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Properties

Label 16.0.96826519964...0000.9
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 3^{8}\cdot 5^{8}\cdot 7^{8}$
Root discriminant $20.49$
Ramified primes $2, 3, 5, 7$
Class number $4$
Class group $[4]$
Galois group $Q_8 : C_2^2$ (as 16T23)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![144, -576, 768, -288, 148, -440, 568, -664, 489, -218, 141, -24, 28, -4, 8, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 8*x^14 - 4*x^13 + 28*x^12 - 24*x^11 + 141*x^10 - 218*x^9 + 489*x^8 - 664*x^7 + 568*x^6 - 440*x^5 + 148*x^4 - 288*x^3 + 768*x^2 - 576*x + 144)
 
gp: K = bnfinit(x^16 + 8*x^14 - 4*x^13 + 28*x^12 - 24*x^11 + 141*x^10 - 218*x^9 + 489*x^8 - 664*x^7 + 568*x^6 - 440*x^5 + 148*x^4 - 288*x^3 + 768*x^2 - 576*x + 144, 1)
 

Normalized defining polynomial

\( x^{16} + 8 x^{14} - 4 x^{13} + 28 x^{12} - 24 x^{11} + 141 x^{10} - 218 x^{9} + 489 x^{8} - 664 x^{7} + 568 x^{6} - 440 x^{5} + 148 x^{4} - 288 x^{3} + 768 x^{2} - 576 x + 144 \)

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:  \(968265199641600000000=2^{16}\cdot 3^{8}\cdot 5^{8}\cdot 7^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.49$
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:  $2, 3, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{6} a^{10} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{6} a^{4} - \frac{1}{3} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{6} a^{11} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{6} a^{5} - \frac{1}{3} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{1608} a^{12} - \frac{1}{12} a^{11} + \frac{1}{268} a^{10} - \frac{32}{201} a^{9} + \frac{1}{6} a^{8} + \frac{2}{67} a^{7} + \frac{79}{536} a^{6} - \frac{13}{201} a^{5} + \frac{515}{1608} a^{4} + \frac{145}{804} a^{3} + \frac{391}{804} a^{2} - \frac{19}{134} a + \frac{51}{134}$, $\frac{1}{4824} a^{13} - \frac{1}{4824} a^{12} + \frac{35}{1206} a^{11} + \frac{137}{2412} a^{10} - \frac{137}{1206} a^{9} - \frac{323}{1206} a^{8} - \frac{1955}{4824} a^{7} + \frac{329}{4824} a^{6} - \frac{419}{1608} a^{5} + \frac{109}{536} a^{4} - \frac{413}{1206} a^{3} - \frac{347}{804} a^{2} - \frac{32}{201} a - \frac{17}{134}$, $\frac{1}{9648} a^{14} - \frac{1}{4824} a^{12} + \frac{1}{804} a^{11} - \frac{79}{2412} a^{10} + \frac{61}{1206} a^{9} + \frac{2381}{9648} a^{8} + \frac{209}{1608} a^{7} - \frac{2185}{9648} a^{6} - \frac{275}{804} a^{5} - \frac{61}{4824} a^{4} - \frac{176}{603} a^{3} - \frac{127}{268} a^{2} - \frac{125}{402} a + \frac{33}{67}$, $\frac{1}{22474639728} a^{15} - \frac{248399}{5618659932} a^{14} - \frac{156535}{1872886644} a^{13} + \frac{1575253}{11237319864} a^{12} - \frac{29001841}{624295548} a^{11} - \frac{145443173}{1872886644} a^{10} + \frac{376936477}{2497182192} a^{9} - \frac{5612596903}{11237319864} a^{8} - \frac{10857099515}{22474639728} a^{7} - \frac{192864377}{416197032} a^{6} + \frac{7155442}{20965149} a^{5} + \frac{2292729865}{11237319864} a^{4} + \frac{379102421}{1872886644} a^{3} - \frac{775112947}{1872886644} a^{2} - \frac{947809}{52024629} a + \frac{87796993}{312147774}$

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

Class group and class number

$C_{4}$, which has order $4$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 7388.19103958 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$Q_8:C_2^2$ (as 16T23):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 17 conjugacy class representatives for $Q_8 : C_2^2$
Character table for $Q_8 : C_2^2$

Intermediate fields

\(\Q(\sqrt{7}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-35}) \), \(\Q(\sqrt{-105}) \), \(\Q(\sqrt{3}, \sqrt{-35})\), \(\Q(\sqrt{3}, \sqrt{7})\), \(\Q(\sqrt{3}, \sqrt{-5})\), \(\Q(\sqrt{-5}, \sqrt{7})\), \(\Q(\sqrt{-15}, \sqrt{21})\), \(\Q(\sqrt{7}, \sqrt{-15})\), \(\Q(\sqrt{-5}, \sqrt{21})\), 8.0.31116960000.5, 8.4.1244678400.1 x2, 8.0.3457440000.1 x2

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 R R R ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$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}$
$3$3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$7$7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$