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Properties

Label 16.0.19131876000...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 3^{14}\cdot 5^{14}$
Root discriminant $21.39$
Ramified primes $2, 3, 5$
Class number $8$
Class group $[8]$
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![1, -16, 128, -372, 626, -716, 520, -156, -101, 108, 40, -148, 146, -84, 32, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 32*x^14 - 84*x^13 + 146*x^12 - 148*x^11 + 40*x^10 + 108*x^9 - 101*x^8 - 156*x^7 + 520*x^6 - 716*x^5 + 626*x^4 - 372*x^3 + 128*x^2 - 16*x + 1)
 
gp: K = bnfinit(x^16 - 8*x^15 + 32*x^14 - 84*x^13 + 146*x^12 - 148*x^11 + 40*x^10 + 108*x^9 - 101*x^8 - 156*x^7 + 520*x^6 - 716*x^5 + 626*x^4 - 372*x^3 + 128*x^2 - 16*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 32 x^{14} - 84 x^{13} + 146 x^{12} - 148 x^{11} + 40 x^{10} + 108 x^{9} - 101 x^{8} - 156 x^{7} + 520 x^{6} - 716 x^{5} + 626 x^{4} - 372 x^{3} + 128 x^{2} - 16 x + 1 \)

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:  \(1913187600000000000000=2^{16}\cdot 3^{14}\cdot 5^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.39$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2} - \frac{1}{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{5} - \frac{1}{2} a^{3} - \frac{1}{4} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{6} + \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{7} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{56} a^{12} - \frac{3}{28} a^{11} - \frac{1}{56} a^{10} + \frac{1}{14} a^{9} - \frac{1}{14} a^{8} + \frac{5}{28} a^{7} + \frac{3}{56} a^{6} + \frac{1}{7} a^{5} + \frac{1}{7} a^{4} - \frac{13}{28} a^{3} + \frac{3}{56} a^{2} + \frac{19}{56}$, $\frac{1}{56} a^{13} + \frac{5}{56} a^{11} - \frac{1}{28} a^{10} + \frac{3}{28} a^{9} - \frac{1}{8} a^{7} - \frac{1}{28} a^{6} - \frac{1}{4} a^{5} + \frac{1}{7} a^{4} + \frac{1}{56} a^{3} - \frac{5}{28} a^{2} + \frac{5}{56} a + \frac{2}{7}$, $\frac{1}{56} a^{14} - \frac{3}{56} a^{10} - \frac{3}{28} a^{9} - \frac{1}{56} a^{8} + \frac{1}{14} a^{7} + \frac{13}{56} a^{6} + \frac{5}{28} a^{5} + \frac{3}{56} a^{4} + \frac{1}{7} a^{3} - \frac{3}{7} a^{2} - \frac{13}{28} a + \frac{3}{56}$, $\frac{1}{20104} a^{15} + \frac{43}{5026} a^{14} + \frac{59}{10052} a^{13} - \frac{25}{20104} a^{12} + \frac{1031}{20104} a^{11} + \frac{1983}{20104} a^{10} - \frac{943}{20104} a^{9} + \frac{267}{10052} a^{8} - \frac{193}{20104} a^{7} - \frac{1509}{20104} a^{6} + \frac{4253}{20104} a^{5} + \frac{795}{10052} a^{4} + \frac{1249}{10052} a^{3} - \frac{4149}{20104} a^{2} - \frac{3921}{20104} a + \frac{1075}{20104}$

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

Class group and class number

$C_{8}$, which has order $8$

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:  \( \frac{4323}{10052} a^{15} - \frac{64845}{20104} a^{14} + \frac{245865}{20104} a^{13} - \frac{76830}{2513} a^{12} + \frac{994845}{20104} a^{11} - \frac{874269}{20104} a^{10} + \frac{4990}{2513} a^{9} + \frac{124605}{2872} a^{8} - \frac{485055}{20104} a^{7} - \frac{1461525}{20104} a^{6} + \frac{134643}{718} a^{5} - \frac{4588785}{20104} a^{4} + \frac{3645195}{20104} a^{3} - \frac{480765}{5026} a^{2} + \frac{512415}{20104} a - \frac{38473}{20104} \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 4515.69639126 \)
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

\(\Q(\sqrt{15}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-1}) \), 4.2.54000.1 x2, 4.0.13500.1 x2, \(\Q(i, \sqrt{15})\), 8.2.43740000000.1 x2, 8.2.43740000000.2 x2, 8.0.2916000000.1

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 R R R ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$2$2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
3Data not computed
$5$5.8.7.3$x^{8} + 10$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
5.8.7.3$x^{8} + 10$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$