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Properties

Label 16.0.17391168553...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 5^{14}\cdot 19^{8}$
Root discriminant $50.41$
Ramified primes $2, 5, 19$
Class number $32$ (GRH)
Class group $[2, 2, 2, 4]$ (GRH)
Galois group $C_2^3.C_4$ (as 16T36)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![429025, 0, -264950, 0, 142175, 0, -24500, 0, -215, 0, -120, 0, 155, 0, 10, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 10*x^14 + 155*x^12 - 120*x^10 - 215*x^8 - 24500*x^6 + 142175*x^4 - 264950*x^2 + 429025)
 
gp: K = bnfinit(x^16 + 10*x^14 + 155*x^12 - 120*x^10 - 215*x^8 - 24500*x^6 + 142175*x^4 - 264950*x^2 + 429025, 1)
 

Normalized defining polynomial

\( x^{16} + 10 x^{14} + 155 x^{12} - 120 x^{10} - 215 x^{8} - 24500 x^{6} + 142175 x^{4} - 264950 x^{2} + 429025 \)

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:  \(1739116855398400000000000000=2^{24}\cdot 5^{14}\cdot 19^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $50.41$
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:  $2, 5, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{10} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{10} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{10} a^{10} - \frac{1}{2}$, $\frac{1}{10} a^{11} - \frac{1}{2} a$, $\frac{1}{220} a^{12} - \frac{1}{20} a^{11} - \frac{1}{110} a^{10} + \frac{9}{220} a^{8} - \frac{1}{2} a^{7} + \frac{7}{44} a^{6} - \frac{15}{44} a^{4} - \frac{7}{22} a^{2} - \frac{1}{4} a - \frac{21}{44}$, $\frac{1}{220} a^{13} + \frac{9}{220} a^{11} + \frac{9}{220} a^{9} - \frac{1}{20} a^{8} - \frac{15}{44} a^{7} - \frac{1}{4} a^{6} - \frac{15}{44} a^{5} + \frac{1}{4} a^{4} - \frac{7}{22} a^{3} + \frac{1}{4} a^{2} - \frac{5}{22} a + \frac{1}{4}$, $\frac{1}{9134781358533820} a^{14} + \frac{3794013460197}{9134781358533820} a^{12} + \frac{41200056476499}{9134781358533820} a^{10} - \frac{1}{20} a^{9} + \frac{53873831374119}{1826956271706764} a^{8} - \frac{1}{4} a^{7} - \frac{6379409615717}{1826956271706764} a^{6} + \frac{1}{4} a^{5} - \frac{166041010754722}{456739067926691} a^{4} + \frac{1}{4} a^{3} + \frac{102017300102084}{456739067926691} a^{2} + \frac{1}{4} a - \frac{155771780491089}{456739067926691}$, $\frac{1}{5983281789839652100} a^{15} + \frac{2226323715498461}{1196656357967930420} a^{13} + \frac{32278931246988413}{1196656357967930420} a^{11} - \frac{1}{20} a^{10} - \frac{20291775558087571}{1196656357967930420} a^{9} - \frac{1}{20} a^{8} - \frac{451015448120499139}{1196656357967930420} a^{7} + \frac{1}{4} a^{6} - \frac{8628197841254951}{119665635796793042} a^{5} + \frac{1}{4} a^{4} + \frac{3345831792254061}{10878694163344822} a^{3} + \frac{1}{4} a^{2} + \frac{27655537506948833}{59832817898396521} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{4}$, which has order $32$ (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:  \( 246558.133646 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_4$ (as 16T36):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $C_2^3.C_4$
Character table for $C_2^3.C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{95}) \), \(\Q(\sqrt{19}) \), 4.4.45125.1, \(\Q(\zeta_{20})^+\), \(\Q(\sqrt{5}, \sqrt{19})\), 8.0.2606420000000.1 x2, 8.0.115520000000.1 x2, 8.8.521284000000.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: data not computed
Degree 8 siblings: data not computed
Degree 16 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 ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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$5.8.7.2$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}$
$19$19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$