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Properties

Label 16.0.35248405205...0864.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 7^{6}\cdot 17^{8}$
Root discriminant $34.21$
Ramified primes $2, 7, 17$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $Q_8:C_2^2.D_6$ (as 16T754)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![109, -604, 2390, -3832, 4286, -844, -2214, 2924, -1033, 32, 342, -244, 164, -92, 36, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 36*x^14 - 92*x^13 + 164*x^12 - 244*x^11 + 342*x^10 + 32*x^9 - 1033*x^8 + 2924*x^7 - 2214*x^6 - 844*x^5 + 4286*x^4 - 3832*x^3 + 2390*x^2 - 604*x + 109)
 
gp: K = bnfinit(x^16 - 8*x^15 + 36*x^14 - 92*x^13 + 164*x^12 - 244*x^11 + 342*x^10 + 32*x^9 - 1033*x^8 + 2924*x^7 - 2214*x^6 - 844*x^5 + 4286*x^4 - 3832*x^3 + 2390*x^2 - 604*x + 109, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 36 x^{14} - 92 x^{13} + 164 x^{12} - 244 x^{11} + 342 x^{10} + 32 x^{9} - 1033 x^{8} + 2924 x^{7} - 2214 x^{6} - 844 x^{5} + 4286 x^{4} - 3832 x^{3} + 2390 x^{2} - 604 x + 109 \)

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:  \(3524840520547043444260864=2^{32}\cdot 7^{6}\cdot 17^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.21$
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:  $2, 7, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{51} a^{13} - \frac{4}{51} a^{12} - \frac{8}{51} a^{11} + \frac{8}{51} a^{10} + \frac{6}{17} a^{8} - \frac{4}{51} a^{7} + \frac{22}{51} a^{6} + \frac{1}{17} a^{5} + \frac{13}{51} a^{4} - \frac{7}{17} a^{3} + \frac{3}{17} a^{2} + \frac{2}{51} a + \frac{3}{17}$, $\frac{1}{28407} a^{14} - \frac{125}{28407} a^{13} - \frac{92}{1671} a^{12} - \frac{826}{28407} a^{11} + \frac{4438}{28407} a^{10} + \frac{1735}{28407} a^{9} + \frac{13645}{28407} a^{8} + \frac{4790}{28407} a^{7} - \frac{13267}{28407} a^{6} - \frac{9547}{28407} a^{5} - \frac{13171}{28407} a^{4} - \frac{256}{1671} a^{3} - \frac{300}{9469} a^{2} + \frac{10987}{28407} a + \frac{3212}{28407}$, $\frac{1}{78074976699418204671} a^{15} + \frac{598906018843184}{78074976699418204671} a^{14} - \frac{704338314817082188}{78074976699418204671} a^{13} + \frac{1494271640816086437}{26024992233139401557} a^{12} - \frac{2243584495169007196}{26024992233139401557} a^{11} + \frac{5264470217114399279}{78074976699418204671} a^{10} - \frac{1248809426364575915}{78074976699418204671} a^{9} - \frac{34312771955666043850}{78074976699418204671} a^{8} - \frac{601495537021315165}{26024992233139401557} a^{7} - \frac{10471578998165907070}{26024992233139401557} a^{6} + \frac{15279962769065379478}{78074976699418204671} a^{5} + \frac{35470573646073280006}{78074976699418204671} a^{4} - \frac{4757531370517236890}{78074976699418204671} a^{3} + \frac{10741362943805094489}{26024992233139401557} a^{2} - \frac{18021971519636868140}{78074976699418204671} a - \frac{1364790334042865897}{78074976699418204671}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( \frac{37047708773092}{8245324395334059} a^{15} - \frac{306502302067078}{8245324395334059} a^{14} + \frac{1404039819175900}{8245324395334059} a^{13} - \frac{3684546822812833}{8245324395334059} a^{12} + \frac{6608046299532820}{8245324395334059} a^{11} - \frac{9672850504271785}{8245324395334059} a^{10} + \frac{13369572248424368}{8245324395334059} a^{9} + \frac{279132040917461}{8245324395334059} a^{8} - \frac{42221297584361860}{8245324395334059} a^{7} + \frac{39528487038181373}{2748441465111353} a^{6} - \frac{99350022553071986}{8245324395334059} a^{5} - \frac{40305603592046603}{8245324395334059} a^{4} + \frac{63479802762411142}{2748441465111353} a^{3} - \frac{162807516981138857}{8245324395334059} a^{2} + \frac{26529166515543440}{2748441465111353} a - \frac{4598435985114987}{2748441465111353} \) (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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 540377.995509 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$Q_8:C_2^2.D_6$ (as 16T754):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 384
The 23 conjugacy class representatives for $Q_8:C_2^2.D_6$
Character table for $Q_8:C_2^2.D_6$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), 4.4.32368.1, 8.0.16762998784.2

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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$7$7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$17$$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.3.2.1$x^{3} - 17$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
17.3.2.1$x^{3} - 17$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
17.6.4.1$x^{6} + 136 x^{3} + 7803$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$