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Properties

Label 16.16.1638350662...6256.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{40}\cdot 3^{8}\cdot 17^{6}\cdot 97^{2}$
Root discriminant $50.22$
Ramified primes $2, 3, 17, 97$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^3.C_2^4.C_2$ (as 16T595)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1724, -7040, 29264, 105760, 6952, -179488, -83472, 91072, 53464, -19392, -13032, 1744, 1412, -48, -64, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 64*x^14 - 48*x^13 + 1412*x^12 + 1744*x^11 - 13032*x^10 - 19392*x^9 + 53464*x^8 + 91072*x^7 - 83472*x^6 - 179488*x^5 + 6952*x^4 + 105760*x^3 + 29264*x^2 - 7040*x - 1724)
 
gp: K = bnfinit(x^16 - 64*x^14 - 48*x^13 + 1412*x^12 + 1744*x^11 - 13032*x^10 - 19392*x^9 + 53464*x^8 + 91072*x^7 - 83472*x^6 - 179488*x^5 + 6952*x^4 + 105760*x^3 + 29264*x^2 - 7040*x - 1724, 1)
 

Normalized defining polynomial

\( x^{16} - 64 x^{14} - 48 x^{13} + 1412 x^{12} + 1744 x^{11} - 13032 x^{10} - 19392 x^{9} + 53464 x^{8} + 91072 x^{7} - 83472 x^{6} - 179488 x^{5} + 6952 x^{4} + 105760 x^{3} + 29264 x^{2} - 7040 x - 1724 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1638350662595178231031136256=2^{40}\cdot 3^{8}\cdot 17^{6}\cdot 97^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $50.22$
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:  $2, 3, 17, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{8} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4}$, $\frac{1}{8} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a$, $\frac{1}{8} a^{10} - \frac{1}{2} a^{7} - \frac{1}{4} a^{6} + \frac{1}{4} a^{2}$, $\frac{1}{8} a^{11} - \frac{1}{4} a^{7} + \frac{1}{4} a^{3}$, $\frac{1}{136} a^{12} - \frac{5}{136} a^{11} - \frac{1}{17} a^{10} - \frac{1}{34} a^{9} + \frac{1}{68} a^{8} - \frac{11}{68} a^{7} - \frac{8}{17} a^{6} + \frac{8}{17} a^{5} + \frac{1}{68} a^{4} - \frac{21}{68} a^{3} + \frac{3}{17} a^{2} + \frac{5}{17}$, $\frac{1}{136} a^{13} + \frac{1}{136} a^{11} + \frac{7}{136} a^{10} - \frac{1}{136} a^{9} + \frac{5}{136} a^{8} + \frac{15}{68} a^{7} + \frac{25}{68} a^{6} - \frac{13}{34} a^{5} - \frac{33}{68} a^{4} + \frac{9}{68} a^{3} - \frac{25}{68} a^{2} - \frac{31}{68} a - \frac{19}{68}$, $\frac{1}{26384} a^{14} + \frac{7}{13192} a^{13} - \frac{7}{13192} a^{12} + \frac{199}{3298} a^{11} - \frac{223}{13192} a^{10} + \frac{1}{194} a^{9} + \frac{111}{6596} a^{8} + \frac{118}{1649} a^{7} + \frac{2589}{13192} a^{6} + \frac{1661}{6596} a^{5} + \frac{599}{6596} a^{4} + \frac{87}{3298} a^{3} - \frac{13}{194} a^{2} + \frac{1013}{3298} a + \frac{27}{1649}$, $\frac{1}{9362708882810617314704} a^{15} - \frac{84265697795493745}{4681354441405308657352} a^{14} - \frac{1723578337107392379}{585169305175663582169} a^{13} + \frac{4767416538290750049}{4681354441405308657352} a^{12} - \frac{264269446516511043839}{4681354441405308657352} a^{11} - \frac{28075255361935150793}{2340677220702654328676} a^{10} + \frac{130814971522880347519}{4681354441405308657352} a^{9} + \frac{11661420102039400989}{275373790670900509256} a^{8} + \frac{447024699575904198977}{4681354441405308657352} a^{7} + \frac{880939545971428835599}{2340677220702654328676} a^{6} + \frac{385590345505767583113}{2340677220702654328676} a^{5} - \frac{69454091688785342205}{585169305175663582169} a^{4} - \frac{327317089969464895181}{1170338610351327164338} a^{3} - \frac{288057169846586431286}{585169305175663582169} a^{2} - \frac{1153813493772564312969}{2340677220702654328676} a + \frac{686307925300490547583}{2340677220702654328676}$

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:  $15$
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:  \( 202459645.634 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_2^4.C_2$ (as 16T595):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 40 conjugacy class representatives for $C_2^3.C_2^4.C_2$
Character table for $C_2^3.C_2^4.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{6}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{3}) \), 4.4.4352.1, 4.4.9792.1, \(\Q(\sqrt{2}, \sqrt{3})\), 8.8.1534132224.1

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 R R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
2Data not computed
3Data not computed
$17$17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.8.6.2$x^{8} + 85 x^{4} + 2601$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$97$97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.1.1$x^{2} - 97$$2$$1$$1$$C_2$$[\ ]_{2}$
97.2.1.1$x^{2} - 97$$2$$1$$1$$C_2$$[\ ]_{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$