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Properties

Label 16.0.40392862252...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 3^{4}\cdot 5^{10}\cdot 29^{4}\cdot 41^{2}$
Root discriminant $53.14$
Ramified primes $2, 3, 5, 29, 41$
Class number $1184$ (GRH)
Class group $[2, 2, 2, 148]$ (GRH)
Galois group 16T799

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![311609, -242772, 422098, -393256, 366131, -290664, 178960, -95208, 46796, -16676, 7508, -1472, 756, -52, 42, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 42*x^14 - 52*x^13 + 756*x^12 - 1472*x^11 + 7508*x^10 - 16676*x^9 + 46796*x^8 - 95208*x^7 + 178960*x^6 - 290664*x^5 + 366131*x^4 - 393256*x^3 + 422098*x^2 - 242772*x + 311609)
 
gp: K = bnfinit(x^16 + 42*x^14 - 52*x^13 + 756*x^12 - 1472*x^11 + 7508*x^10 - 16676*x^9 + 46796*x^8 - 95208*x^7 + 178960*x^6 - 290664*x^5 + 366131*x^4 - 393256*x^3 + 422098*x^2 - 242772*x + 311609, 1)
 

Normalized defining polynomial

\( x^{16} + 42 x^{14} - 52 x^{13} + 756 x^{12} - 1472 x^{11} + 7508 x^{10} - 16676 x^{9} + 46796 x^{8} - 95208 x^{7} + 178960 x^{6} - 290664 x^{5} + 366131 x^{4} - 393256 x^{3} + 422098 x^{2} - 242772 x + 311609 \)

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:  \(4039286225255792640000000000=2^{32}\cdot 3^{4}\cdot 5^{10}\cdot 29^{4}\cdot 41^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $53.14$
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:  $2, 3, 5, 29, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} + \frac{1}{5} a^{11} - \frac{2}{5} a^{10} - \frac{1}{5} a^{9} - \frac{1}{5} a^{8} + \frac{2}{5} a^{7} + \frac{1}{5} a^{4} - \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{13} + \frac{2}{5} a^{11} + \frac{1}{5} a^{10} - \frac{2}{5} a^{8} - \frac{2}{5} a^{7} + \frac{1}{5} a^{5} - \frac{1}{5} a^{4} - \frac{1}{5} a^{2} + \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{405} a^{14} + \frac{7}{135} a^{13} - \frac{4}{405} a^{12} + \frac{167}{405} a^{11} - \frac{107}{405} a^{10} + \frac{164}{405} a^{9} - \frac{38}{405} a^{8} + \frac{101}{405} a^{7} + \frac{31}{405} a^{6} - \frac{10}{81} a^{5} + \frac{1}{135} a^{4} - \frac{166}{405} a^{3} + \frac{86}{405} a^{2} + \frac{1}{15} a + \frac{83}{405}$, $\frac{1}{48574572372830537117049032392183979715} a^{15} - \frac{498583202411055502661387923688768}{5397174708092281901894336932464886635} a^{14} - \frac{325315158693455881322908435154600491}{48574572372830537117049032392183979715} a^{13} + \frac{3281254240404714110151906375910400732}{48574572372830537117049032392183979715} a^{12} - \frac{10985401904711039354097712292427982463}{48574572372830537117049032392183979715} a^{11} - \frac{4159971486562165774489633705878622439}{9714914474566107423409806478436795943} a^{10} + \frac{20837208886823163752244645058772639317}{48574572372830537117049032392183979715} a^{9} - \frac{1735917645000072473364739552644637190}{9714914474566107423409806478436795943} a^{8} - \frac{2374721559511978748273606051295648113}{48574572372830537117049032392183979715} a^{7} + \frac{3092763730849788895161625189873488766}{6939224624690076731007004627454854245} a^{6} - \frac{152620925360213847405381708024280949}{5397174708092281901894336932464886635} a^{5} + \frac{12690228239287267054467408711445207292}{48574572372830537117049032392183979715} a^{4} - \frac{6600665853822752981216425603468234531}{48574572372830537117049032392183979715} a^{3} - \frac{1137432609193444456107839010726387887}{2313074874896692243669001542484951415} a^{2} - \frac{11622365476515706995776664007605935719}{48574572372830537117049032392183979715} a + \frac{5103473439187366266288152972572399981}{16191524124276845705683010797394659905}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{148}$, which has order $1184$ (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:  \( 9192.49654065 \) (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

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{2}) \), 4.4.725.1, 4.4.46400.1, \(\Q(\sqrt{2}, \sqrt{5})\), 8.8.2152960000.1

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 R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
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$$[\ ]_{2}^{4}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$29$29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41Data not computed