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Properties

Label 16.0.34336147906...0625.6
Degree $16$
Signature $[0, 8]$
Discriminant $3^{8}\cdot 5^{12}\cdot 11^{8}$
Root discriminant $19.21$
Ramified primes $3, 5, 11$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $C_2 \times (C_2^2:C_4)$ (as 16T21)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![256, -128, 448, -320, 720, -272, 104, 400, -271, 200, 26, -34, 45, -10, 7, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 7*x^14 - 10*x^13 + 45*x^12 - 34*x^11 + 26*x^10 + 200*x^9 - 271*x^8 + 400*x^7 + 104*x^6 - 272*x^5 + 720*x^4 - 320*x^3 + 448*x^2 - 128*x + 256)
 
gp: K = bnfinit(x^16 - x^15 + 7*x^14 - 10*x^13 + 45*x^12 - 34*x^11 + 26*x^10 + 200*x^9 - 271*x^8 + 400*x^7 + 104*x^6 - 272*x^5 + 720*x^4 - 320*x^3 + 448*x^2 - 128*x + 256, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 7 x^{14} - 10 x^{13} + 45 x^{12} - 34 x^{11} + 26 x^{10} + 200 x^{9} - 271 x^{8} + 400 x^{7} + 104 x^{6} - 272 x^{5} + 720 x^{4} - 320 x^{3} + 448 x^{2} - 128 x + 256 \)

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:  \(343361479062744140625=3^{8}\cdot 5^{12}\cdot 11^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.21$
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:  $3, 5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{2}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{4} a^{8} - \frac{3}{8} a^{7} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5} + \frac{1}{8} a^{3}$, $\frac{1}{176} a^{12} - \frac{1}{16} a^{11} - \frac{1}{16} a^{10} + \frac{9}{44} a^{9} - \frac{1}{16} a^{8} + \frac{1}{4} a^{7} - \frac{39}{88} a^{6} - \frac{1}{4} a^{5} - \frac{5}{16} a^{4} - \frac{21}{88} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a + \frac{4}{11}$, $\frac{1}{352} a^{13} - \frac{1}{352} a^{12} + \frac{1}{32} a^{11} - \frac{15}{176} a^{10} + \frac{41}{352} a^{9} + \frac{1}{16} a^{8} - \frac{17}{176} a^{7} + \frac{9}{22} a^{6} - \frac{5}{32} a^{5} + \frac{7}{22} a^{4} + \frac{27}{88} a^{3} + \frac{2}{11} a - \frac{2}{11}$, $\frac{1}{704} a^{14} - \frac{1}{704} a^{13} - \frac{1}{704} a^{12} + \frac{7}{352} a^{11} + \frac{85}{704} a^{10} - \frac{73}{352} a^{9} + \frac{49}{352} a^{8} - \frac{15}{88} a^{7} + \frac{177}{704} a^{6} - \frac{15}{44} a^{5} + \frac{1}{11} a^{4} + \frac{1}{11} a^{3} + \frac{1}{11} a^{2} - \frac{1}{11} a - \frac{1}{11}$, $\frac{1}{336512} a^{15} - \frac{141}{336512} a^{14} - \frac{329}{336512} a^{13} - \frac{397}{168256} a^{12} - \frac{4471}{336512} a^{11} - \frac{6351}{168256} a^{10} - \frac{32909}{168256} a^{9} + \frac{5427}{21032} a^{8} - \frac{74823}{336512} a^{7} + \frac{4245}{84128} a^{6} + \frac{18283}{84128} a^{5} + \frac{13863}{42064} a^{4} - \frac{272}{2629} a^{3} - \frac{43}{2629} a^{2} + \frac{2487}{5258} a - \frac{338}{2629}$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{1519}{21032} a^{15} - \frac{1055}{84128} a^{14} - \frac{35125}{84128} a^{13} + \frac{24985}{84128} a^{12} - \frac{47775}{21032} a^{11} - \frac{28969}{84128} a^{10} + \frac{38515}{42064} a^{9} - \frac{240595}{21032} a^{8} + \frac{61625}{10516} a^{7} - \frac{482935}{84128} a^{6} - \frac{329411}{21032} a^{5} + \frac{427645}{42064} a^{4} - \frac{153885}{10516} a^{3} + \frac{12995}{5258} a^{2} - \frac{39905}{5258} a + \frac{12033}{2629} \) (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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 3350.16978018 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^2:C_4$ (as 16T21):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_2 \times (C_2^2:C_4)$
Character table for $C_2 \times (C_2^2:C_4)$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-3}) \), 4.2.12375.1, 4.2.1375.1, 4.4.15125.1, 4.0.136125.2, \(\Q(\sqrt{-3}, \sqrt{5})\), 4.2.2475.1, 4.2.275.1, 8.0.6125625.1, 8.0.153140625.1, 8.0.18530015625.2, 8.4.18530015625.1, 8.0.18530015625.7, 8.4.228765625.1, 8.0.18530015625.8

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 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$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}$
$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}$