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Properties

Label 16.0.6926910400390625.1
Degree $16$
Signature $[0, 8]$
Discriminant $6.927\times 10^{15}$
Root discriminant $9.77$
Ramified primes $5, 61$
Class number $1$
Class group trivial
Galois group 16T1192

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 25*x^14 - 55*x^13 + 75*x^12 - 46*x^11 - 43*x^10 + 140*x^9 - 160*x^8 + 80*x^7 + 37*x^6 - 109*x^5 + 110*x^4 - 70*x^3 + 30*x^2 - 8*x + 1)
 
gp: K = bnfinit(x^16 - 7*x^15 + 25*x^14 - 55*x^13 + 75*x^12 - 46*x^11 - 43*x^10 + 140*x^9 - 160*x^8 + 80*x^7 + 37*x^6 - 109*x^5 + 110*x^4 - 70*x^3 + 30*x^2 - 8*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -8, 30, -70, 110, -109, 37, 80, -160, 140, -43, -46, 75, -55, 25, -7, 1]);
 

\( x^{16} - 7 x^{15} + 25 x^{14} - 55 x^{13} + 75 x^{12} - 46 x^{11} - 43 x^{10} + 140 x^{9} - 160 x^{8} + 80 x^{7} + 37 x^{6} - 109 x^{5} + 110 x^{4} - 70 x^{3} + 30 x^{2} - 8 x + 1 \)

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

Invariants

Degree:  $16$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 8]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(6926910400390625\)\(\medspace = 5^{15}\cdot 61^{3}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $9.77$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $5, 61$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $4$
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{1231} a^{15} - \frac{252}{1231} a^{14} + \frac{215}{1231} a^{13} + \frac{203}{1231} a^{12} - \frac{420}{1231} a^{11} - \frac{550}{1231} a^{10} + \frac{528}{1231} a^{9} + \frac{35}{1231} a^{8} - \frac{118}{1231} a^{7} - \frac{554}{1231} a^{6} + \frac{357}{1231} a^{5} - \frac{173}{1231} a^{4} - \frac{590}{1231} a^{3} + \frac{453}{1231} a^{2} - \frac{165}{1231} a - \frac{206}{1231}$

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

Class group and class number

Trivial group, which has order $1$

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $7$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( \frac{6623}{1231} a^{15} - \frac{44076}{1231} a^{14} + \frac{149860}{1231} a^{13} - \frac{309995}{1231} a^{12} + \frac{383241}{1231} a^{11} - \frac{163844}{1231} a^{10} - \frac{343776}{1231} a^{9} + \frac{796834}{1231} a^{8} - \frac{764280}{1231} a^{7} + \frac{252824}{1231} a^{6} + \frac{327106}{1231} a^{5} - \frac{595522}{1231} a^{4} + \frac{512951}{1231} a^{3} - \frac{281627}{1231} a^{2} + \frac{100044}{1231} a - \frac{17624}{1231} \) (order $10$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 65.2827160145 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{8}\cdot 65.2827160145 \cdot 1}{10\sqrt{6926910400390625}}\approx 0.190531758727$

Galois group

16T1192:

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 1024
The 88 conjugacy class representatives for t16n1192 are not computed
Character table for t16n1192 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 8.0.4765625.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ $16$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
5Data not computed
$61$61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.4.3.1$x^{4} - 61$$4$$1$$3$$C_4$$[\ ]_{4}$