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Properties

Label 16.0.26843545600...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{40}\cdot 5^{12}$
Root discriminant $18.91$
Ramified primes $2, 5$
Class number $2$
Class group $[2]$
Galois group $C_2^2.D_4$ (as 16T33)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 8, 0, -40, 204, -312, 368, -460, 691, -712, 408, -76, -36, 16, 4, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 4*x^14 + 16*x^13 - 36*x^12 - 76*x^11 + 408*x^10 - 712*x^9 + 691*x^8 - 460*x^7 + 368*x^6 - 312*x^5 + 204*x^4 - 40*x^3 + 8*x + 1)
 
gp: K = bnfinit(x^16 - 4*x^15 + 4*x^14 + 16*x^13 - 36*x^12 - 76*x^11 + 408*x^10 - 712*x^9 + 691*x^8 - 460*x^7 + 368*x^6 - 312*x^5 + 204*x^4 - 40*x^3 + 8*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 4 x^{14} + 16 x^{13} - 36 x^{12} - 76 x^{11} + 408 x^{10} - 712 x^{9} + 691 x^{8} - 460 x^{7} + 368 x^{6} - 312 x^{5} + 204 x^{4} - 40 x^{3} + 8 x + 1 \)

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:  \(268435456000000000000=2^{40}\cdot 5^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.91$
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:  $2, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{183} a^{14} - \frac{16}{61} a^{13} + \frac{32}{183} a^{12} + \frac{1}{61} a^{11} - \frac{1}{3} a^{10} + \frac{16}{183} a^{9} + \frac{3}{61} a^{8} - \frac{16}{61} a^{7} - \frac{32}{183} a^{6} - \frac{12}{61} a^{5} + \frac{5}{61} a^{4} - \frac{21}{61} a^{3} + \frac{27}{61} a^{2} - \frac{46}{183} a - \frac{67}{183}$, $\frac{1}{728544871155513} a^{15} - \frac{920020633696}{728544871155513} a^{14} - \frac{104395696526782}{728544871155513} a^{13} + \frac{113112391562500}{728544871155513} a^{12} + \frac{56051748475433}{728544871155513} a^{11} + \frac{231170961923684}{728544871155513} a^{10} + \frac{45057103093637}{728544871155513} a^{9} - \frac{11886173159772}{242848290385171} a^{8} + \frac{132502758313810}{728544871155513} a^{7} - \frac{162648913552984}{728544871155513} a^{6} + \frac{95872164717581}{242848290385171} a^{5} + \frac{120825673225489}{242848290385171} a^{4} + \frac{16615762940880}{242848290385171} a^{3} + \frac{75181305638816}{728544871155513} a^{2} + \frac{36947810420051}{242848290385171} a - \frac{65752878945761}{728544871155513}$

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:  \( -\frac{42721846858}{103736988631} a^{15} + \frac{553185840820}{311210965893} a^{14} - \frac{226275078280}{103736988631} a^{13} - \frac{1877178247045}{311210965893} a^{12} + \frac{1753457451480}{103736988631} a^{11} + \frac{8215497001418}{311210965893} a^{10} - \frac{55314769018880}{311210965893} a^{9} + \frac{36060036446920}{103736988631} a^{8} - \frac{39518439769580}{103736988631} a^{7} + \frac{88337624721220}{311210965893} a^{6} - \frac{22359251571144}{103736988631} a^{5} + \frac{18717569141210}{103736988631} a^{4} - \frac{13338462962210}{103736988631} a^{3} + \frac{4557878596410}{103736988631} a^{2} - \frac{1984935653800}{311210965893} a - \frac{878668725724}{311210965893} \) (order $4$)
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:  \( 6060.05248189 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2.D_4$ (as 16T33):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $C_2^2.D_4$
Character table for $C_2^2.D_4$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-5}) \), 4.0.32000.1 x2, \(\Q(i, \sqrt{5})\), 4.2.8000.1 x2, 8.0.16384000000.2 x2, 8.0.204800000.2 x2, 8.0.1024000000.6

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
2Data not computed
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$