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Properties

Label 16.0.17618734784...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 5^{4}\cdot 17^{8}\cdot 97^{2}$
Root discriminant $43.69$
Ramified primes $2, 5, 17, 97$
Class number $8$ (GRH)
Class group $[2, 2, 2]$ (GRH)
Galois group $C_2^3.C_2^4.C_2$ (as 16T608)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1178948, 2259152, 1745248, 1269776, 749640, 165320, -8556, -48464, -16099, -4200, 1466, 944, 138, -32, -26, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 26*x^14 - 32*x^13 + 138*x^12 + 944*x^11 + 1466*x^10 - 4200*x^9 - 16099*x^8 - 48464*x^7 - 8556*x^6 + 165320*x^5 + 749640*x^4 + 1269776*x^3 + 1745248*x^2 + 2259152*x + 1178948)
 
gp: K = bnfinit(x^16 - 26*x^14 - 32*x^13 + 138*x^12 + 944*x^11 + 1466*x^10 - 4200*x^9 - 16099*x^8 - 48464*x^7 - 8556*x^6 + 165320*x^5 + 749640*x^4 + 1269776*x^3 + 1745248*x^2 + 2259152*x + 1178948, 1)
 

Normalized defining polynomial

\( x^{16} - 26 x^{14} - 32 x^{13} + 138 x^{12} + 944 x^{11} + 1466 x^{10} - 4200 x^{9} - 16099 x^{8} - 48464 x^{7} - 8556 x^{6} + 165320 x^{5} + 749640 x^{4} + 1269776 x^{3} + 1745248 x^{2} + 2259152 x + 1178948 \)

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:  \(176187347840967261552640000=2^{32}\cdot 5^{4}\cdot 17^{8}\cdot 97^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.69$
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:  $2, 5, 17, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{8} + \frac{1}{8} a^{6} - \frac{1}{2} a^{5} - \frac{3}{8} a^{4} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{9} + \frac{1}{8} a^{7} - \frac{1}{2} a^{6} - \frac{3}{8} a^{5} - \frac{1}{2} a^{2} - \frac{1}{4} a$, $\frac{1}{8} a^{12} - \frac{1}{2} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{3}{8} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{8} a^{13} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{3}{8} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a$, $\frac{1}{16} a^{14} - \frac{1}{4} a^{9} - \frac{1}{8} a^{8} - \frac{1}{4} a^{7} + \frac{5}{16} a^{6} - \frac{1}{4} a^{5} - \frac{1}{8} a^{4} - \frac{1}{4} a^{3} + \frac{3}{8} a^{2}$, $\frac{1}{1070462712039320793716890385195834286128} a^{15} - \frac{3073833165888306622197458031083402331}{1070462712039320793716890385195834286128} a^{14} - \frac{25700426759357430899514020164443386439}{535231356019660396858445192597917143064} a^{13} - \frac{4027605622281988543862373782543529442}{66903919502457549607305649074739642883} a^{12} - \frac{10695721257657173526793901571300707271}{535231356019660396858445192597917143064} a^{11} + \frac{4108015732483508649469669717379635132}{66903919502457549607305649074739642883} a^{10} - \frac{28751909219483156212771122685768952603}{133807839004915099214611298149479285766} a^{9} + \frac{56141501973851591799682375535129109011}{535231356019660396858445192597917143064} a^{8} + \frac{393388070034014311237262608962663756751}{1070462712039320793716890385195834286128} a^{7} - \frac{519798066707640840908578316583231268631}{1070462712039320793716890385195834286128} a^{6} - \frac{17989177281354127119903751964971205617}{535231356019660396858445192597917143064} a^{5} + \frac{248357395946309345173277443508212761135}{535231356019660396858445192597917143064} a^{4} - \frac{252597005336179324438845106146643201665}{535231356019660396858445192597917143064} a^{3} - \frac{135184270601906610383908293082421815161}{535231356019660396858445192597917143064} a^{2} + \frac{7501607620498221142368648971545997375}{133807839004915099214611298149479285766} a + \frac{1290807446238233833136419896182196415}{2846975297976917004566197832967644378}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}$, which has order $8$ (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:  $7$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 3337708.26788 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_2^4.C_2$ (as 16T608):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 34 conjugacy class representatives for $C_2^3.C_2^4.C_2$
Character table for $C_2^3.C_2^4.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{34}) \), 4.0.2312.1 x2, 4.0.1088.2 x2, \(\Q(\sqrt{2}, \sqrt{17})\), 8.0.342102016.2

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 R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$2$2.4.8.2$x^{4} + 6 x^{2} + 1$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.8.2$x^{4} + 6 x^{2} + 1$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.8.3$x^{4} + 6 x^{2} + 4 x + 14$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.8.3$x^{4} + 6 x^{2} + 4 x + 14$$4$$1$$8$$C_2^2$$[2, 3]$
$5$5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$17$17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$97$97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.2.2$x^{4} - 97 x^{2} + 47045$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$