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Properties

Label 16.8.17142224957...4976.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{58}\cdot 3^{4}\cdot 7^{6}\cdot 79^{2}$
Root discriminant $58.16$
Ramified primes $2, 3, 7, 79$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T781

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-9281, 210216, -422432, 585152, -662896, 355344, 78280, -213624, 120834, -27544, -3040, 3872, -1056, 80, 24, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 24*x^14 + 80*x^13 - 1056*x^12 + 3872*x^11 - 3040*x^10 - 27544*x^9 + 120834*x^8 - 213624*x^7 + 78280*x^6 + 355344*x^5 - 662896*x^4 + 585152*x^3 - 422432*x^2 + 210216*x - 9281)
 
gp: K = bnfinit(x^16 - 8*x^15 + 24*x^14 + 80*x^13 - 1056*x^12 + 3872*x^11 - 3040*x^10 - 27544*x^9 + 120834*x^8 - 213624*x^7 + 78280*x^6 + 355344*x^5 - 662896*x^4 + 585152*x^3 - 422432*x^2 + 210216*x - 9281, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 24 x^{14} + 80 x^{13} - 1056 x^{12} + 3872 x^{11} - 3040 x^{10} - 27544 x^{9} + 120834 x^{8} - 213624 x^{7} + 78280 x^{6} + 355344 x^{5} - 662896 x^{4} + 585152 x^{3} - 422432 x^{2} + 210216 x - 9281 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(17142224957643668854696574976=2^{58}\cdot 3^{4}\cdot 7^{6}\cdot 79^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $58.16$
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:  $2, 3, 7, 79$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{17} a^{12} + \frac{3}{17} a^{11} + \frac{5}{17} a^{10} + \frac{4}{17} a^{9} + \frac{1}{17} a^{8} - \frac{1}{17} a^{7} - \frac{5}{17} a^{6} - \frac{1}{17} a^{5} - \frac{8}{17} a^{4} - \frac{1}{17} a^{3} - \frac{5}{17} a^{2} + \frac{3}{17} a + \frac{2}{17}$, $\frac{1}{17} a^{13} - \frac{4}{17} a^{11} + \frac{6}{17} a^{10} + \frac{6}{17} a^{9} - \frac{4}{17} a^{8} - \frac{2}{17} a^{7} - \frac{3}{17} a^{6} - \frac{5}{17} a^{5} + \frac{6}{17} a^{4} - \frac{2}{17} a^{3} + \frac{1}{17} a^{2} - \frac{7}{17} a - \frac{6}{17}$, $\frac{1}{17} a^{14} + \frac{1}{17} a^{11} - \frac{8}{17} a^{10} - \frac{5}{17} a^{9} + \frac{2}{17} a^{8} - \frac{7}{17} a^{7} - \frac{8}{17} a^{6} + \frac{2}{17} a^{5} - \frac{3}{17} a^{3} + \frac{7}{17} a^{2} + \frac{6}{17} a + \frac{8}{17}$, $\frac{1}{246993258887673872352184262575768501087} a^{15} - \frac{5273926568374479575121142716645368888}{246993258887673872352184262575768501087} a^{14} - \frac{5240703249902209187748591233101772983}{246993258887673872352184262575768501087} a^{13} - \frac{1301133768029973996255304049296062444}{246993258887673872352184262575768501087} a^{12} - \frac{75423914026135578886364581005941911222}{246993258887673872352184262575768501087} a^{11} - \frac{80094521421985988603335066973706900246}{246993258887673872352184262575768501087} a^{10} - \frac{93255782363797383175668388596442511419}{246993258887673872352184262575768501087} a^{9} + \frac{21590992234854566874187586757555874771}{246993258887673872352184262575768501087} a^{8} + \frac{26611082304804242421089193989350795288}{246993258887673872352184262575768501087} a^{7} + \frac{39445895688416853780831186477949499140}{246993258887673872352184262575768501087} a^{6} - \frac{101500256198994363462793664843391647580}{246993258887673872352184262575768501087} a^{5} - \frac{37607341082176878382060802867474956584}{246993258887673872352184262575768501087} a^{4} - \frac{42460704134342798121592521984687890075}{246993258887673872352184262575768501087} a^{3} - \frac{118495948891123918673526963707657320307}{246993258887673872352184262575768501087} a^{2} - \frac{85047046666350299738420704940090922843}{246993258887673872352184262575768501087} a - \frac{122749307921515624415678669642819609468}{246993258887673872352184262575768501087}$

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:  $11$
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:  \( 240000295.454 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T781:

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 t16n781 are not computed
Character table for t16n781 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 4.4.14336.1, 4.4.7168.1, 8.8.3288334336.1

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 R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
$3$3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$7$7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
79Data not computed