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Properties

Label 16.2.58553107207...0000.2
Degree $16$
Signature $[2, 7]$
Discriminant $-\,2^{28}\cdot 5^{8}\cdot 89^{5}$
Root discriminant $30.58$
Ramified primes $2, 5, 89$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1778

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, -520, 526, 458, 195, -618, -188, -346, 248, -66, 66, -110, 54, -38, 14, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 14*x^14 - 38*x^13 + 54*x^12 - 110*x^11 + 66*x^10 - 66*x^9 + 248*x^8 - 346*x^7 - 188*x^6 - 618*x^5 + 195*x^4 + 458*x^3 + 526*x^2 - 520*x + 4)
 
gp: K = bnfinit(x^16 - 4*x^15 + 14*x^14 - 38*x^13 + 54*x^12 - 110*x^11 + 66*x^10 - 66*x^9 + 248*x^8 - 346*x^7 - 188*x^6 - 618*x^5 + 195*x^4 + 458*x^3 + 526*x^2 - 520*x + 4, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 14 x^{14} - 38 x^{13} + 54 x^{12} - 110 x^{11} + 66 x^{10} - 66 x^{9} + 248 x^{8} - 346 x^{7} - 188 x^{6} - 618 x^{5} + 195 x^{4} + 458 x^{3} + 526 x^{2} - 520 x + 4 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-585531072079462400000000=-\,2^{28}\cdot 5^{8}\cdot 89^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.58$
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:  $2, 5, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{6} a^{13} - \frac{1}{6} a^{12} - \frac{1}{6} a^{11} - \frac{1}{6} a^{10} - \frac{1}{6} a^{9} - \frac{1}{2} a^{8} - \frac{1}{6} a^{7} - \frac{1}{6} a^{6} - \frac{1}{6} a^{5} + \frac{1}{6} a^{4} - \frac{1}{2} a^{3} - \frac{1}{6} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{126} a^{14} - \frac{4}{63} a^{13} + \frac{4}{21} a^{12} + \frac{1}{3} a^{11} - \frac{2}{21} a^{10} + \frac{11}{63} a^{9} - \frac{5}{63} a^{8} - \frac{1}{21} a^{7} - \frac{2}{21} a^{6} - \frac{20}{63} a^{5} - \frac{8}{63} a^{4} - \frac{26}{63} a^{3} - \frac{1}{126} a^{2} - \frac{16}{63} a + \frac{23}{63}$, $\frac{1}{3384440826468906755754} a^{15} - \frac{4134791315522781409}{1128146942156302251918} a^{14} - \frac{113307967363021334359}{3384440826468906755754} a^{13} + \frac{42758990335116584}{200666478505212069} a^{12} - \frac{24852644221227480251}{376048980718767417306} a^{11} + \frac{294797333196706609592}{1692220413234453377877} a^{10} + \frac{597875199663821742811}{3384440826468906755754} a^{9} + \frac{775376733914863268165}{1692220413234453377877} a^{8} - \frac{372417932119413197687}{1128146942156302251918} a^{7} - \frac{13267610706318765323}{1692220413234453377877} a^{6} + \frac{47308531410802351621}{125349660239589139102} a^{5} + \frac{281912111376101696144}{564073471078151125959} a^{4} - \frac{14801465259862884250}{62674830119794569551} a^{3} + \frac{926160758554947700931}{3384440826468906755754} a^{2} + \frac{73403455165237711820}{564073471078151125959} a - \frac{608002189885630781660}{1692220413234453377877}$

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:  $8$
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:  \( 214592.070238 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1778:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16384
The 148 conjugacy class representatives for t16n1778 are not computed
Character table for t16n1778 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.2225.1, 8.8.5069440000.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$2$2.4.4.4$x^{4} - 5$$2$$2$$4$$D_{4}$$[2, 2]^{2}$
2.4.4.4$x^{4} - 5$$2$$2$$4$$D_{4}$$[2, 2]^{2}$
2.8.20.33$x^{8} + 8 x^{6} + 112$$4$$2$$20$$(((C_4 \times C_2): C_2):C_2):C_2$$[2, 2, 3, 7/2, 7/2]^{2}$
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$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$$[\ ]_{2}^{4}$
$89$89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.4.3.3$x^{4} + 267$$4$$1$$3$$C_4$$[\ ]_{4}$
89.4.0.1$x^{4} - x + 27$$1$$4$$0$$C_4$$[\ ]^{4}$
89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$