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Properties

Label 16.8.45919170820...8224.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{40}\cdot 3^{12}\cdot 17^{4}\cdot 97^{2}$
Root discriminant $46.38$
Ramified primes $2, 3, 17, 97$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^3.C_2^4.C_2$ (as 16T595)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-503, 9636, -44728, 25776, 15796, -29244, 7580, 2592, 3145, 1044, -1888, 456, -170, -24, 20, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 20*x^14 - 24*x^13 - 170*x^12 + 456*x^11 - 1888*x^10 + 1044*x^9 + 3145*x^8 + 2592*x^7 + 7580*x^6 - 29244*x^5 + 15796*x^4 + 25776*x^3 - 44728*x^2 + 9636*x - 503)
 
gp: K = bnfinit(x^16 + 20*x^14 - 24*x^13 - 170*x^12 + 456*x^11 - 1888*x^10 + 1044*x^9 + 3145*x^8 + 2592*x^7 + 7580*x^6 - 29244*x^5 + 15796*x^4 + 25776*x^3 - 44728*x^2 + 9636*x - 503, 1)
 

Normalized defining polynomial

\( x^{16} + 20 x^{14} - 24 x^{13} - 170 x^{12} + 456 x^{11} - 1888 x^{10} + 1044 x^{9} + 3145 x^{8} + 2592 x^{7} + 7580 x^{6} - 29244 x^{5} + 15796 x^{4} + 25776 x^{3} - 44728 x^{2} + 9636 x - 503 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(459191708201416736032948224=2^{40}\cdot 3^{12}\cdot 17^{4}\cdot 97^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $46.38$
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:  $2, 3, 17, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{9} + \frac{1}{3} a^{7} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{7} + \frac{1}{3} a^{5} - \frac{1}{3} a$, $\frac{1}{9} a^{12} + \frac{1}{3} a^{7} + \frac{2}{9} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a + \frac{1}{9}$, $\frac{1}{9} a^{13} + \frac{2}{9} a^{7} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2} + \frac{1}{9} a - \frac{1}{3}$, $\frac{1}{901809} a^{14} - \frac{18998}{901809} a^{13} - \frac{3980}{100201} a^{12} + \frac{34622}{300603} a^{11} + \frac{10787}{100201} a^{10} + \frac{6670}{300603} a^{9} - \frac{58234}{901809} a^{8} + \frac{151874}{901809} a^{7} - \frac{125306}{300603} a^{6} + \frac{6100}{300603} a^{5} + \frac{89114}{300603} a^{4} + \frac{101159}{300603} a^{3} + \frac{74554}{901809} a^{2} + \frac{187141}{901809} a - \frac{46952}{100201}$, $\frac{1}{584817641946841020205822284771} a^{15} + \frac{270484567404749211621725}{584817641946841020205822284771} a^{14} - \frac{8294340572262155960172899203}{584817641946841020205822284771} a^{13} - \frac{14580391503501220444117545007}{584817641946841020205822284771} a^{12} + \frac{6264192908926252571596066349}{64979737994093446689535809419} a^{11} + \frac{8672927790597030476674151308}{194939213982280340068607428257} a^{10} - \frac{71484505417593659259833487031}{584817641946841020205822284771} a^{9} + \frac{56513958646744450643838954022}{584817641946841020205822284771} a^{8} - \frac{166814813835298692247758456497}{584817641946841020205822284771} a^{7} - \frac{229348465656828604758946455620}{584817641946841020205822284771} a^{6} + \frac{20709803479690840303217485721}{64979737994093446689535809419} a^{5} - \frac{9679013120046767996129533324}{64979737994093446689535809419} a^{4} - \frac{272394803058607194763947138425}{584817641946841020205822284771} a^{3} + \frac{157931951815297303277706936896}{584817641946841020205822284771} a^{2} - \frac{187684009180538099524567598215}{584817641946841020205822284771} a + \frac{252994170107472567026686048073}{584817641946841020205822284771}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $11$
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:  \( 21264663.1427 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_2^4.C_2$ (as 16T595):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 40 conjugacy class representatives for $C_2^3.C_2^4.C_2$
Character table for $C_2^3.C_2^4.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{6}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{3}) \), 4.4.4352.1, 4.4.9792.1, \(\Q(\sqrt{2}, \sqrt{3})\), 8.8.1534132224.1

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 R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
2Data not computed
3Data not computed
$17$17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
97Data not computed