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Properties

Label 16.0.33751961368...3344.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 17^{4}\cdot 97^{2}$
Root discriminant $14.39$
Ramified primes $2, 17, 97$
Class number $1$
Class group Trivial
Galois group $C_2\times D_4^2.C_2$ (as 16T509)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -8, 70, -148, 178, -244, 316, -256, 135, -84, 68, -24, -14, 12, 2, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 2*x^14 + 12*x^13 - 14*x^12 - 24*x^11 + 68*x^10 - 84*x^9 + 135*x^8 - 256*x^7 + 316*x^6 - 244*x^5 + 178*x^4 - 148*x^3 + 70*x^2 - 8*x + 1)
 
gp: K = bnfinit(x^16 - 4*x^15 + 2*x^14 + 12*x^13 - 14*x^12 - 24*x^11 + 68*x^10 - 84*x^9 + 135*x^8 - 256*x^7 + 316*x^6 - 244*x^5 + 178*x^4 - 148*x^3 + 70*x^2 - 8*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 2 x^{14} + 12 x^{13} - 14 x^{12} - 24 x^{11} + 68 x^{10} - 84 x^{9} + 135 x^{8} - 256 x^{7} + 316 x^{6} - 244 x^{5} + 178 x^{4} - 148 x^{3} + 70 x^{2} - 8 x + 1 \)

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:  \(3375196136846393344=2^{32}\cdot 17^{4}\cdot 97^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $14.39$
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:  $2, 17, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{12} a^{12} + \frac{1}{6} a^{11} + \frac{1}{12} a^{10} - \frac{1}{6} a^{9} - \frac{1}{6} a^{8} - \frac{1}{6} a^{7} - \frac{1}{12} a^{6} - \frac{1}{2} a^{5} + \frac{1}{6} a^{4} - \frac{1}{2} a^{3} + \frac{1}{12} a^{2} + \frac{1}{6} a - \frac{1}{12}$, $\frac{1}{12} a^{13} - \frac{1}{4} a^{11} + \frac{1}{6} a^{10} + \frac{1}{6} a^{9} + \frac{1}{6} a^{8} + \frac{1}{4} a^{7} + \frac{1}{6} a^{6} + \frac{1}{6} a^{5} + \frac{1}{6} a^{4} + \frac{1}{12} a^{3} - \frac{1}{2} a^{2} - \frac{5}{12} a + \frac{1}{6}$, $\frac{1}{24} a^{14} - \frac{1}{24} a^{13} - \frac{1}{24} a^{11} - \frac{1}{8} a^{10} - \frac{1}{4} a^{9} + \frac{1}{24} a^{8} + \frac{5}{24} a^{7} - \frac{3}{8} a^{6} + \frac{1}{4} a^{5} + \frac{11}{24} a^{4} - \frac{1}{24} a^{3} + \frac{5}{12} a^{2} - \frac{11}{24} a + \frac{1}{24}$, $\frac{1}{304248} a^{15} - \frac{1969}{101416} a^{14} - \frac{193}{10866} a^{13} - \frac{8185}{304248} a^{12} - \frac{8683}{304248} a^{11} - \frac{1895}{25354} a^{10} + \frac{73597}{304248} a^{9} - \frac{795}{101416} a^{8} - \frac{56065}{304248} a^{7} - \frac{687}{3622} a^{6} + \frac{32327}{304248} a^{5} - \frac{139091}{304248} a^{4} - \frac{20613}{50708} a^{3} - \frac{86083}{304248} a^{2} + \frac{28505}{304248} a + \frac{23753}{152124}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( \frac{15787}{76062} a^{15} - \frac{70573}{304248} a^{14} - \frac{43411}{43464} a^{13} + \frac{63429}{50708} a^{12} + \frac{739463}{304248} a^{11} - \frac{1291399}{304248} a^{10} + \frac{14967}{50708} a^{9} - \frac{17897}{304248} a^{8} + \frac{2435893}{304248} a^{7} - \frac{100289}{14488} a^{6} - \frac{421219}{50708} a^{5} + \frac{2818661}{304248} a^{4} - \frac{130175}{101416} a^{3} + \frac{250918}{38031} a^{2} - \frac{1913071}{304248} a + \frac{113511}{101416} \) (order $8$)
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:  \( 2099.90373907 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times D_4^2.C_2$ (as 16T509):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 40 conjugacy class representatives for $C_2\times D_4^2.C_2$
Character table for $C_2\times D_4^2.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), 4.0.1088.2, 4.4.4352.1, \(\Q(\zeta_{8})\), 8.4.1837170688.2, 8.4.1837170688.1, 8.0.18939904.2

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

Sibling fields

Degree 16 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$2$2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
$17$17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
$97$$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
97.2.1.2$x^{2} + 485$$2$$1$$1$$C_2$$[\ ]_{2}$
97.2.1.2$x^{2} + 485$$2$$1$$1$$C_2$$[\ ]_{2}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$