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Properties

Label 16.0.23933974895...4736.4
Degree $16$
Signature $[0, 8]$
Discriminant $2^{52}\cdot 3^{12}$
Root discriminant $21.69$
Ramified primes $2, 3$
Class number $2$
Class group $[2]$
Galois group $C_2^2:Q_8$ (as 16T31)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9, 0, 72, -72, 144, -120, 156, -144, 64, -64, 80, -16, -20, -8, 20, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 20*x^14 - 8*x^13 - 20*x^12 - 16*x^11 + 80*x^10 - 64*x^9 + 64*x^8 - 144*x^7 + 156*x^6 - 120*x^5 + 144*x^4 - 72*x^3 + 72*x^2 + 9)
 
gp: K = bnfinit(x^16 - 8*x^15 + 20*x^14 - 8*x^13 - 20*x^12 - 16*x^11 + 80*x^10 - 64*x^9 + 64*x^8 - 144*x^7 + 156*x^6 - 120*x^5 + 144*x^4 - 72*x^3 + 72*x^2 + 9, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 20 x^{14} - 8 x^{13} - 20 x^{12} - 16 x^{11} + 80 x^{10} - 64 x^{9} + 64 x^{8} - 144 x^{7} + 156 x^{6} - 120 x^{5} + 144 x^{4} - 72 x^{3} + 72 x^{2} + 9 \)

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:  \(2393397489569403764736=2^{52}\cdot 3^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.69$
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:  $2, 3$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{8} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{4}$, $\frac{1}{2481329953521651} a^{15} - \frac{225797236898626}{2481329953521651} a^{14} - \frac{402749530483634}{2481329953521651} a^{13} - \frac{35171747431640}{827109984507217} a^{12} + \frac{129449076179967}{827109984507217} a^{11} - \frac{888629951014549}{2481329953521651} a^{10} + \frac{194277726274711}{2481329953521651} a^{9} - \frac{444387000203902}{2481329953521651} a^{8} + \frac{317350144078945}{827109984507217} a^{7} - \frac{337239106667012}{2481329953521651} a^{6} + \frac{281440153713935}{2481329953521651} a^{5} - \frac{533207526243751}{2481329953521651} a^{4} - \frac{75673042662693}{827109984507217} a^{3} - \frac{32897791704431}{827109984507217} a^{2} + \frac{406414644119396}{827109984507217} a + \frac{338547577658481}{827109984507217}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( 6435.15910617 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:Q_8$ (as 16T31):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_2^2:Q_8$
Character table for $C_2^2:Q_8$

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{6}) \), 4.2.2048.1, \(\Q(\sqrt{2}, \sqrt{3})\), 4.2.18432.3, 8.4.1358954496.2, 8.0.12230590464.1, 8.4.764411904.3

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 16 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 R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
$3$3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$