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Properties

Label 16.16.2214429531...8144.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{48}\cdot 3^{12}\cdot 23^{6}$
Root discriminant $59.10$
Ramified primes $2, 3, 23$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_4.D_4$ (as 16T30)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8433, -67104, 158652, -31752, -285732, 195744, 149568, -135384, -32168, 38968, 2236, -5432, 184, 352, -32, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 - 32*x^14 + 352*x^13 + 184*x^12 - 5432*x^11 + 2236*x^10 + 38968*x^9 - 32168*x^8 - 135384*x^7 + 149568*x^6 + 195744*x^5 - 285732*x^4 - 31752*x^3 + 158652*x^2 - 67104*x + 8433)
 
gp: K = bnfinit(x^16 - 8*x^15 - 32*x^14 + 352*x^13 + 184*x^12 - 5432*x^11 + 2236*x^10 + 38968*x^9 - 32168*x^8 - 135384*x^7 + 149568*x^6 + 195744*x^5 - 285732*x^4 - 31752*x^3 + 158652*x^2 - 67104*x + 8433, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} - 32 x^{14} + 352 x^{13} + 184 x^{12} - 5432 x^{11} + 2236 x^{10} + 38968 x^{9} - 32168 x^{8} - 135384 x^{7} + 149568 x^{6} + 195744 x^{5} - 285732 x^{4} - 31752 x^{3} + 158652 x^{2} - 67104 x + 8433 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(22144295318673432094540038144=2^{48}\cdot 3^{12}\cdot 23^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $59.10$
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:  $2, 3, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{7} + \frac{1}{3} a^{4}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4}$, $\frac{1}{897} a^{12} - \frac{25}{299} a^{11} - \frac{98}{897} a^{10} + \frac{17}{897} a^{9} + \frac{38}{299} a^{8} - \frac{100}{299} a^{7} + \frac{89}{299} a^{6} - \frac{105}{299} a^{5} + \frac{38}{897} a^{4} + \frac{147}{299} a^{3} - \frac{102}{299} a^{2} + \frac{122}{299} a - \frac{140}{299}$, $\frac{1}{897} a^{13} - \frac{14}{299} a^{11} + \frac{142}{897} a^{10} - \frac{106}{897} a^{9} - \frac{122}{897} a^{8} - \frac{107}{897} a^{7} - \frac{323}{897} a^{6} - \frac{265}{897} a^{5} - \frac{99}{299} a^{4} - \frac{140}{299} a^{3} - \frac{53}{299} a^{2} + \frac{40}{299} a - \frac{35}{299}$, $\frac{1}{897} a^{14} - \frac{6}{299} a^{11} - \frac{12}{299} a^{10} - \frac{2}{299} a^{9} - \frac{103}{897} a^{8} - \frac{365}{897} a^{7} - \frac{38}{299} a^{6} - \frac{24}{299} a^{5} - \frac{319}{897} a^{4} + \frac{141}{299} a^{3} - \frac{58}{299} a^{2} + \frac{6}{299} a + \frac{100}{299}$, $\frac{1}{88346192360654196033} a^{15} - \frac{17831086352144075}{88346192360654196033} a^{14} + \frac{1204529351801656}{3841138798289312871} a^{13} - \frac{11079253614080924}{29448730786884732011} a^{12} + \frac{6743331073348584973}{88346192360654196033} a^{11} + \frac{129164334784880306}{1210221813159646521} a^{10} - \frac{1471574486265717744}{29448730786884732011} a^{9} - \frac{4583955408623617083}{29448730786884732011} a^{8} - \frac{9419326407330949268}{29448730786884732011} a^{7} + \frac{38498344666704472925}{88346192360654196033} a^{6} + \frac{34943929691810619292}{88346192360654196033} a^{5} + \frac{44005886094005595821}{88346192360654196033} a^{4} + \frac{7580474185221335484}{29448730786884732011} a^{3} - \frac{13785563359116221353}{29448730786884732011} a^{2} + \frac{4399146585217511502}{29448730786884732011} a + \frac{13592785778276293234}{29448730786884732011}$

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:  $15$
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:  \( 1875974037.6 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4.D_4$ (as 16T30):

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 $C_4.D_4$
Character table for $C_4.D_4$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{6}) \), 4.4.105984.1, 4.4.105984.2, \(\Q(\sqrt{2}, \sqrt{3})\), 8.8.404373897216.3, 8.8.1617495588864.1, 8.8.179721732096.1

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 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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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}$
$23$23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$