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Properties

Label 16.0.26893256677...0000.5
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 5^{12}\cdot 29^{6}\cdot 41^{4}$
Root discriminant $59.82$
Ramified primes $2, 5, 29, 41$
Class number $64$ (GRH)
Class group $[2, 4, 8]$ (GRH)
Galois group $C_2^2.C_2^5.C_2$ (as 16T610)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -24, 462, -1926, 13276, 33150, 83373, -2460, 44647, -12190, 9133, -1800, 876, -136, 42, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 42*x^14 - 136*x^13 + 876*x^12 - 1800*x^11 + 9133*x^10 - 12190*x^9 + 44647*x^8 - 2460*x^7 + 83373*x^6 + 33150*x^5 + 13276*x^4 - 1926*x^3 + 462*x^2 - 24*x + 1)
 
gp: K = bnfinit(x^16 - 4*x^15 + 42*x^14 - 136*x^13 + 876*x^12 - 1800*x^11 + 9133*x^10 - 12190*x^9 + 44647*x^8 - 2460*x^7 + 83373*x^6 + 33150*x^5 + 13276*x^4 - 1926*x^3 + 462*x^2 - 24*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 42 x^{14} - 136 x^{13} + 876 x^{12} - 1800 x^{11} + 9133 x^{10} - 12190 x^{9} + 44647 x^{8} - 2460 x^{7} + 83373 x^{6} + 33150 x^{5} + 13276 x^{4} - 1926 x^{3} + 462 x^{2} - 24 x + 1 \)

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:  \(26893256677956496000000000000=2^{16}\cdot 5^{12}\cdot 29^{6}\cdot 41^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $59.82$
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:  $2, 5, 29, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{19} a^{12} - \frac{3}{19} a^{11} + \frac{4}{19} a^{10} + \frac{9}{19} a^{9} + \frac{9}{19} a^{8} - \frac{2}{19} a^{7} - \frac{3}{19} a^{6} - \frac{9}{19} a^{5} - \frac{7}{19} a^{4} + \frac{3}{19} a^{3} + \frac{4}{19} a^{2} - \frac{7}{19} a + \frac{1}{19}$, $\frac{1}{779} a^{13} + \frac{13}{779} a^{12} + \frac{317}{779} a^{11} - \frac{174}{779} a^{10} - \frac{189}{779} a^{9} + \frac{161}{779} a^{8} + \frac{60}{779} a^{7} + \frac{19}{41} a^{6} + \frac{210}{779} a^{5} - \frac{71}{779} a^{4} - \frac{347}{779} a^{3} + \frac{8}{41} a^{2} - \frac{187}{779} a + \frac{244}{779}$, $\frac{1}{779} a^{14} - \frac{16}{779} a^{12} + \frac{92}{779} a^{11} - \frac{141}{779} a^{10} + \frac{363}{779} a^{9} + \frac{386}{779} a^{8} - \frac{91}{779} a^{7} - \frac{96}{779} a^{6} + \frac{233}{779} a^{5} + \frac{166}{779} a^{4} + \frac{276}{779} a^{3} + \frac{297}{779} a^{2} - \frac{72}{779} a - \frac{220}{779}$, $\frac{1}{4166560472107089246225989129} a^{15} + \frac{2286005015852665935100557}{4166560472107089246225989129} a^{14} + \frac{1839033572273656003827356}{4166560472107089246225989129} a^{13} - \frac{58022958698755938403498823}{4166560472107089246225989129} a^{12} - \frac{212442782593138228914545040}{4166560472107089246225989129} a^{11} - \frac{1445169230836621083486684125}{4166560472107089246225989129} a^{10} + \frac{1204354493067872733368454210}{4166560472107089246225989129} a^{9} + \frac{48871148882640987150931448}{4166560472107089246225989129} a^{8} - \frac{39185224395759388762131276}{219292656426688907696104691} a^{7} + \frac{599225422019700268093182503}{4166560472107089246225989129} a^{6} + \frac{1760963493085686891454866495}{4166560472107089246225989129} a^{5} + \frac{1167802550357132774506541631}{4166560472107089246225989129} a^{4} + \frac{184495855252407401960678106}{378778224737008113293271739} a^{3} - \frac{1193912584258860683773106684}{4166560472107089246225989129} a^{2} - \frac{1410820396741623056001471309}{4166560472107089246225989129} a - \frac{943404943423509705573576576}{4166560472107089246225989129}$

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

Class group and class number

$C_{2}\times C_{4}\times C_{8}$, which has order $64$ (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:  \( 811646.528599 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2.C_2^5.C_2$ (as 16T610):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 40 conjugacy class representatives for $C_2^2.C_2^5.C_2$
Character table for $C_2^2.C_2^5.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.29725.2, \(\Q(\zeta_{20})^+\), 4.0.2378000.5, 8.0.5654884000000.2

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 R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
5Data not computed
$29$29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.8.6.2$x^{8} + 145 x^{4} + 7569$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$41$$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$