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Properties

Label 16.8.23027322604...0625.1
Degree $16$
Signature $[8, 4]$
Discriminant $5^{8}\cdot 11^{8}\cdot 229^{4}$
Root discriminant $28.85$
Ramified primes $5, 11, 229$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1744

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![41, -7443, -14481, 4038, 12143, 135, -4495, -1980, 1093, 356, -347, -5, 107, 6, -17, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 17*x^14 + 6*x^13 + 107*x^12 - 5*x^11 - 347*x^10 + 356*x^9 + 1093*x^8 - 1980*x^7 - 4495*x^6 + 135*x^5 + 12143*x^4 + 4038*x^3 - 14481*x^2 - 7443*x + 41)
 
gp: K = bnfinit(x^16 - x^15 - 17*x^14 + 6*x^13 + 107*x^12 - 5*x^11 - 347*x^10 + 356*x^9 + 1093*x^8 - 1980*x^7 - 4495*x^6 + 135*x^5 + 12143*x^4 + 4038*x^3 - 14481*x^2 - 7443*x + 41, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 17 x^{14} + 6 x^{13} + 107 x^{12} - 5 x^{11} - 347 x^{10} + 356 x^{9} + 1093 x^{8} - 1980 x^{7} - 4495 x^{6} + 135 x^{5} + 12143 x^{4} + 4038 x^{3} - 14481 x^{2} - 7443 x + 41 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(230273226043640531640625=5^{8}\cdot 11^{8}\cdot 229^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.85$
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:  $5, 11, 229$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{229} a^{14} - \frac{51}{229} a^{13} + \frac{76}{229} a^{12} - \frac{86}{229} a^{11} - \frac{41}{229} a^{10} - \frac{81}{229} a^{9} + \frac{16}{229} a^{8} + \frac{30}{229} a^{7} - \frac{102}{229} a^{6} - \frac{58}{229} a^{5} + \frac{96}{229} a^{4} - \frac{17}{229} a^{3} - \frac{62}{229} a^{2} - \frac{99}{229} a - \frac{93}{229}$, $\frac{1}{87809171531752858649847339343} a^{15} + \frac{117254705321724513993351366}{87809171531752858649847339343} a^{14} - \frac{7447760906057611105035519093}{87809171531752858649847339343} a^{13} + \frac{11749802613486526485533490332}{87809171531752858649847339343} a^{12} - \frac{36644473654579058212134799963}{87809171531752858649847339343} a^{11} - \frac{28367360101468058011590475840}{87809171531752858649847339343} a^{10} - \frac{12262128491397380756888305289}{87809171531752858649847339343} a^{9} + \frac{26961652756708918185215007944}{87809171531752858649847339343} a^{8} - \frac{32094961382731361143839036804}{87809171531752858649847339343} a^{7} + \frac{22935297931238384388361572465}{87809171531752858649847339343} a^{6} - \frac{34381586940773605023978533454}{87809171531752858649847339343} a^{5} + \frac{31578605640702227180739654691}{87809171531752858649847339343} a^{4} + \frac{22662059680634810902799686970}{87809171531752858649847339343} a^{3} - \frac{11197621020116398466395859916}{87809171531752858649847339343} a^{2} - \frac{31542041501520057181786576091}{87809171531752858649847339343} a + \frac{19291407706795512971290084079}{87809171531752858649847339343}$

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:  $11$
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:  \( 235773.081481 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1744:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 8192
The 95 conjugacy class representatives for t16n1744 are not computed
Character table for t16n1744 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.2.275.1, 8.4.17318125.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.8.6.2$x^{8} - 781 x^{4} + 290521$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
229Data not computed