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Properties

Label 16.0.23712129003...0569.9
Degree $16$
Signature $[0, 8]$
Discriminant $13^{6}\cdot 53^{12}$
Root discriminant $51.40$
Ramified primes $13, 53$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1263

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![937, -2116, 7618, -9722, 13848, -10964, 9431, -4977, 3145, -1224, 716, -252, 150, -47, 21, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 21*x^14 - 47*x^13 + 150*x^12 - 252*x^11 + 716*x^10 - 1224*x^9 + 3145*x^8 - 4977*x^7 + 9431*x^6 - 10964*x^5 + 13848*x^4 - 9722*x^3 + 7618*x^2 - 2116*x + 937)
 
gp: K = bnfinit(x^16 - 4*x^15 + 21*x^14 - 47*x^13 + 150*x^12 - 252*x^11 + 716*x^10 - 1224*x^9 + 3145*x^8 - 4977*x^7 + 9431*x^6 - 10964*x^5 + 13848*x^4 - 9722*x^3 + 7618*x^2 - 2116*x + 937, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 21 x^{14} - 47 x^{13} + 150 x^{12} - 252 x^{11} + 716 x^{10} - 1224 x^{9} + 3145 x^{8} - 4977 x^{7} + 9431 x^{6} - 10964 x^{5} + 13848 x^{4} - 9722 x^{3} + 7618 x^{2} - 2116 x + 937 \)

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:  \(2371212900396504113756570569=13^{6}\cdot 53^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $51.40$
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:  $13, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{9} a^{10} - \frac{1}{9} a^{9} - \frac{1}{3} a^{7} + \frac{4}{9} a^{6} - \frac{1}{3} a^{5} + \frac{1}{9} a^{4} - \frac{1}{3} a^{3} + \frac{4}{9} a^{2} - \frac{4}{9} a + \frac{4}{9}$, $\frac{1}{27} a^{11} + \frac{1}{27} a^{10} - \frac{2}{27} a^{9} + \frac{2}{9} a^{8} - \frac{2}{27} a^{7} - \frac{4}{27} a^{6} + \frac{4}{27} a^{5} + \frac{8}{27} a^{4} - \frac{2}{27} a^{3} - \frac{5}{27} a^{2} - \frac{4}{27} a - \frac{1}{27}$, $\frac{1}{1053} a^{12} - \frac{1}{351} a^{11} + \frac{1}{351} a^{10} + \frac{140}{1053} a^{9} + \frac{514}{1053} a^{8} - \frac{320}{1053} a^{7} + \frac{515}{1053} a^{6} - \frac{62}{1053} a^{5} + \frac{218}{1053} a^{4} - \frac{44}{351} a^{3} + \frac{4}{81} a^{2} - \frac{34}{351} a - \frac{95}{1053}$, $\frac{1}{3159} a^{13} - \frac{1}{3159} a^{12} - \frac{1}{1053} a^{11} + \frac{146}{3159} a^{10} - \frac{259}{3159} a^{9} + \frac{236}{1053} a^{8} + \frac{928}{3159} a^{7} - \frac{1138}{3159} a^{6} - \frac{959}{3159} a^{5} + \frac{304}{3159} a^{4} - \frac{1265}{3159} a^{3} - \frac{1051}{3159} a^{2} - \frac{23}{243} a - \frac{1243}{3159}$, $\frac{1}{862407} a^{14} - \frac{23}{862407} a^{13} + \frac{397}{862407} a^{12} - \frac{6187}{862407} a^{11} - \frac{779}{287469} a^{10} + \frac{134089}{862407} a^{9} + \frac{182803}{862407} a^{8} + \frac{228142}{862407} a^{7} - \frac{55033}{862407} a^{6} - \frac{3970}{95823} a^{5} + \frac{427}{3159} a^{4} + \frac{145363}{862407} a^{3} + \frac{10889}{862407} a^{2} + \frac{402721}{862407} a - \frac{67532}{862407}$, $\frac{1}{2587221} a^{15} - \frac{44}{862407} a^{13} + \frac{487}{2587221} a^{12} - \frac{3188}{199017} a^{11} - \frac{118679}{2587221} a^{10} - \frac{143645}{862407} a^{9} - \frac{189128}{862407} a^{8} - \frac{1112429}{2587221} a^{7} - \frac{92912}{369603} a^{6} - \frac{56531}{862407} a^{5} + \frac{182764}{2587221} a^{4} + \frac{37288}{2587221} a^{3} - \frac{1103587}{2587221} a^{2} - \frac{45545}{862407} a - \frac{840706}{2587221}$

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:  \( 3644382.66642 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1263:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 34 conjugacy class representatives for t16n1263
Character table for t16n1263 is not computed

Intermediate fields

\(\Q(\sqrt{53}) \), 4.0.148877.1, 8.0.288136694677.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$13$13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$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$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.4.3.1$x^{4} - 13$$4$$1$$3$$C_4$$[\ ]_{4}$
$53$53.8.6.1$x^{8} - 1643 x^{4} + 1755625$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
53.8.6.1$x^{8} - 1643 x^{4} + 1755625$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$