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Properties

Label 16.0.42764397425...0736.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{44}\cdot 79^{6}$
Root discriminant $34.63$
Ramified primes $2, 79$
Class number $32$ (GRH)
Class group $[2, 16]$ (GRH)
Galois group $\GL(2,Z/4)$ (as 16T186)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1511, -600, 352, -536, 1244, -1544, 1060, -288, -108, 432, -88, -128, 172, -96, 36, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 36*x^14 - 96*x^13 + 172*x^12 - 128*x^11 - 88*x^10 + 432*x^9 - 108*x^8 - 288*x^7 + 1060*x^6 - 1544*x^5 + 1244*x^4 - 536*x^3 + 352*x^2 - 600*x + 1511)
 
gp: K = bnfinit(x^16 - 8*x^15 + 36*x^14 - 96*x^13 + 172*x^12 - 128*x^11 - 88*x^10 + 432*x^9 - 108*x^8 - 288*x^7 + 1060*x^6 - 1544*x^5 + 1244*x^4 - 536*x^3 + 352*x^2 - 600*x + 1511, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 36 x^{14} - 96 x^{13} + 172 x^{12} - 128 x^{11} - 88 x^{10} + 432 x^{9} - 108 x^{8} - 288 x^{7} + 1060 x^{6} - 1544 x^{5} + 1244 x^{4} - 536 x^{3} + 352 x^{2} - 600 x + 1511 \)

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:  \(4276439742589131330420736=2^{44}\cdot 79^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.63$
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:  $2, 79$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{3}$, $\frac{1}{6} a^{12} + \frac{1}{6} a^{10} - \frac{1}{6} a^{9} - \frac{1}{6} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{2} a^{4} + \frac{1}{3} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{6}$, $\frac{1}{6} a^{13} + \frac{1}{6} a^{11} - \frac{1}{6} a^{10} - \frac{1}{6} a^{9} + \frac{1}{6} a^{8} + \frac{1}{3} a^{7} - \frac{1}{2} a^{5} + \frac{1}{3} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{6} a - \frac{1}{2}$, $\frac{1}{258} a^{14} - \frac{1}{129} a^{13} - \frac{4}{129} a^{12} + \frac{1}{43} a^{11} + \frac{17}{129} a^{10} - \frac{5}{43} a^{9} + \frac{21}{86} a^{8} + \frac{25}{129} a^{7} + \frac{37}{86} a^{6} + \frac{34}{129} a^{5} - \frac{53}{129} a^{4} + \frac{5}{43} a^{3} + \frac{1}{3} a^{2} - \frac{17}{129} a + \frac{35}{86}$, $\frac{1}{40329997776325542545862} a^{15} + \frac{6651159825831394571}{20164998888162771272931} a^{14} - \frac{1417241299141061843093}{40329997776325542545862} a^{13} - \frac{731162473473795649301}{20164998888162771272931} a^{12} - \frac{912860027661873930419}{40329997776325542545862} a^{11} + \frac{2943632092112431020733}{20164998888162771272931} a^{10} + \frac{2489178441160336776722}{20164998888162771272931} a^{9} - \frac{1614365733166642015655}{13443332592108514181954} a^{8} + \frac{419061458514082825523}{40329997776325542545862} a^{7} + \frac{1676144212853378759318}{6721666296054257090977} a^{6} - \frac{1222916660188214875105}{40329997776325542545862} a^{5} - \frac{3210828449583576922174}{6721666296054257090977} a^{4} - \frac{892955654193740693113}{1920476084586930597422} a^{3} + \frac{2305235721443870860993}{20164998888162771272931} a^{2} + \frac{33168251304621561194}{960238042293465298711} a - \frac{15128407488780416402843}{40329997776325542545862}$

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

Class group and class number

$C_{2}\times C_{16}$, 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:  \( 24443.2023421 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$\GL(2,Z/4)$ (as 16T186):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 96
The 14 conjugacy class representatives for $\GL(2,Z/4)$
Character table for $\GL(2,Z/4)$

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.80896.1, 4.0.80896.3, 8.8.6544162816.1

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

Sibling fields

Degree 12 siblings: 12.0.2039165374083104768.1, 12.0.206497759400820736.1, 12.0.825991037603282944.1, 12.0.206497759400820736.2
Degree 16 sibling: Deg 16
Degree 24 siblings: data not computed
Degree 32 sibling: 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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$79$79.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
79.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.4.2.1$x^{4} + 395 x^{2} + 56169$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
79.4.2.1$x^{4} + 395 x^{2} + 56169$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$