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Properties

Label 16.0.13742275226...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{30}\cdot 5^{10}\cdot 41^{6}\cdot 359^{8}$
Root discriminant $764.94$
Ramified primes $2, 5, 41, 359$
Class number $6912$ (GRH)
Class group $[2, 2, 2, 6, 144]$ (GRH)
Galois group 16T1220

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![13594580577640000, 0, 408443341848000, 0, 6640477030400, 0, 71673084000, 0, 581558100, 0, 1705860, 0, -9166, 0, -6, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^14 - 9166*x^12 + 1705860*x^10 + 581558100*x^8 + 71673084000*x^6 + 6640477030400*x^4 + 408443341848000*x^2 + 13594580577640000)
 
gp: K = bnfinit(x^16 - 6*x^14 - 9166*x^12 + 1705860*x^10 + 581558100*x^8 + 71673084000*x^6 + 6640477030400*x^4 + 408443341848000*x^2 + 13594580577640000, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{14} - 9166 x^{12} + 1705860 x^{10} + 581558100 x^{8} + 71673084000 x^{6} + 6640477030400 x^{4} + 408443341848000 x^{2} + 13594580577640000 \)

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:  \(13742275226577042645713355852595855360000000000=2^{30}\cdot 5^{10}\cdot 41^{6}\cdot 359^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $764.94$
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:  $2, 5, 41, 359$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{6} a^{4} + \frac{1}{3}$, $\frac{1}{6} a^{5} + \frac{1}{3} a$, $\frac{1}{6} a^{6} + \frac{1}{3} a^{2}$, $\frac{1}{6} a^{7} + \frac{1}{3} a^{3}$, $\frac{1}{180} a^{8} - \frac{1}{30} a^{6} + \frac{1}{45} a^{4} + \frac{1}{3} a^{2} + \frac{2}{9}$, $\frac{1}{180} a^{9} - \frac{1}{30} a^{7} + \frac{1}{45} a^{5} + \frac{1}{3} a^{3} + \frac{2}{9} a$, $\frac{1}{720} a^{10} - \frac{1}{360} a^{9} + \frac{1}{60} a^{7} - \frac{1}{360} a^{6} + \frac{13}{180} a^{5} + \frac{1}{30} a^{4} - \frac{1}{6} a^{3} - \frac{13}{36} a^{2} + \frac{1}{18} a + \frac{1}{6}$, $\frac{1}{720} a^{11} - \frac{1}{360} a^{7} + \frac{1}{30} a^{5} - \frac{13}{36} a^{3} + \frac{1}{6} a$, $\frac{1}{1771200} a^{12} - \frac{1}{295200} a^{10} - \frac{571}{295200} a^{8} - \frac{1663}{29520} a^{6} + \frac{707}{9840} a^{4} + \frac{17}{36} a^{2} - \frac{11}{108}$, $\frac{1}{3542400} a^{13} - \frac{1}{590400} a^{11} - \frac{571}{590400} a^{9} - \frac{1663}{59040} a^{7} - \frac{311}{6560} a^{5} + \frac{17}{72} a^{3} + \frac{61}{216} a$, $\frac{1}{84306583326248976957479252526915936000} a^{14} - \frac{852549832380979612260218250919}{21076645831562244239369813131728984000} a^{12} - \frac{2637712687242442418088377108453341}{14051097221041496159579875421152656000} a^{10} - \frac{1}{360} a^{9} + \frac{1324980077354185996745971823713219}{702554861052074807978993771057632800} a^{8} + \frac{1}{60} a^{7} + \frac{290185792410030090511519282132179}{6244932098240664959813277964956736} a^{6} + \frac{13}{180} a^{5} - \frac{8966337161475913633051504875064493}{140510972210414961595798754211526560} a^{4} - \frac{1}{6} a^{3} + \frac{127511564963304249250817826466231}{5140645324771279082773125154080240} a^{2} + \frac{1}{18} a - \frac{124505034416793479425720368560767}{514064532477127908277312515408024}$, $\frac{1}{11987553083159342033583974916802176939840000} a^{15} - \frac{42889048700220766324636607552041981}{332987585643315056488443747688949359440000} a^{13} - \frac{1}{3542400} a^{12} - \frac{1266277428376568621610742128900527802701}{1997925513859890338930662486133696156640000} a^{11} + \frac{1}{590400} a^{10} + \frac{136523199257180842974222578173160290981}{99896275692994516946533124306684807832000} a^{9} + \frac{571}{590400} a^{8} - \frac{1421196517560152432085645824744018449817}{39958510277197806778613249722673923132800} a^{7} + \frac{1663}{59040} a^{6} + \frac{35045970863631568504952689115481646123}{3995851027719780677861324972267392313280} a^{5} + \frac{311}{6560} a^{4} + \frac{143824959612312343613212336014095807611}{730948358729228172779510665658669325600} a^{3} - \frac{17}{72} a^{2} + \frac{5784637831627047102280889380020236627}{24364945290974272425983688855288977520} a - \frac{61}{216}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{6}\times C_{144}$, which has order $6912$ (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:  \( 8590327656330 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1220:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 55 conjugacy class representatives for t16n1220 are not computed
Character table for t16n1220 is not computed

Intermediate fields

\(\Q(\sqrt{359}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{1795}) \), 4.0.2113648400.1, 4.0.1025.1, \(\Q(\sqrt{5}, \sqrt{359})\), 8.0.4467509558822560000.1

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{12}$

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$2.8.12.20$x^{8} + 8 x^{6} + 12 x^{4} + 80$$4$$2$$12$$C_2^3: C_4$$[2, 2, 2]^{4}$
2.8.18.20$x^{8} + 4 x^{7} + 2 x^{6} + 10 x^{4} + 20 x^{2} + 4$$4$$2$$18$$C_2^3: C_4$$[2, 2, 3, 7/2]^{2}$
$5$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$41$41.4.3.3$x^{4} + 246$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.3.4$x^{4} + 8856$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
359Data not computed