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Properties

Label 16.4.24109722907...601.24
Degree $16$
Signature $[4, 6]$
Discriminant $13^{10}\cdot 53^{10}$
Root discriminant $59.41$
Ramified primes $13, 53$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T875

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2187, -2376, 15417, -9702, 13705, 6504, -46181, 41789, -20469, 12394, -6918, 2063, -309, 23, -2, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 2*x^14 + 23*x^13 - 309*x^12 + 2063*x^11 - 6918*x^10 + 12394*x^9 - 20469*x^8 + 41789*x^7 - 46181*x^6 + 6504*x^5 + 13705*x^4 - 9702*x^3 + 15417*x^2 - 2376*x + 2187)
 
gp: K = bnfinit(x^16 - x^15 - 2*x^14 + 23*x^13 - 309*x^12 + 2063*x^11 - 6918*x^10 + 12394*x^9 - 20469*x^8 + 41789*x^7 - 46181*x^6 + 6504*x^5 + 13705*x^4 - 9702*x^3 + 15417*x^2 - 2376*x + 2187, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 2 x^{14} + 23 x^{13} - 309 x^{12} + 2063 x^{11} - 6918 x^{10} + 12394 x^{9} - 20469 x^{8} + 41789 x^{7} - 46181 x^{6} + 6504 x^{5} + 13705 x^{4} - 9702 x^{3} + 15417 x^{2} - 2376 x + 2187 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(24109722907876309716269637601=13^{10}\cdot 53^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $59.41$
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:  $13, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{13} a^{9} + \frac{5}{13} a^{8} + \frac{1}{13} a^{7} - \frac{5}{13} a^{6} + \frac{2}{13} a^{5} - \frac{5}{13} a^{4} + \frac{2}{13} a^{3} + \frac{1}{13} a^{2} + \frac{2}{13} a - \frac{4}{13}$, $\frac{1}{13} a^{10} + \frac{2}{13} a^{8} + \frac{3}{13} a^{7} + \frac{1}{13} a^{6} - \frac{2}{13} a^{5} + \frac{1}{13} a^{4} + \frac{4}{13} a^{3} - \frac{3}{13} a^{2} - \frac{1}{13} a - \frac{6}{13}$, $\frac{1}{13} a^{11} + \frac{6}{13} a^{8} - \frac{1}{13} a^{7} - \frac{5}{13} a^{6} - \frac{3}{13} a^{5} + \frac{1}{13} a^{4} + \frac{6}{13} a^{3} - \frac{3}{13} a^{2} + \frac{3}{13} a - \frac{5}{13}$, $\frac{1}{13} a^{12} - \frac{5}{13} a^{8} + \frac{2}{13} a^{7} + \frac{1}{13} a^{6} + \frac{2}{13} a^{5} - \frac{3}{13} a^{4} - \frac{2}{13} a^{3} - \frac{3}{13} a^{2} - \frac{4}{13} a - \frac{2}{13}$, $\frac{1}{195} a^{13} + \frac{2}{195} a^{12} + \frac{1}{195} a^{11} - \frac{4}{195} a^{10} + \frac{1}{65} a^{9} + \frac{19}{39} a^{8} + \frac{1}{5} a^{7} - \frac{71}{195} a^{6} - \frac{23}{65} a^{5} + \frac{14}{195} a^{4} - \frac{53}{195} a^{3} - \frac{2}{65} a^{2} - \frac{1}{3} a + \frac{16}{65}$, $\frac{1}{26325} a^{14} - \frac{49}{26325} a^{13} - \frac{251}{26325} a^{12} - \frac{41}{5265} a^{11} + \frac{23}{2925} a^{10} + \frac{857}{26325} a^{9} + \frac{263}{8775} a^{8} - \frac{142}{5265} a^{7} + \frac{1549}{8775} a^{6} - \frac{11257}{26325} a^{5} + \frac{10558}{26325} a^{4} + \frac{1309}{8775} a^{3} + \frac{7126}{26325} a^{2} - \frac{193}{675} a - \frac{48}{325}$, $\frac{1}{102881203211196039353914125} a^{15} - \frac{177108772158679158788}{20576240642239207870782825} a^{14} + \frac{154692742685060010259183}{102881203211196039353914125} a^{13} - \frac{2657708576095827031006789}{102881203211196039353914125} a^{12} + \frac{1013060940562057134295804}{34293734403732013117971375} a^{11} + \frac{388429797915928351846264}{20576240642239207870782825} a^{10} + \frac{39650870088290869304917}{1270138311249333819184125} a^{9} + \frac{6382223647906238857474966}{102881203211196039353914125} a^{8} - \frac{1062957777860053621797109}{3810414933748001457552375} a^{7} + \frac{47690162125354255925436266}{102881203211196039353914125} a^{6} - \frac{1531069291459472650381981}{20576240642239207870782825} a^{5} + \frac{792589479019163282570611}{11431244801244004372657125} a^{4} - \frac{40840051084789290637367531}{102881203211196039353914125} a^{3} + \frac{10293365458094068303939444}{34293734403732013117971375} a^{2} + \frac{291461100833281872177466}{11431244801244004372657125} a - \frac{52282264979696591570698}{141126479027703757687125}$

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:  $9$
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:  \( 147323170.261 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T875:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 32 conjugacy class representatives for t16n875
Character table for t16n875 is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), 4.4.8957.1, 8.4.225360027841.2

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$13$13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
53Data not computed