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Properties

Label 16.0.35256881742...0625.3
Degree $16$
Signature $[0, 8]$
Discriminant $3^{12}\cdot 5^{8}\cdot 19^{8}$
Root discriminant $22.22$
Ramified primes $3, 5, 19$
Class number $4$
Class group $[4]$
Galois group $C_8:C_2^2$ (as 16T35)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![99, 90, 1260, -1494, 3021, -3018, 2781, -1662, 724, -221, 116, -125, 112, -64, 26, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 26*x^14 - 64*x^13 + 112*x^12 - 125*x^11 + 116*x^10 - 221*x^9 + 724*x^8 - 1662*x^7 + 2781*x^6 - 3018*x^5 + 3021*x^4 - 1494*x^3 + 1260*x^2 + 90*x + 99)
 
gp: K = bnfinit(x^16 - 7*x^15 + 26*x^14 - 64*x^13 + 112*x^12 - 125*x^11 + 116*x^10 - 221*x^9 + 724*x^8 - 1662*x^7 + 2781*x^6 - 3018*x^5 + 3021*x^4 - 1494*x^3 + 1260*x^2 + 90*x + 99, 1)
 

Normalized defining polynomial

\( x^{16} - 7 x^{15} + 26 x^{14} - 64 x^{13} + 112 x^{12} - 125 x^{11} + 116 x^{10} - 221 x^{9} + 724 x^{8} - 1662 x^{7} + 2781 x^{6} - 3018 x^{5} + 3021 x^{4} - 1494 x^{3} + 1260 x^{2} + 90 x + 99 \)

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:  \(3525688174246906640625=3^{12}\cdot 5^{8}\cdot 19^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.22$
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:  $3, 5, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{4389} a^{12} - \frac{2125}{4389} a^{11} - \frac{1699}{4389} a^{10} - \frac{1186}{4389} a^{9} - \frac{92}{4389} a^{8} + \frac{571}{4389} a^{7} - \frac{769}{4389} a^{6} + \frac{1675}{4389} a^{5} - \frac{281}{627} a^{4} + \frac{358}{1463} a^{3} + \frac{383}{1463} a^{2} - \frac{641}{1463} a + \frac{50}{133}$, $\frac{1}{4389} a^{13} - \frac{149}{627} a^{11} + \frac{586}{4389} a^{10} - \frac{32}{133} a^{9} - \frac{259}{627} a^{8} + \frac{414}{1463} a^{7} + \frac{86}{1463} a^{6} - \frac{109}{231} a^{5} - \frac{43}{399} a^{4} + \frac{373}{1463} a^{3} - \frac{194}{1463} a^{2} + \frac{478}{1463} a - \frac{17}{133}$, $\frac{1}{917301} a^{14} + \frac{17}{917301} a^{13} + \frac{3}{43681} a^{12} + \frac{50944}{917301} a^{11} + \frac{37880}{305767} a^{10} + \frac{47741}{305767} a^{9} + \frac{134709}{305767} a^{8} + \frac{187664}{917301} a^{7} + \frac{1828}{27797} a^{6} + \frac{31106}{83391} a^{5} + \frac{433420}{917301} a^{4} + \frac{110958}{305767} a^{3} - \frac{41530}{305767} a^{2} - \frac{14860}{305767} a - \frac{5105}{27797}$, $\frac{1}{2327192637} a^{15} + \frac{92}{2327192637} a^{14} - \frac{23171}{775730879} a^{13} + \frac{264878}{2327192637} a^{12} + \frac{24515399}{54120759} a^{11} + \frac{2929118}{17497689} a^{10} - \frac{69947689}{332456091} a^{9} + \frac{70111667}{775730879} a^{8} - \frac{353429414}{2327192637} a^{7} - \frac{383385}{915857} a^{6} + \frac{5883586}{39443943} a^{5} + \frac{187789675}{2327192637} a^{4} - \frac{318120623}{775730879} a^{3} - \frac{6232916}{70520989} a^{2} + \frac{238447009}{775730879} a - \frac{23492005}{70520989}$

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

Class group and class number

$C_{4}$, which has order $4$

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:  \( -1 \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 6810.46306433 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_8:C_2^2$ (as 16T35):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $C_8:C_2^2$
Character table for $C_8:C_2^2$

Intermediate fields

\(\Q(\sqrt{-19}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-95}) \), 4.0.16245.1 x2, 4.2.4275.1 x2, \(\Q(\sqrt{5}, \sqrt{-19})\), 8.2.3125131875.2 x2, 8.2.3125131875.1 x2, 8.0.6597500625.3

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 8 siblings: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ R R ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$3.8.6.3$x^{8} - 3 x^{4} + 18$$4$$2$$6$$C_8:C_2$$[\ ]_{4}^{4}$
3.8.6.3$x^{8} - 3 x^{4} + 18$$4$$2$$6$$C_8:C_2$$[\ ]_{4}^{4}$
$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}$
$19$19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$