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Properties

Label 16.0.79471930692...0625.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{8}\cdot 5^{8}\cdot 1327^{4}$
Root discriminant $23.38$
Ramified primes $3, 5, 1327$
Class number $2$
Class group $[2]$
Galois group $C_2\times C_2^2:S_4:C_2$ (as 16T724)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![25, -50, -20, 0, 241, -246, 90, -51, -32, 112, -9, -56, 20, 13, -6, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 6*x^14 + 13*x^13 + 20*x^12 - 56*x^11 - 9*x^10 + 112*x^9 - 32*x^8 - 51*x^7 + 90*x^6 - 246*x^5 + 241*x^4 - 20*x^2 - 50*x + 25)
 
gp: K = bnfinit(x^16 - x^15 - 6*x^14 + 13*x^13 + 20*x^12 - 56*x^11 - 9*x^10 + 112*x^9 - 32*x^8 - 51*x^7 + 90*x^6 - 246*x^5 + 241*x^4 - 20*x^2 - 50*x + 25, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 6 x^{14} + 13 x^{13} + 20 x^{12} - 56 x^{11} - 9 x^{10} + 112 x^{9} - 32 x^{8} - 51 x^{7} + 90 x^{6} - 246 x^{5} + 241 x^{4} - 20 x^{2} - 50 x + 25 \)

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:  \(7947193069254687890625=3^{8}\cdot 5^{8}\cdot 1327^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.38$
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:  $3, 5, 1327$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{6} a^{12} - \frac{1}{3} a^{11} - \frac{1}{2} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{6} a^{6} - \frac{1}{3} a^{4} - \frac{1}{6} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{6}$, $\frac{1}{6} a^{13} + \frac{1}{3} a^{11} - \frac{1}{2} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{2} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{6} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{690} a^{14} + \frac{7}{345} a^{13} - \frac{7}{230} a^{12} + \frac{113}{690} a^{11} + \frac{1}{3} a^{10} + \frac{93}{230} a^{9} - \frac{103}{230} a^{8} - \frac{1}{5} a^{7} + \frac{11}{230} a^{6} + \frac{33}{230} a^{5} - \frac{10}{69} a^{4} - \frac{101}{690} a^{3} - \frac{49}{690} a^{2} + \frac{14}{69} a - \frac{47}{138}$, $\frac{1}{2489540094870} a^{15} + \frac{77655733}{829846698290} a^{14} + \frac{136401747059}{2489540094870} a^{13} + \frac{4920479461}{108240873690} a^{12} + \frac{97599323039}{497908018974} a^{11} + \frac{161277385799}{2489540094870} a^{10} - \frac{200486543323}{829846698290} a^{9} + \frac{367607481529}{829846698290} a^{8} - \frac{124806059419}{829846698290} a^{7} - \frac{197950208287}{829846698290} a^{6} - \frac{157583845457}{497908018974} a^{5} + \frac{410780925913}{829846698290} a^{4} + \frac{32482610817}{829846698290} a^{3} - \frac{226778557201}{497908018974} a^{2} - \frac{33512776585}{497908018974} a - \frac{110638833829}{248954009487}$

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:  \( \frac{2852920022}{54120436845} a^{15} - \frac{929538031}{54120436845} a^{14} - \frac{36526166531}{108240873690} a^{13} + \frac{3325666865}{7216058246} a^{12} + \frac{76859270018}{54120436845} a^{11} - \frac{225018097799}{108240873690} a^{10} - \frac{75831211061}{36080291230} a^{9} + \frac{52281658724}{10824087369} a^{8} + \frac{195722636221}{108240873690} a^{7} - \frac{241195841273}{108240873690} a^{6} + \frac{166110347134}{54120436845} a^{5} - \frac{387100998593}{36080291230} a^{4} + \frac{172527549509}{36080291230} a^{3} + \frac{277639303931}{54120436845} a^{2} + \frac{15684296857}{21648174738} a - \frac{33426682505}{21648174738} \) (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:  \( 32330.2911008 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^2:S_4:C_2$ (as 16T724):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 384
The 28 conjugacy class representatives for $C_2\times C_2^2:S_4:C_2$
Character table for $C_2\times C_2^2:S_4:C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-3}, \sqrt{5})\), 8.0.9905225625.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 12 siblings: data not computed
Degree 16 siblings: data not computed
Degree 24 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$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$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
1327Data not computed