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Properties

Label 16.4.54566491401...0000.1
Degree $16$
Signature $[4, 6]$
Discriminant $2^{24}\cdot 5^{8}\cdot 13^{6}\cdot 29^{7}$
Root discriminant $72.20$
Ramified primes $2, 5, 13, 29$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group 16T1558

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![153781, 555822, 299738, 686428, 750695, -218166, -385816, -34630, 48234, 8092, -828, 222, -96, -116, 16, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 16*x^14 - 116*x^13 - 96*x^12 + 222*x^11 - 828*x^10 + 8092*x^9 + 48234*x^8 - 34630*x^7 - 385816*x^6 - 218166*x^5 + 750695*x^4 + 686428*x^3 + 299738*x^2 + 555822*x + 153781)
 
gp: K = bnfinit(x^16 - 6*x^15 + 16*x^14 - 116*x^13 - 96*x^12 + 222*x^11 - 828*x^10 + 8092*x^9 + 48234*x^8 - 34630*x^7 - 385816*x^6 - 218166*x^5 + 750695*x^4 + 686428*x^3 + 299738*x^2 + 555822*x + 153781, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 16 x^{14} - 116 x^{13} - 96 x^{12} + 222 x^{11} - 828 x^{10} + 8092 x^{9} + 48234 x^{8} - 34630 x^{7} - 385816 x^{6} - 218166 x^{5} + 750695 x^{4} + 686428 x^{3} + 299738 x^{2} + 555822 x + 153781 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(545664914012032080281600000000=2^{24}\cdot 5^{8}\cdot 13^{6}\cdot 29^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $72.20$
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:  $2, 5, 13, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{62548198477022244654351618274765286161842608079629} a^{15} - \frac{1176615204658696205322438846183217717022230786882}{5686199861547476786759238024978662378349328007239} a^{14} - \frac{22682901754371224756119846601630213988408114039354}{62548198477022244654351618274765286161842608079629} a^{13} + \frac{20525846269171988880855291562070692076608504473422}{62548198477022244654351618274765286161842608079629} a^{12} + \frac{40818419662261023754061904409265402711312892815}{62548198477022244654351618274765286161842608079629} a^{11} + \frac{29509277337426236811096514679716739904073820721701}{62548198477022244654351618274765286161842608079629} a^{10} + \frac{29451914215047539021376190809820939558384946617155}{62548198477022244654351618274765286161842608079629} a^{9} - \frac{29088153664852087214900808796944364083895205980011}{62548198477022244654351618274765286161842608079629} a^{8} + \frac{2209004461349453204587641551150624435651702759570}{62548198477022244654351618274765286161842608079629} a^{7} - \frac{29891839694732499666198086390923482069490609960843}{62548198477022244654351618274765286161842608079629} a^{6} + \frac{16900498768900553164610445063375257494574517340140}{62548198477022244654351618274765286161842608079629} a^{5} + \frac{20269423279434060685970692167604812023915225852979}{62548198477022244654351618274765286161842608079629} a^{4} - \frac{26719402825704520823478639314755667968761918568314}{62548198477022244654351618274765286161842608079629} a^{3} - \frac{5768590351178805670438530215582956881237162889870}{62548198477022244654351618274765286161842608079629} a^{2} + \frac{21860071932545380871575267920526238007564826548883}{62548198477022244654351618274765286161842608079629} a + \frac{490778124950491129581394310400085186534262581192}{1025380302902004010727075709422381740358075542289}$

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:  $9$
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:  \( 111038529.177 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1558:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4096
The 94 conjugacy class representatives for t16n1558 are not computed
Character table for t16n1558 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.725.1, 8.4.659478560000.2

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 $16$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ R $16$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ $16$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$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}$
$13$13.8.0.1$x^{8} + 4 x^{2} - x + 6$$1$$8$$0$$C_8$$[\ ]^{8}$
13.8.6.3$x^{8} + 65 x^{4} + 1352$$4$$2$$6$$C_8$$[\ ]_{4}^{2}$
$29$29.8.7.3$x^{8} + 58$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
29.8.0.1$x^{8} + x^{2} - 3 x + 3$$1$$8$$0$$C_8$$[\ ]^{8}$