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Properties

Label 16.0.55383364000...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 5^{14}\cdot 61^{4}$
Root discriminant $22.85$
Ramified primes $2, 5, 61$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $C_2^4:C_2^2.C_2$ (as 16T317)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![121301, 197210, 150223, 57540, 14203, -19470, 8291, -5230, 1870, -2270, 481, -200, 93, -30, 8, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 8*x^14 - 30*x^13 + 93*x^12 - 200*x^11 + 481*x^10 - 2270*x^9 + 1870*x^8 - 5230*x^7 + 8291*x^6 - 19470*x^5 + 14203*x^4 + 57540*x^3 + 150223*x^2 + 197210*x + 121301)
 
gp: K = bnfinit(x^16 + 8*x^14 - 30*x^13 + 93*x^12 - 200*x^11 + 481*x^10 - 2270*x^9 + 1870*x^8 - 5230*x^7 + 8291*x^6 - 19470*x^5 + 14203*x^4 + 57540*x^3 + 150223*x^2 + 197210*x + 121301, 1)
 

Normalized defining polynomial

\( x^{16} + 8 x^{14} - 30 x^{13} + 93 x^{12} - 200 x^{11} + 481 x^{10} - 2270 x^{9} + 1870 x^{8} - 5230 x^{7} + 8291 x^{6} - 19470 x^{5} + 14203 x^{4} + 57540 x^{3} + 150223 x^{2} + 197210 x + 121301 \)

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:  \(5538336400000000000000=2^{16}\cdot 5^{14}\cdot 61^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.85$
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:  $2, 5, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{10} a^{12} - \frac{1}{5} a^{11} + \frac{3}{10} a^{10} + \frac{2}{5} a^{9} + \frac{3}{10} a^{8} - \frac{2}{5} a^{7} - \frac{3}{10} a^{6} + \frac{2}{5} a^{5} - \frac{1}{5} a^{4} - \frac{2}{5} a^{3} - \frac{1}{5} a^{2} + \frac{1}{5} a + \frac{1}{10}$, $\frac{1}{10} a^{13} - \frac{1}{10} a^{11} + \frac{1}{10} a^{9} + \frac{1}{5} a^{8} - \frac{1}{10} a^{7} - \frac{1}{5} a^{6} - \frac{2}{5} a^{5} + \frac{1}{5} a^{4} - \frac{1}{5} a^{2} - \frac{1}{2} a + \frac{1}{5}$, $\frac{1}{410} a^{14} - \frac{1}{205} a^{13} + \frac{11}{410} a^{12} - \frac{66}{205} a^{11} - \frac{23}{410} a^{10} - \frac{11}{205} a^{9} - \frac{149}{410} a^{8} + \frac{91}{205} a^{7} - \frac{13}{205} a^{6} - \frac{56}{205} a^{5} - \frac{14}{205} a^{4} + \frac{5}{41} a^{3} - \frac{11}{82} a^{2} + \frac{18}{205} a - \frac{91}{205}$, $\frac{1}{49627595437985249361279846804412490} a^{15} + \frac{20936783146217650047998775140897}{49627595437985249361279846804412490} a^{14} + \frac{479152834681508156413219726255077}{24813797718992624680639923402206245} a^{13} + \frac{1236762021687075865598767367105257}{24813797718992624680639923402206245} a^{12} + \frac{1363703683235883664253127330524927}{24813797718992624680639923402206245} a^{11} - \frac{11822122513433919041516487353335824}{24813797718992624680639923402206245} a^{10} + \frac{7417209346525622305631492059869031}{24813797718992624680639923402206245} a^{9} + \frac{12333703289103267793486677985329002}{24813797718992624680639923402206245} a^{8} - \frac{9908496678150283306530027257371817}{49627595437985249361279846804412490} a^{7} + \frac{21641233268218726676257027171444841}{49627595437985249361279846804412490} a^{6} - \frac{9089464497499313143921275398152351}{24813797718992624680639923402206245} a^{5} + \frac{6971869693635560910164531762780053}{24813797718992624680639923402206245} a^{4} - \frac{334604791545026646124815183267713}{49627595437985249361279846804412490} a^{3} - \frac{686101405172829696858882524469157}{9925519087597049872255969360882498} a^{2} - \frac{11802473806736625755416391918567049}{49627595437985249361279846804412490} a - \frac{52462246531355403368275242439109}{491362331069160884765146998063490}$

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

Class group and class number

$C_{2}\times C_{2}$, 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:  \( \frac{407940625124688565301660195839}{24813797718992624680639923402206245} a^{15} - \frac{943499185966773065055292973707}{49627595437985249361279846804412490} a^{14} + \frac{731324009257466521401579349609}{4962759543798524936127984680441249} a^{13} - \frac{30902795104869378747795460376147}{49627595437985249361279846804412490} a^{12} + \frac{52770759876692960729397158657804}{24813797718992624680639923402206245} a^{11} - \frac{257144143395315264286171577526013}{49627595437985249361279846804412490} a^{10} + \frac{59890210577946528212111742852324}{4962759543798524936127984680441249} a^{9} - \frac{434468918573082114308417412886007}{9925519087597049872255969360882498} a^{8} + \frac{346694227145244211220305896201820}{4962759543798524936127984680441249} a^{7} - \frac{615089323914179334938171394530938}{4962759543798524936127984680441249} a^{6} + \frac{4720772993194651702826041441714034}{24813797718992624680639923402206245} a^{5} - \frac{9671133724592264262617622052030011}{24813797718992624680639923402206245} a^{4} + \frac{2429074325281164626299876802365677}{4962759543798524936127984680441249} a^{3} + \frac{28755824678414324518876912239946793}{49627595437985249361279846804412490} a^{2} + \frac{20807618799094578626758600831501264}{24813797718992624680639923402206245} a + \frac{173600839000613458088422632018281}{245681165534580442382573499031745} \) (order $20$)
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:  \( 48078.5125638 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4:C_2^2.C_2$ (as 16T317):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 23 conjugacy class representatives for $C_2^4:C_2^2.C_2$
Character table for $C_2^4:C_2^2.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{-1}) \), \(\Q(\zeta_{5})\), \(\Q(\zeta_{20})^+\), \(\Q(i, \sqrt{5})\), \(\Q(\zeta_{20})\)

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
$5$5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
61Data not computed