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Properties

Label 16.8.52222277247...0000.3
Degree $16$
Signature $[8, 4]$
Discriminant $2^{8}\cdot 5^{8}\cdot 11\cdot 29^{4}\cdot 509^{4}$
Root discriminant $40.49$
Ramified primes $2, 5, 11, 29, 509$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1719

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![68249, 80375, -118200, -98427, 73962, 34586, -15935, -8828, 1066, 2630, -210, -447, 40, 47, -3, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 - 3*x^14 + 47*x^13 + 40*x^12 - 447*x^11 - 210*x^10 + 2630*x^9 + 1066*x^8 - 8828*x^7 - 15935*x^6 + 34586*x^5 + 73962*x^4 - 98427*x^3 - 118200*x^2 + 80375*x + 68249)
 
gp: K = bnfinit(x^16 - 5*x^15 - 3*x^14 + 47*x^13 + 40*x^12 - 447*x^11 - 210*x^10 + 2630*x^9 + 1066*x^8 - 8828*x^7 - 15935*x^6 + 34586*x^5 + 73962*x^4 - 98427*x^3 - 118200*x^2 + 80375*x + 68249, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} - 3 x^{14} + 47 x^{13} + 40 x^{12} - 447 x^{11} - 210 x^{10} + 2630 x^{9} + 1066 x^{8} - 8828 x^{7} - 15935 x^{6} + 34586 x^{5} + 73962 x^{4} - 98427 x^{3} - 118200 x^{2} + 80375 x + 68249 \)

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:  \(52222277247435505100000000=2^{8}\cdot 5^{8}\cdot 11\cdot 29^{4}\cdot 509^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $40.49$
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:  $2, 5, 11, 29, 509$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{29} a^{12} + \frac{13}{29} a^{10} + \frac{2}{29} a^{9} + \frac{6}{29} a^{8} + \frac{12}{29} a^{7} - \frac{12}{29} a^{6} + \frac{3}{29} a^{5} + \frac{14}{29} a^{3} - \frac{5}{29} a^{2} + \frac{12}{29} a - \frac{11}{29}$, $\frac{1}{29} a^{13} + \frac{13}{29} a^{11} + \frac{2}{29} a^{10} + \frac{6}{29} a^{9} + \frac{12}{29} a^{8} - \frac{12}{29} a^{7} + \frac{3}{29} a^{6} + \frac{14}{29} a^{4} - \frac{5}{29} a^{3} + \frac{12}{29} a^{2} - \frac{11}{29} a$, $\frac{1}{9251} a^{14} + \frac{41}{9251} a^{13} - \frac{125}{9251} a^{12} + \frac{4479}{9251} a^{11} + \frac{3775}{9251} a^{10} - \frac{4426}{9251} a^{9} - \frac{135}{319} a^{8} - \frac{4494}{9251} a^{7} - \frac{4166}{9251} a^{6} - \frac{1966}{9251} a^{5} - \frac{2186}{9251} a^{4} - \frac{3662}{9251} a^{3} + \frac{2534}{9251} a^{2} - \frac{1614}{9251} a + \frac{1605}{9251}$, $\frac{1}{5500267707666295762925122121} a^{15} - \frac{78660998100796078868115}{5500267707666295762925122121} a^{14} - \frac{30409125283530796261821841}{5500267707666295762925122121} a^{13} + \frac{14337337230214650921275713}{5500267707666295762925122121} a^{12} - \frac{13617058784102165529698197}{189664403712630888376728349} a^{11} - \frac{1645104588692031202441118918}{5500267707666295762925122121} a^{10} + \frac{1231235333343947218579174423}{5500267707666295762925122121} a^{9} - \frac{781090261216155971250812596}{5500267707666295762925122121} a^{8} + \frac{1496839593719307102700057094}{5500267707666295762925122121} a^{7} - \frac{8109073845290307262509156}{289487774087699776996059059} a^{6} + \frac{507134553959587127994176476}{5500267707666295762925122121} a^{5} - \frac{1719459160875741980172687888}{5500267707666295762925122121} a^{4} + \frac{1558606317145030983569163799}{5500267707666295762925122121} a^{3} + \frac{1319979733757852436859331205}{5500267707666295762925122121} a^{2} + \frac{495145480994619130585728534}{5500267707666295762925122121} a + \frac{1252827054464114930606564729}{5500267707666295762925122121}$

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:  \( 9311584.24898 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1719:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 8192
The 152 conjugacy class representatives for t16n1719 are not computed
Character table for t16n1719 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.369025.1, 4.4.725.1, 4.4.12725.1, 8.8.136179450625.2

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 R ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$2$2.8.8.7$x^{8} + 2 x^{6} + 4 x^{5} + 16$$2$$4$$8$$((C_8 : C_2):C_2):C_2$$[2, 2, 2, 2]^{4}$
2.8.0.1$x^{8} + x^{4} + x^{3} + x + 1$$1$$8$$0$$C_8$$[\ ]^{8}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$29$29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
509Data not computed