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Properties

Label 16.0.93675802669...5625.2
Degree $16$
Signature $[0, 8]$
Discriminant $5^{8}\cdot 13^{4}\cdot 29^{6}\cdot 109^{4}$
Root discriminant $48.50$
Ramified primes $5, 13, 29, 109$
Class number $256$ (GRH)
Class group $[2, 2, 64]$ (GRH)
Galois group 16T1177

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![21327179, -2226847, 242264, 2084554, -390428, -482089, 124859, 66040, -8307, 1967, 2407, -637, -110, 49, -9, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 9*x^14 + 49*x^13 - 110*x^12 - 637*x^11 + 2407*x^10 + 1967*x^9 - 8307*x^8 + 66040*x^7 + 124859*x^6 - 482089*x^5 - 390428*x^4 + 2084554*x^3 + 242264*x^2 - 2226847*x + 21327179)
 
gp: K = bnfinit(x^16 - 2*x^15 - 9*x^14 + 49*x^13 - 110*x^12 - 637*x^11 + 2407*x^10 + 1967*x^9 - 8307*x^8 + 66040*x^7 + 124859*x^6 - 482089*x^5 - 390428*x^4 + 2084554*x^3 + 242264*x^2 - 2226847*x + 21327179, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 9 x^{14} + 49 x^{13} - 110 x^{12} - 637 x^{11} + 2407 x^{10} + 1967 x^{9} - 8307 x^{8} + 66040 x^{7} + 124859 x^{6} - 482089 x^{5} - 390428 x^{4} + 2084554 x^{3} + 242264 x^{2} - 2226847 x + 21327179 \)

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:  \(936758026692429703922265625=5^{8}\cdot 13^{4}\cdot 29^{6}\cdot 109^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $48.50$
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:  $5, 13, 29, 109$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} + \frac{1}{5} a^{11} + \frac{1}{5} a^{10} + \frac{2}{5} a^{9} - \frac{1}{5} a^{7} - \frac{1}{5} a^{6} - \frac{2}{5} a^{4} + \frac{2}{5} a^{3} + \frac{2}{5} a^{2} - \frac{1}{5}$, $\frac{1}{5} a^{13} + \frac{1}{5} a^{10} - \frac{2}{5} a^{9} - \frac{1}{5} a^{8} + \frac{1}{5} a^{6} - \frac{2}{5} a^{5} - \frac{1}{5} a^{4} - \frac{2}{5} a^{2} - \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{715} a^{14} - \frac{4}{715} a^{13} - \frac{2}{65} a^{12} + \frac{34}{715} a^{11} - \frac{288}{715} a^{10} + \frac{83}{715} a^{9} - \frac{291}{715} a^{8} - \frac{4}{55} a^{7} + \frac{196}{715} a^{6} + \frac{102}{715} a^{5} + \frac{133}{715} a^{4} - \frac{12}{55} a^{3} + \frac{18}{65} a^{2} - \frac{16}{143} a + \frac{28}{715}$, $\frac{1}{14944654185554809250741560364504165811298013588305} a^{15} - \frac{4283537656463608841065107505473141565935060757}{14944654185554809250741560364504165811298013588305} a^{14} + \frac{12870825572628694760349023278117574835174296900}{229917756700843219242177851761602550943046362897} a^{13} - \frac{704731477586710720451009283474451235763638224086}{14944654185554809250741560364504165811298013588305} a^{12} + \frac{3472710191064453484140693139457033055849654217}{58150405391263849224675332157603758020614838865} a^{11} - \frac{468312127589568017731211278584845074779482699494}{14944654185554809250741560364504165811298013588305} a^{10} + \frac{286238249758568880139503423719022829899974696481}{1149588783504216096210889258808012754715231814485} a^{9} + \frac{6959071221293191046802528256544291392807043850876}{14944654185554809250741560364504165811298013588305} a^{8} - \frac{591532491173305954018353250660180655789762881962}{14944654185554809250741560364504165811298013588305} a^{7} + \frac{535174649824678825658308361303519723332975506124}{2988930837110961850148312072900833162259602717661} a^{6} - \frac{6275384792899058232985219984469243003494779937658}{14944654185554809250741560364504165811298013588305} a^{5} + \frac{6123405248559166216683249945614231684981395486387}{14944654185554809250741560364504165811298013588305} a^{4} + \frac{3900003214094827002956139606585878516279133638654}{14944654185554809250741560364504165811298013588305} a^{3} + \frac{193207661976944074533082302203666092216450389336}{786560746608147855302187387605482411120948083595} a^{2} + \frac{328338312500843094746929757669157420194414556446}{1149588783504216096210889258808012754715231814485} a + \frac{3956418775943384721742043809479843780763602786147}{14944654185554809250741560364504165811298013588305}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{64}$, which has order $256$ (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:  \( 17025.7429126 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1177:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 76 conjugacy class representatives for t16n1177 are not computed
Character table for t16n1177 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.725.1, 8.4.280793605625.2, 8.0.30606503013125.1, 8.4.57293125.2

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$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}$
$13$13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$29$29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.3.3$x^{4} + 58$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.3.3$x^{4} + 58$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
109Data not computed