{1}{864} a^{9} - \frac{11}{432} a^{8} - \frac{37}{864} a^{7} + \frac{7}{432} a^{6} + \frac{91}{864} a^{5} - \frac{151}{864} a^{4} + \frac{389}{864} a^{3} + \frac{277}{864} a^{2} - \frac{137}{864} a + \frac{61}{144}$, $\frac{1}{6664032} a^{14} - \frac{7}{6664032} a^{13} + \frac{541}{3332016} a^{12} - \frac{6401}{6664032} a^{11} - \frac{5273}{555336} a^{10} + \frac{1555}{2221344} a^{9} + \frac{3131}{6664032} a^{8} + \frac{259585}{6664032} a^{7} + \frac{67535}{740448} a^{6} - \frac{410671}{1666008} a^{5} - \frac{101473}{1666008} a^{4} + \frac{67765}{138834} a^{3} - \frac{1360627}{3332016} a^{2} - \frac{2251277}{6664032} a + \frac{3361}{208251}$, $\frac{1}{23037558624} a^{15} + \frac{1721}{23037558624} a^{14} + \frac{3480769}{11518779312} a^{13} + \frac{9869389}{23037558624} a^{12} - \frac{8087809}{959898276} a^{11} - \frac{14622773}{7679186208} a^{10} - \frac{256198555}{23037558624} a^{9} - \frac{871190165}{23037558624} a^{8} - \frac{42199955}{853242912} a^{7} - \frac{2730208097}{11518779312} a^{6} - \frac{101848579}{5759389656} a^{5} - \frac{624700609}{3839593104} a^{4} - \frac{392244767}{5759389656} a^{3} + \frac{1859755573}{23037558624} a^{2} - \frac{1450576835}{11518779312} a - \frac{38037961}{142207152}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $11$
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:  \( 17569347.9344 \) (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

\(\Q(\sqrt{61}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{793}) \), 4.4.48373.1 x2, 4.4.10309.1 x2, \(\Q(\sqrt{13}, \sqrt{61})\), 8.4.5140863842413.2 x2, 8.8.395451064801.1, 8.4.395451064801.1 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$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}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
$61$61.4.2.1$x^{4} + 183 x^{2} + 14884$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
61.4.2.1$x^{4} + 183 x^{2} + 14884$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
61.4.2.1$x^{4} + 183 x^{2} + 14884$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
61.4.2.1$x^{4} + 183 x^{2} + 14884$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$