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Properties

Label 16.12.2588240309...5625.2
Degree $16$
Signature $[12, 2]$
Discriminant $5^{8}\cdot 13^{2}\cdot 29^{8}\cdot 97^{8}$
Root discriminant $163.42$
Ramified primes $5, 13, 29, 97$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1220

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3389, 1178370, 312549, -4573696, 291102, 3272810, 28717, -620902, -292439, 58828, 41679, -2362, -1228, 70, -27, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 27*x^14 + 70*x^13 - 1228*x^12 - 2362*x^11 + 41679*x^10 + 58828*x^9 - 292439*x^8 - 620902*x^7 + 28717*x^6 + 3272810*x^5 + 291102*x^4 - 4573696*x^3 + 312549*x^2 + 1178370*x - 3389)
 
gp: K = bnfinit(x^16 - 2*x^15 - 27*x^14 + 70*x^13 - 1228*x^12 - 2362*x^11 + 41679*x^10 + 58828*x^9 - 292439*x^8 - 620902*x^7 + 28717*x^6 + 3272810*x^5 + 291102*x^4 - 4573696*x^3 + 312549*x^2 + 1178370*x - 3389, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 27 x^{14} + 70 x^{13} - 1228 x^{12} - 2362 x^{11} + 41679 x^{10} + 58828 x^{9} - 292439 x^{8} - 620902 x^{7} + 28717 x^{6} + 3272810 x^{5} + 291102 x^{4} - 4573696 x^{3} + 312549 x^{2} + 1178370 x - 3389 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(258824030924531989915946237128515625=5^{8}\cdot 13^{2}\cdot 29^{8}\cdot 97^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $163.42$
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:  $5, 13, 29, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{7} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{638} a^{14} - \frac{41}{638} a^{13} - \frac{24}{319} a^{12} + \frac{48}{319} a^{11} + \frac{28}{319} a^{10} + \frac{36}{319} a^{9} + \frac{74}{319} a^{8} + \frac{108}{319} a^{7} - \frac{237}{638} a^{6} - \frac{113}{638} a^{5} - \frac{94}{319} a^{4} - \frac{181}{638} a^{3} - \frac{189}{638} a^{2} + \frac{219}{638} a + \frac{130}{319}$, $\frac{1}{1428264832096810907625579686983572535373195321086} a^{15} - \frac{19947646858805194083588153250202177038845052}{102018916578343636258969977641683752526656808649} a^{14} + \frac{177861691589910536454162906640360944007265072575}{1428264832096810907625579686983572535373195321086} a^{13} + \frac{133154129922982415149280038912616696353897878221}{714132416048405453812789843491786267686597660543} a^{12} - \frac{130826409550965606966236946739202476928646005359}{1428264832096810907625579686983572535373195321086} a^{11} - \frac{77489056209133008455301767858145285589464096223}{1428264832096810907625579686983572535373195321086} a^{10} - \frac{209650535114760363664199412223285865497916135903}{1428264832096810907625579686983572535373195321086} a^{9} + \frac{554137764349093517885989770602508498722805222}{2238659611437007692203102957654502406541058497} a^{8} - \frac{202038004263113513641971453272888393130313735489}{714132416048405453812789843491786267686597660543} a^{7} + \frac{80730777353330419998221610060282564147209714299}{714132416048405453812789843491786267686597660543} a^{6} + \frac{220756783019719469062668064436227480541119583093}{714132416048405453812789843491786267686597660543} a^{5} - \frac{445030652739475977359443122779945273211904879479}{1428264832096810907625579686983572535373195321086} a^{4} - \frac{533227055618612652296779928175970766280739901053}{1428264832096810907625579686983572535373195321086} a^{3} + \frac{429719353267935998762679062665687951564407797831}{1428264832096810907625579686983572535373195321086} a^{2} - \frac{7310835615991737700885267016299622866167421556}{102018916578343636258969977641683752526656808649} a - \frac{316583706148730054545862746686651616745653067118}{714132416048405453812789843491786267686597660543}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $13$
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:  \( 1075738472980 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1220:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 55 conjugacy class representatives for t16n1220 are not computed
Character table for t16n1220 is not computed

Intermediate fields

\(\Q(\sqrt{2813}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{14065}) \), 4.4.725.1, \(\Q(\sqrt{5}, \sqrt{2813})\), 4.4.6821525.1, 8.8.39134423996850625.3

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$5.4.2.1$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}$
5.8.4.1$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$$[\ ]^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$29$29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$97$97.8.4.1$x^{8} + 432814 x^{4} - 912673 x^{2} + 46831989649$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
97.8.4.1$x^{8} + 432814 x^{4} - 912673 x^{2} + 46831989649$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$