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Properties

Label 16.16.3307735506...0000.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{16}\cdot 3^{8}\cdot 5^{12}\cdot 29^{2}\cdot 6121^{2}$
Root discriminant $52.48$
Ramified primes $2, 3, 5, 29, 6121$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T797

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-312839, 21526, 2334136, -2393916, -1862220, 2722820, 293460, -975158, 46897, 158732, -17357, -12730, 1777, 472, -74, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 - 74*x^14 + 472*x^13 + 1777*x^12 - 12730*x^11 - 17357*x^10 + 158732*x^9 + 46897*x^8 - 975158*x^7 + 293460*x^6 + 2722820*x^5 - 1862220*x^4 - 2393916*x^3 + 2334136*x^2 + 21526*x - 312839)
 
gp: K = bnfinit(x^16 - 6*x^15 - 74*x^14 + 472*x^13 + 1777*x^12 - 12730*x^11 - 17357*x^10 + 158732*x^9 + 46897*x^8 - 975158*x^7 + 293460*x^6 + 2722820*x^5 - 1862220*x^4 - 2393916*x^3 + 2334136*x^2 + 21526*x - 312839, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} - 74 x^{14} + 472 x^{13} + 1777 x^{12} - 12730 x^{11} - 17357 x^{10} + 158732 x^{9} + 46897 x^{8} - 975158 x^{7} + 293460 x^{6} + 2722820 x^{5} - 1862220 x^{4} - 2393916 x^{3} + 2334136 x^{2} + 21526 x - 312839 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3307735506823056000000000000=2^{16}\cdot 3^{8}\cdot 5^{12}\cdot 29^{2}\cdot 6121^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $52.48$
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:  $2, 3, 5, 29, 6121$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{11} a^{14} - \frac{4}{11} a^{13} - \frac{1}{11} a^{12} + \frac{3}{11} a^{11} - \frac{3}{11} a^{10} + \frac{3}{11} a^{9} - \frac{5}{11} a^{8} + \frac{4}{11} a^{7} + \frac{3}{11} a^{6} + \frac{3}{11} a^{5} - \frac{2}{11} a^{4} - \frac{2}{11} a^{3} + \frac{2}{11} a^{2} - \frac{1}{11} a - \frac{3}{11}$, $\frac{1}{1984420445853581910670661899603860646909} a^{15} + \frac{55941787901348327025469909011376342174}{1984420445853581910670661899603860646909} a^{14} - \frac{575139368589842139451545047267617939766}{1984420445853581910670661899603860646909} a^{13} - \frac{402732180851402824989718614700446918932}{1984420445853581910670661899603860646909} a^{12} - \frac{255622508830810988732570838794307185425}{1984420445853581910670661899603860646909} a^{11} + \frac{374145456738657847978613878785871154461}{1984420445853581910670661899603860646909} a^{10} + \frac{104813368372470313687448908778256715692}{1984420445853581910670661899603860646909} a^{9} + \frac{241648993438660804012945283437709342030}{1984420445853581910670661899603860646909} a^{8} + \frac{755991238531003390088063565541048059520}{1984420445853581910670661899603860646909} a^{7} - \frac{841281377432373301139727024314563207254}{1984420445853581910670661899603860646909} a^{6} + \frac{898469349310124122659153006420154447275}{1984420445853581910670661899603860646909} a^{5} - \frac{553674063228044490769990329174627636799}{1984420445853581910670661899603860646909} a^{4} - \frac{337318858234283763622206622531717396889}{1984420445853581910670661899603860646909} a^{3} - \frac{24719227577465015580463142553656341855}{180401858713961991879151081782169149719} a^{2} - \frac{645897461038540025957491473883882204712}{1984420445853581910670661899603860646909} a - \frac{870957779134299499475205348758890651859}{1984420445853581910670661899603860646909}$

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:  $15$
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:  \( 256550726.18 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T797:

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 t16n797 are not computed
Character table for t16n797 is not computed

Intermediate fields

\(\Q(\sqrt{15}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{3}) \), \(\Q(\zeta_{20})^+\), \(\Q(\zeta_{15})^+\), \(\Q(\sqrt{3}, \sqrt{5})\), \(\Q(\zeta_{60})^+\)

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 ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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
3Data not computed
5Data not computed
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{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}$
6121Data not computed