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Properties

Label 16.12.5074426245...8441.1
Degree $16$
Signature $[12, 2]$
Discriminant $11^{10}\cdot 89^{14}$
Root discriminant $227.29$
Ramified primes $11, 89$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T817

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-352528, -4996640, 7184468, 63277728, 59143279, -6174233, -15451735, -44849, 721609, 54030, 49484, -13831, -1849, 597, -57, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 57*x^14 + 597*x^13 - 1849*x^12 - 13831*x^11 + 49484*x^10 + 54030*x^9 + 721609*x^8 - 44849*x^7 - 15451735*x^6 - 6174233*x^5 + 59143279*x^4 + 63277728*x^3 + 7184468*x^2 - 4996640*x - 352528)
 
gp: K = bnfinit(x^16 - 2*x^15 - 57*x^14 + 597*x^13 - 1849*x^12 - 13831*x^11 + 49484*x^10 + 54030*x^9 + 721609*x^8 - 44849*x^7 - 15451735*x^6 - 6174233*x^5 + 59143279*x^4 + 63277728*x^3 + 7184468*x^2 - 4996640*x - 352528, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 57 x^{14} + 597 x^{13} - 1849 x^{12} - 13831 x^{11} + 49484 x^{10} + 54030 x^{9} + 721609 x^{8} - 44849 x^{7} - 15451735 x^{6} - 6174233 x^{5} + 59143279 x^{4} + 63277728 x^{3} + 7184468 x^{2} - 4996640 x - 352528 \)

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:  \(50744262454554467455972799262367678441=11^{10}\cdot 89^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $227.29$
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:  $11, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4}$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{9} + \frac{1}{8} a^{8} - \frac{1}{4} a^{7} + \frac{3}{8} a^{6} - \frac{1}{2} a^{4} + \frac{1}{8} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{16} a^{13} - \frac{1}{16} a^{12} - \frac{1}{4} a^{11} + \frac{3}{16} a^{10} - \frac{1}{8} a^{9} + \frac{1}{16} a^{8} - \frac{7}{16} a^{7} + \frac{1}{16} a^{6} - \frac{1}{4} a^{5} - \frac{7}{16} a^{4} - \frac{1}{8} a^{3} - \frac{5}{16} a^{2} - \frac{1}{8} a - \frac{1}{4}$, $\frac{1}{518368} a^{14} + \frac{4573}{259184} a^{13} - \frac{18939}{518368} a^{12} + \frac{367}{518368} a^{11} + \frac{41527}{518368} a^{10} - \frac{115185}{518368} a^{9} - \frac{2542}{16199} a^{8} - \frac{16447}{129592} a^{7} + \frac{1211}{5344} a^{6} + \frac{167293}{518368} a^{5} - \frac{207399}{518368} a^{4} - \frac{45559}{518368} a^{3} + \frac{118851}{518368} a^{2} + \frac{15755}{259184} a + \frac{42625}{129592}$, $\frac{1}{16031015189208118310640363914694765992446504359075008} a^{15} + \frac{7483159542996651574086657515537524388321777737}{8015507594604059155320181957347382996223252179537504} a^{14} + \frac{496756917775104701018583866254062575016199797253907}{16031015189208118310640363914694765992446504359075008} a^{13} - \frac{716868635603514039014302215717192150431790932999967}{16031015189208118310640363914694765992446504359075008} a^{12} - \frac{2237802184480237771812380228349747210123758487203545}{16031015189208118310640363914694765992446504359075008} a^{11} - \frac{3503763113712744851923975739081751052213699501773375}{16031015189208118310640363914694765992446504359075008} a^{10} + \frac{626310747888369558478448022607637894631284352825205}{4007753797302029577660090978673691498111626089768752} a^{9} - \frac{991599763649084150123046527889225113108885136615163}{8015507594604059155320181957347382996223252179537504} a^{8} + \frac{2946117387240839142725316068791401558277186195487}{95994102929389930003834514459250095763152720713024} a^{7} + \frac{856888660527673193578321714213694059795388169663867}{16031015189208118310640363914694765992446504359075008} a^{6} + \frac{6744527015367493674151435499854964242984768118042233}{16031015189208118310640363914694765992446504359075008} a^{5} - \frac{328481661524583926884884117253333069243393971472721}{16031015189208118310640363914694765992446504359075008} a^{4} + \frac{4786783942555787376475800312797964588880778968041223}{16031015189208118310640363914694765992446504359075008} a^{3} + \frac{88767532381272197361561475389360600576154630695062}{250484612331376848603755686167105718631976630610547} a^{2} + \frac{133998268764595569911047350582691787657206204288883}{2003876898651014788830045489336845749055813044884376} a + \frac{177424461028711729002371806661628647631404326347225}{2003876898651014788830045489336845749055813044884376}$

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:  \( 70783447361700 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T817:

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 t16n817
Character table for t16n817 is not computed

Intermediate fields

\(\Q(\sqrt{89}) \), 4.4.704969.1, 8.8.647590974205440089.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$11$11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.4.3.2$x^{4} - 11$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.3.2$x^{4} - 11$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
89Data not computed