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Properties

Label 16.0.22176987219...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{28}\cdot 5^{8}\cdot 59^{2}\cdot 157^{4}$
Root discriminant $44.32$
Ramified primes $2, 5, 59, 157$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group 16T1765

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![37696, -64256, 30016, 18432, -24352, 2784, 8384, -4416, -472, 1392, -744, 184, 44, -68, 32, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 32*x^14 - 68*x^13 + 44*x^12 + 184*x^11 - 744*x^10 + 1392*x^9 - 472*x^8 - 4416*x^7 + 8384*x^6 + 2784*x^5 - 24352*x^4 + 18432*x^3 + 30016*x^2 - 64256*x + 37696)
 
gp: K = bnfinit(x^16 - 8*x^15 + 32*x^14 - 68*x^13 + 44*x^12 + 184*x^11 - 744*x^10 + 1392*x^9 - 472*x^8 - 4416*x^7 + 8384*x^6 + 2784*x^5 - 24352*x^4 + 18432*x^3 + 30016*x^2 - 64256*x + 37696, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 32 x^{14} - 68 x^{13} + 44 x^{12} + 184 x^{11} - 744 x^{10} + 1392 x^{9} - 472 x^{8} - 4416 x^{7} + 8384 x^{6} + 2784 x^{5} - 24352 x^{4} + 18432 x^{3} + 30016 x^{2} - 64256 x + 37696 \)

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:  \(221769872198179225600000000=2^{28}\cdot 5^{8}\cdot 59^{2}\cdot 157^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $44.32$
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:  $2, 5, 59, 157$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{4} a^{6}$, $\frac{1}{4} a^{7}$, $\frac{1}{8} a^{8}$, $\frac{1}{8} a^{9}$, $\frac{1}{8} a^{10}$, $\frac{1}{16} a^{11}$, $\frac{1}{80} a^{12} + \frac{1}{40} a^{11} - \frac{1}{20} a^{10} + \frac{1}{20} a^{8} - \frac{1}{20} a^{7} + \frac{1}{10} a^{6} + \frac{1}{5} a^{5} - \frac{1}{5} a^{4} - \frac{1}{5} a^{3} - \frac{2}{5} a^{2} + \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{80} a^{13} + \frac{1}{40} a^{11} - \frac{1}{40} a^{10} + \frac{1}{20} a^{9} - \frac{1}{40} a^{8} - \frac{1}{20} a^{7} - \frac{1}{10} a^{5} + \frac{1}{5} a^{4} + \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{160} a^{14} + \frac{1}{40} a^{11} - \frac{1}{20} a^{10} + \frac{1}{20} a^{9} + \frac{1}{20} a^{8} + \frac{1}{20} a^{7} + \frac{1}{10} a^{6} - \frac{1}{10} a^{5} + \frac{1}{5} a^{4} + \frac{1}{5} a^{3} - \frac{2}{5} a^{2} + \frac{1}{5}$, $\frac{1}{12739844536608840480} a^{15} - \frac{8973474204376127}{4246614845536280160} a^{14} - \frac{912906215403193}{796240283538052530} a^{13} + \frac{13116578345456087}{3184961134152210120} a^{12} - \frac{3528029223890443}{424661484553628016} a^{11} + \frac{70445293764526759}{3184961134152210120} a^{10} + \frac{170871190706977157}{3184961134152210120} a^{9} + \frac{152222732522167573}{3184961134152210120} a^{8} + \frac{74668669306070243}{1592480567076105060} a^{7} + \frac{17148268970937097}{159248056707610506} a^{6} + \frac{9300524760665899}{88471142615339170} a^{5} + \frac{58049393822267551}{265413427846017510} a^{4} - \frac{35115212326160555}{159248056707610506} a^{3} - \frac{195996586222719112}{398120141769026265} a^{2} + \frac{8768969108683328}{26541342784601751} a + \frac{56340550880909924}{398120141769026265}$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  \( 1692319.12636 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1765:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 12288
The 74 conjugacy class representatives for t16n1765 are not computed
Character table for t16n1765 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 8.4.63101440000.3

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ R

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
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
59Data not computed
$157$157.4.0.1$x^{4} - x + 15$$1$$4$$0$$C_4$$[\ ]^{4}$
157.4.2.2$x^{4} - 157 x^{2} + 147894$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
157.4.2.2$x^{4} - 157 x^{2} + 147894$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
157.4.0.1$x^{4} - x + 15$$1$$4$$0$$C_4$$[\ ]^{4}$