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Properties

Label 16.0.11445391768...321.55
Degree $16$
Signature $[0, 8]$
Discriminant $13^{12}\cdot 53^{12}$
Root discriminant $134.48$
Ramified primes $13, 53$
Class number $100352$ (GRH)
Class group $[2, 2, 2, 4, 4, 28, 28]$ (GRH)
Galois group $Q_{16}$ (as 16T14)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![15497228633, -18857570867, 8997200644, -3260668645, 1187076972, -243806269, 42987317, -15443166, 2239826, -395763, 129015, -5825, 6081, -103, 133, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 133*x^14 - 103*x^13 + 6081*x^12 - 5825*x^11 + 129015*x^10 - 395763*x^9 + 2239826*x^8 - 15443166*x^7 + 42987317*x^6 - 243806269*x^5 + 1187076972*x^4 - 3260668645*x^3 + 8997200644*x^2 - 18857570867*x + 15497228633)
 
gp: K = bnfinit(x^16 - x^15 + 133*x^14 - 103*x^13 + 6081*x^12 - 5825*x^11 + 129015*x^10 - 395763*x^9 + 2239826*x^8 - 15443166*x^7 + 42987317*x^6 - 243806269*x^5 + 1187076972*x^4 - 3260668645*x^3 + 8997200644*x^2 - 18857570867*x + 15497228633, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 133 x^{14} - 103 x^{13} + 6081 x^{12} - 5825 x^{11} + 129015 x^{10} - 395763 x^{9} + 2239826 x^{8} - 15443166 x^{7} + 42987317 x^{6} - 243806269 x^{5} + 1187076972 x^{4} - 3260668645 x^{3} + 8997200644 x^{2} - 18857570867 x + 15497228633 \)

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:  \(11445391768549949624817238631584321=13^{12}\cdot 53^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $134.48$
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:  $13, 53$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{141311} a^{14} + \frac{30345}{141311} a^{13} + \frac{47075}{141311} a^{12} - \frac{42168}{141311} a^{11} + \frac{875}{141311} a^{10} - \frac{49301}{141311} a^{9} + \frac{23679}{141311} a^{8} + \frac{12350}{141311} a^{7} - \frac{45557}{141311} a^{6} - \frac{28292}{141311} a^{5} - \frac{15022}{141311} a^{4} + \frac{52730}{141311} a^{3} + \frac{1701}{141311} a^{2} - \frac{27165}{141311} a - \frac{35154}{141311}$, $\frac{1}{714422766695225015424142786766545632319637286960802275244077991896937} a^{15} + \frac{299309828185863915452844443423887867133947268154081435172858078}{714422766695225015424142786766545632319637286960802275244077991896937} a^{14} - \frac{294947335027921396337467168800957903800502898578146441169452041234219}{714422766695225015424142786766545632319637286960802275244077991896937} a^{13} - \frac{243002037282467206817951504612498466353802217194509742655519375342863}{714422766695225015424142786766545632319637286960802275244077991896937} a^{12} - \frac{50605342040176721515488196378828540207019090229227612907533976339152}{714422766695225015424142786766545632319637286960802275244077991896937} a^{11} + \frac{215088558031875318380223404813683645611286484334226362249874251836900}{714422766695225015424142786766545632319637286960802275244077991896937} a^{10} + \frac{322221375630073478306029221118841201211538704930379816291609039735190}{714422766695225015424142786766545632319637286960802275244077991896937} a^{9} - \frac{224627052003957260356207862113766236917092762472269791508090298395511}{714422766695225015424142786766545632319637286960802275244077991896937} a^{8} + \frac{112120847513900269354047502344238779566737323372782303892793983292157}{714422766695225015424142786766545632319637286960802275244077991896937} a^{7} - \frac{158993141509573172017946148307045521612299962695567839886610655064991}{714422766695225015424142786766545632319637286960802275244077991896937} a^{6} + \frac{951158802322693612568033211840129756965689183725754249151723130035}{714422766695225015424142786766545632319637286960802275244077991896937} a^{5} + \frac{281251906750256047067226021693663691372669587403985526399568461176911}{714422766695225015424142786766545632319637286960802275244077991896937} a^{4} + \frac{285945134258461851858748434773816446633505568841068148830582188580031}{714422766695225015424142786766545632319637286960802275244077991896937} a^{3} - \frac{281176274021501850095021237673277792305719589382872873391813520218767}{714422766695225015424142786766545632319637286960802275244077991896937} a^{2} - \frac{330684322583581673424563653893863458289688655655545654158333096177987}{714422766695225015424142786766545632319637286960802275244077991896937} a + \frac{82414037848429016293632707118683134618925699526758575042449666700962}{714422766695225015424142786766545632319637286960802275244077991896937}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{4}\times C_{4}\times C_{28}\times C_{28}$, which has order $100352$ (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:  \( 165651.191156 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$Q_{16}$ (as 16T14):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 7 conjugacy class representatives for $Q_{16}$
Character table for $Q_{16}$

Intermediate fields

\(\Q(\sqrt{689}) \), \(\Q(\sqrt{53}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{13}, \sqrt{53})\), 4.4.36517.1 x2, 4.4.8957.1 x2, 8.8.225360027841.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ 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$13.4.3.1$x^{4} - 13$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.1$x^{4} - 13$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.1$x^{4} - 13$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.1$x^{4} - 13$$4$$1$$3$$C_4$$[\ ]_{4}$
$53$53.4.3.2$x^{4} - 212$$4$$1$$3$$C_4$$[\ ]_{4}$
53.4.3.2$x^{4} - 212$$4$$1$$3$$C_4$$[\ ]_{4}$
53.4.3.2$x^{4} - 212$$4$$1$$3$$C_4$$[\ ]_{4}$
53.4.3.2$x^{4} - 212$$4$$1$$3$$C_4$$[\ ]_{4}$