↑

Properties

Label 16.12.1274602568...9401.1
Degree $16$
Signature $[12, 2]$
Discriminant $31^{10}\cdot 41^{15}$
Root discriminant $278.03$
Ramified primes $31, 41$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1223

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![20892667, -5205126238, -10822844427, -2559956558, 5102097565, 2180463570, -387385807, -200346915, 16528758, 7483195, -678393, -140131, 18632, 1346, -233, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 - 233*x^14 + 1346*x^13 + 18632*x^12 - 140131*x^11 - 678393*x^10 + 7483195*x^9 + 16528758*x^8 - 200346915*x^7 - 387385807*x^6 + 2180463570*x^5 + 5102097565*x^4 - 2559956558*x^3 - 10822844427*x^2 - 5205126238*x + 20892667)
 
gp: K = bnfinit(x^16 - 5*x^15 - 233*x^14 + 1346*x^13 + 18632*x^12 - 140131*x^11 - 678393*x^10 + 7483195*x^9 + 16528758*x^8 - 200346915*x^7 - 387385807*x^6 + 2180463570*x^5 + 5102097565*x^4 - 2559956558*x^3 - 10822844427*x^2 - 5205126238*x + 20892667, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} - 233 x^{14} + 1346 x^{13} + 18632 x^{12} - 140131 x^{11} - 678393 x^{10} + 7483195 x^{9} + 16528758 x^{8} - 200346915 x^{7} - 387385807 x^{6} + 2180463570 x^{5} + 5102097565 x^{4} - 2559956558 x^{3} - 10822844427 x^{2} - 5205126238 x + 20892667 \)

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:  \(1274602568003244304847660149105336749401=31^{10}\cdot 41^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $278.03$
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:  $31, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{4864001327165085251051352066747889542368095276700848702577534602488619036157} a^{15} + \frac{2191349033151362515207913876345052526823716980570251772735829247984692468925}{4864001327165085251051352066747889542368095276700848702577534602488619036157} a^{14} + \frac{2089136833814346746243065496316079478172172113465793846924767731357601822397}{4864001327165085251051352066747889542368095276700848702577534602488619036157} a^{13} + \frac{1875652229742680775075737353695815125716967847907519676450592059776720862004}{4864001327165085251051352066747889542368095276700848702577534602488619036157} a^{12} + \frac{47464221053142536962186545925557486926301474136096250707275491063721467555}{4864001327165085251051352066747889542368095276700848702577534602488619036157} a^{11} - \frac{1774656907130081074461423763065158217614085091475877030912623419000121711403}{4864001327165085251051352066747889542368095276700848702577534602488619036157} a^{10} + \frac{2149908097586889882406017499711584134214846508768993526075288030616305672486}{4864001327165085251051352066747889542368095276700848702577534602488619036157} a^{9} - \frac{1162528490591442818404106642860288384903283105855924693896575999852741811704}{4864001327165085251051352066747889542368095276700848702577534602488619036157} a^{8} - \frac{94660166526285118401000809265011943307974128520107589674955339652000109899}{4864001327165085251051352066747889542368095276700848702577534602488619036157} a^{7} - \frac{491583858442528336782638581993031902040395448862057602041311300728606497586}{4864001327165085251051352066747889542368095276700848702577534602488619036157} a^{6} + \frac{1646593885801206249397480898304503362856208826113009084303392061902864764981}{4864001327165085251051352066747889542368095276700848702577534602488619036157} a^{5} - \frac{1375647034378697901284521733802237905212678876477830880704291791090332561933}{4864001327165085251051352066747889542368095276700848702577534602488619036157} a^{4} - \frac{107833198569608192288162282348167015725860651360821166421282736970033475603}{4864001327165085251051352066747889542368095276700848702577534602488619036157} a^{3} - \frac{17842191069257341439781126750172168748611370431633710537085759671226361958}{4864001327165085251051352066747889542368095276700848702577534602488619036157} a^{2} + \frac{1193618620185152432901584597957199378332204402231708060646052273804637804675}{4864001327165085251051352066747889542368095276700848702577534602488619036157} a + \frac{2431115296962797809304812886896536792383111210305853674549082661355271752966}{4864001327165085251051352066747889542368095276700848702577534602488619036157}$

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:  $13$
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:  \( 114073970907000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1223:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 40 conjugacy class representatives for t16n1223
Character table for t16n1223 is not computed

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.68921.1, 8.8.179859661768855001.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{8}$ $16$ R ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ $16$ $16$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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
$31$31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.4.3.2$x^{4} - 31$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
31.4.3.2$x^{4} - 31$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
41Data not computed