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Properties

Label 16.0.11945283218...0096.3
Degree $16$
Signature $[0, 8]$
Discriminant $2^{20}\cdot 3^{12}\cdot 11^{8}$
Root discriminant $17.98$
Ramified primes $2, 3, 11$
Class number $2$
Class group $[2]$
Galois group $C_2^2\wr C_2$ (as 16T46)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16, 0, -144, 0, 524, 0, -828, 0, 705, 0, -345, 0, 98, 0, -15, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 15*x^14 + 98*x^12 - 345*x^10 + 705*x^8 - 828*x^6 + 524*x^4 - 144*x^2 + 16)
 
gp: K = bnfinit(x^16 - 15*x^14 + 98*x^12 - 345*x^10 + 705*x^8 - 828*x^6 + 524*x^4 - 144*x^2 + 16, 1)
 

Normalized defining polynomial

\( x^{16} - 15 x^{14} + 98 x^{12} - 345 x^{10} + 705 x^{8} - 828 x^{6} + 524 x^{4} - 144 x^{2} + 16 \)

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:  \(119452832185734660096=2^{20}\cdot 3^{12}\cdot 11^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.98$
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:  $2, 3, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5}$, $\frac{1}{88} a^{12} - \frac{1}{4} a^{11} - \frac{21}{88} a^{10} - \frac{1}{4} a^{9} - \frac{2}{11} a^{8} - \frac{5}{88} a^{6} + \frac{1}{4} a^{5} - \frac{1}{88} a^{4} - \frac{1}{4} a^{3} + \frac{13}{44} a^{2} - \frac{1}{2} a + \frac{9}{22}$, $\frac{1}{88} a^{13} - \frac{21}{88} a^{11} - \frac{2}{11} a^{9} - \frac{5}{88} a^{7} - \frac{1}{88} a^{5} + \frac{13}{44} a^{3} + \frac{9}{22} a$, $\frac{1}{4048} a^{14} - \frac{3}{4048} a^{12} - \frac{1}{4} a^{11} + \frac{353}{2024} a^{10} - \frac{1}{4} a^{9} + \frac{675}{4048} a^{8} - \frac{1}{2} a^{7} - \frac{1499}{4048} a^{6} + \frac{1}{4} a^{5} - \frac{449}{1012} a^{4} - \frac{1}{4} a^{3} + \frac{37}{253} a^{2} - \frac{73}{506}$, $\frac{1}{4048} a^{15} - \frac{3}{4048} a^{13} + \frac{353}{2024} a^{11} + \frac{675}{4048} a^{9} - \frac{1499}{4048} a^{7} - \frac{449}{1012} a^{5} + \frac{37}{253} a^{3} - \frac{73}{506} a$

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:  \( -\frac{71}{1012} a^{14} + \frac{509}{506} a^{12} - \frac{6311}{1012} a^{10} + \frac{20661}{1012} a^{8} - \frac{9566}{253} a^{6} + \frac{38667}{1012} a^{4} - \frac{4896}{253} a^{2} + \frac{83}{23} \) (order $6$)
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:  \( 12532.8296063 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\wr C_2$ (as 16T46):

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 $C_2^2\wr C_2$
Character table for $C_2^2\wr C_2$

Intermediate fields

\(\Q(\sqrt{33}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{-3}) \), 4.0.13068.1, 4.0.5808.2, 4.4.13068.1, 4.0.5808.1, 4.0.52272.1, \(\Q(\sqrt{-3}, \sqrt{-11})\), 4.0.432.1, 8.0.2732361984.3, 8.0.170772624.1, 8.0.303595776.3

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 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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$2.4.4.4$x^{4} - 5$$2$$2$$4$$D_{4}$$[2, 2]^{2}$
2.4.4.4$x^{4} - 5$$2$$2$$4$$D_{4}$$[2, 2]^{2}$
2.8.12.14$x^{8} + 12 x^{4} + 144$$4$$2$$12$$D_4$$[2, 2]^{2}$
$3$3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
$11$11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$