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Properties

Label 16.0.68372792983...489.14
Degree $16$
Signature $[0, 8]$
Discriminant $17^{14}\cdot 67^{8}$
Root discriminant $97.65$
Ramified primes $17, 67$
Class number $1152$ (GRH)
Class group $[2, 2, 2, 12, 12]$ (GRH)
Galois group $C_2^2 : C_8$ (as 16T24)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5248717, -2133395, 5300006, 2922833, 2443546, 274328, -227593, -8134, 19212, 10255, 1144, -3436, 357, 224, -30, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 - 30*x^14 + 224*x^13 + 357*x^12 - 3436*x^11 + 1144*x^10 + 10255*x^9 + 19212*x^8 - 8134*x^7 - 227593*x^6 + 274328*x^5 + 2443546*x^4 + 2922833*x^3 + 5300006*x^2 - 2133395*x + 5248717)
 
gp: K = bnfinit(x^16 - 6*x^15 - 30*x^14 + 224*x^13 + 357*x^12 - 3436*x^11 + 1144*x^10 + 10255*x^9 + 19212*x^8 - 8134*x^7 - 227593*x^6 + 274328*x^5 + 2443546*x^4 + 2922833*x^3 + 5300006*x^2 - 2133395*x + 5248717, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} - 30 x^{14} + 224 x^{13} + 357 x^{12} - 3436 x^{11} + 1144 x^{10} + 10255 x^{9} + 19212 x^{8} - 8134 x^{7} - 227593 x^{6} + 274328 x^{5} + 2443546 x^{4} + 2922833 x^{3} + 5300006 x^{2} - 2133395 x + 5248717 \)

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:  \(68372792983010839504523425519489=17^{14}\cdot 67^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $97.65$
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:  $17, 67$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{26} a^{13} - \frac{1}{26} a^{12} - \frac{2}{13} a^{11} - \frac{5}{26} a^{10} + \frac{9}{26} a^{9} - \frac{1}{26} a^{7} + \frac{3}{26} a^{6} + \frac{4}{13} a^{5} + \frac{3}{26} a^{4} - \frac{1}{2} a^{3} + \frac{5}{13} a^{2} + \frac{11}{26} a + \frac{5}{26}$, $\frac{1}{1229278882} a^{14} + \frac{9816871}{1229278882} a^{13} - \frac{132678069}{1229278882} a^{12} + \frac{582503215}{1229278882} a^{11} + \frac{149190557}{1229278882} a^{10} + \frac{14697737}{94559914} a^{9} + \frac{300225327}{1229278882} a^{8} - \frac{61615681}{1229278882} a^{7} - \frac{475815111}{1229278882} a^{6} + \frac{2814841}{1229278882} a^{5} + \frac{44991017}{94559914} a^{4} - \frac{378120343}{1229278882} a^{3} + \frac{545708473}{1229278882} a^{2} + \frac{323882551}{1229278882} a - \frac{190219}{94559914}$, $\frac{1}{25241236618452841528081550123793745709689838} a^{15} + \frac{7339649638444654554513487362180721}{25241236618452841528081550123793745709689838} a^{14} + \frac{298580168930847617333809331197578838501347}{25241236618452841528081550123793745709689838} a^{13} + \frac{3248030188662188950288342940388092078647481}{25241236618452841528081550123793745709689838} a^{12} - \frac{9864430263063541472423553096803938128299573}{25241236618452841528081550123793745709689838} a^{11} + \frac{109196998515344439035300872161717871645603}{25241236618452841528081550123793745709689838} a^{10} - \frac{8729935608403802276471865918999363418231227}{25241236618452841528081550123793745709689838} a^{9} - \frac{9756993213666914470281751369867110749985755}{25241236618452841528081550123793745709689838} a^{8} - \frac{6406595134974671719695565076795648856800041}{25241236618452841528081550123793745709689838} a^{7} - \frac{4494390707076216326454599741412149512804689}{25241236618452841528081550123793745709689838} a^{6} + \frac{7221275720447806383318206007932387550315891}{25241236618452841528081550123793745709689838} a^{5} + \frac{2527026994209702963409334606468051259528623}{25241236618452841528081550123793745709689838} a^{4} - \frac{1488064070749486520373716657156128547777583}{25241236618452841528081550123793745709689838} a^{3} + \frac{2881058404374057442857779571351005821564699}{25241236618452841528081550123793745709689838} a^{2} - \frac{1086768693102356202050014343616847974500061}{25241236618452841528081550123793745709689838} a - \frac{2716023878856026229012893836469773326114552}{12620618309226420764040775061896872854844919}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{12}\times C_{12}$, which has order $1152$ (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:  \( 1910768.12169 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:C_8$ (as 16T24):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_2^2 : C_8$
Character table for $C_2^2 : C_8$

Intermediate fields

\(\Q(\sqrt{17}) \), 4.2.329171.1, 4.4.4913.1, 4.2.19363.1, 8.0.8268784250602433.5, 8.4.1842010303097.2, 8.4.108353547241.1

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

Sibling fields

Galois closure: data not computed
Degree 16 sibling: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{8}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$17$17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$
$67$67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$