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Properties

Label 16.0.40930145344...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 5^{10}\cdot 11^{6}\cdot 19^{2}$
Root discriminant $19.42$
Ramified primes $2, 5, 11, 19$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T799

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![61, -288, 682, -918, 957, -912, 817, -382, 298, -176, 153, -124, 97, -54, 23, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 23*x^14 - 54*x^13 + 97*x^12 - 124*x^11 + 153*x^10 - 176*x^9 + 298*x^8 - 382*x^7 + 817*x^6 - 912*x^5 + 957*x^4 - 918*x^3 + 682*x^2 - 288*x + 61)
 
gp: K = bnfinit(x^16 - 6*x^15 + 23*x^14 - 54*x^13 + 97*x^12 - 124*x^11 + 153*x^10 - 176*x^9 + 298*x^8 - 382*x^7 + 817*x^6 - 912*x^5 + 957*x^4 - 918*x^3 + 682*x^2 - 288*x + 61, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 23 x^{14} - 54 x^{13} + 97 x^{12} - 124 x^{11} + 153 x^{10} - 176 x^{9} + 298 x^{8} - 382 x^{7} + 817 x^{6} - 912 x^{5} + 957 x^{4} - 918 x^{3} + 682 x^{2} - 288 x + 61 \)

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:  \(409301453440000000000=2^{16}\cdot 5^{10}\cdot 11^{6}\cdot 19^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.42$
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:  $2, 5, 11, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{6} a^{13} - \frac{1}{6} a^{12} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{2} a^{7} - \frac{1}{6} a^{6} - \frac{1}{6} a^{5} + \frac{1}{6} a^{4} - \frac{1}{6} a^{3} + \frac{1}{6} a^{2} - \frac{1}{6} a + \frac{1}{6}$, $\frac{1}{6} a^{14} + \frac{1}{6} a^{12} + \frac{1}{3} a^{9} - \frac{1}{2} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{6}$, $\frac{1}{5619503277819001086} a^{15} + \frac{330689197309421011}{5619503277819001086} a^{14} + \frac{135300315430805291}{5619503277819001086} a^{13} + \frac{179693470297958873}{936583879636500181} a^{12} - \frac{1273875986625466832}{2809751638909500543} a^{11} - \frac{60769291564608912}{936583879636500181} a^{10} + \frac{1877622588217493749}{5619503277819001086} a^{9} - \frac{536686340123694865}{5619503277819001086} a^{8} - \frac{267489343975683984}{936583879636500181} a^{7} - \frac{887537023355804645}{1873167759273000362} a^{6} + \frac{1259097149509207168}{2809751638909500543} a^{5} + \frac{1325634466306080829}{5619503277819001086} a^{4} - \frac{18218303727670994}{90637149642241953} a^{3} - \frac{2390666746222992689}{5619503277819001086} a^{2} + \frac{5696169538430033}{26382644496802822} a + \frac{971899815802843264}{2809751638909500543}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( \frac{4834361895}{323488862786} a^{15} - \frac{52107041803}{485233294179} a^{14} + \frac{207435776605}{485233294179} a^{13} - \frac{1062847868671}{970466588358} a^{12} + \frac{959498870651}{485233294179} a^{11} - \frac{1302031948019}{485233294179} a^{10} + \frac{975122077899}{323488862786} a^{9} - \frac{607478270290}{161744431393} a^{8} + \frac{5354876665259}{970466588358} a^{7} - \frac{8494852363999}{970466588358} a^{6} + \frac{13822550375125}{970466588358} a^{5} - \frac{22593584058967}{970466588358} a^{4} + \frac{514593061987}{31305373818} a^{3} - \frac{20426655298585}{970466588358} a^{2} + \frac{102063475576}{6834271749} a - \frac{1792492110081}{323488862786} \) (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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 12014.8060289 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T799:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 62 conjugacy class representatives for t16n799 are not computed
Character table for t16n799 is not computed

Intermediate fields

\(\Q(\sqrt{-5}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-1}) \), 4.2.4400.1, 4.2.275.1, \(\Q(i, \sqrt{5})\), 8.0.19360000.2

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$2.8.8.1$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}$
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$11$11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.4.2.2$x^{4} - 11 x^{2} + 847$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
11.8.4.1$x^{8} + 484 x^{4} - 1331 x^{2} + 58564$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$19$19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$