↑

Properties

Label 16.8.14551885220...0625.1
Degree $16$
Signature $[8, 4]$
Discriminant $5^{8}\cdot 29^{10}\cdot 941^{2}$
Root discriminant $43.17$
Ramified primes $5, 29, 941$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_4^2:C_2^2.C_2$ (as 16T406)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2411, 73117, -156502, 88799, 49275, -108385, 64629, -8900, -8563, 4146, -86, -255, 19, 24, -9, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 9*x^14 + 24*x^13 + 19*x^12 - 255*x^11 - 86*x^10 + 4146*x^9 - 8563*x^8 - 8900*x^7 + 64629*x^6 - 108385*x^5 + 49275*x^4 + 88799*x^3 - 156502*x^2 + 73117*x + 2411)
 
gp: K = bnfinit(x^16 - 2*x^15 - 9*x^14 + 24*x^13 + 19*x^12 - 255*x^11 - 86*x^10 + 4146*x^9 - 8563*x^8 - 8900*x^7 + 64629*x^6 - 108385*x^5 + 49275*x^4 + 88799*x^3 - 156502*x^2 + 73117*x + 2411, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 9 x^{14} + 24 x^{13} + 19 x^{12} - 255 x^{11} - 86 x^{10} + 4146 x^{9} - 8563 x^{8} - 8900 x^{7} + 64629 x^{6} - 108385 x^{5} + 49275 x^{4} + 88799 x^{3} - 156502 x^{2} + 73117 x + 2411 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(145518852206990344406640625=5^{8}\cdot 29^{10}\cdot 941^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.17$
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:  $5, 29, 941$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{67367818826884631933954782580123532143} a^{15} + \frac{31764509441643036508515583629090485964}{67367818826884631933954782580123532143} a^{14} + \frac{7977469974768296433269785938352810700}{67367818826884631933954782580123532143} a^{13} - \frac{22413609226549439455460396557101051058}{67367818826884631933954782580123532143} a^{12} - \frac{8187957155082215321524651169155767597}{67367818826884631933954782580123532143} a^{11} - \frac{3204477379283892811731014427768628785}{9623974118126375990564968940017647449} a^{10} + \frac{23047826998395353705835376113160117291}{67367818826884631933954782580123532143} a^{9} + \frac{32817874264577936631825778195185530909}{67367818826884631933954782580123532143} a^{8} + \frac{797572905527648447119735583803278735}{9623974118126375990564968940017647449} a^{7} - \frac{32257403027738606981817404549817128020}{67367818826884631933954782580123532143} a^{6} + \frac{27144822205131323256223961330117430689}{67367818826884631933954782580123532143} a^{5} + \frac{24823583885658164056502728396224508405}{67367818826884631933954782580123532143} a^{4} + \frac{37856235740882522415331197021414601}{1643117532363039803267189819027403223} a^{3} - \frac{14397571219296152305340781350077805967}{67367818826884631933954782580123532143} a^{2} - \frac{13572541194670662103642880237675773392}{67367818826884631933954782580123532143} a - \frac{33142092909810032107008357799745769328}{67367818826884631933954782580123532143}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $11$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
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:  \( 8499043.31881 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4^2:C_2^2.C_2$ (as 16T406):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 26 conjugacy class representatives for $C_4^2:C_2^2.C_2$
Character table for $C_4^2:C_2^2.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{29}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{145}) \), 4.4.4205.1 x2, 4.4.725.1 x2, \(\Q(\sqrt{5}, \sqrt{29})\), 8.8.442050625.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$29$29.8.6.2$x^{8} + 145 x^{4} + 7569$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
29.8.4.1$x^{8} + 31958 x^{4} - 24389 x^{2} + 255328441$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
941Data not computed