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Properties

Label 16.0.12374161428...9069.6
Degree $16$
Signature $[0, 8]$
Discriminant $13^{14}\cdot 61^{7}$
Root discriminant $56.99$
Ramified primes $13, 61$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 16T1281

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1298079, 770640, -194447, -154765, 106758, 43589, -19423, -3809, 7351, 1547, -945, -130, 149, 0, -19, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 19*x^14 + 149*x^12 - 130*x^11 - 945*x^10 + 1547*x^9 + 7351*x^8 - 3809*x^7 - 19423*x^6 + 43589*x^5 + 106758*x^4 - 154765*x^3 - 194447*x^2 + 770640*x + 1298079)
 
gp: K = bnfinit(x^16 - 19*x^14 + 149*x^12 - 130*x^11 - 945*x^10 + 1547*x^9 + 7351*x^8 - 3809*x^7 - 19423*x^6 + 43589*x^5 + 106758*x^4 - 154765*x^3 - 194447*x^2 + 770640*x + 1298079, 1)
 

Normalized defining polynomial

\( x^{16} - 19 x^{14} + 149 x^{12} - 130 x^{11} - 945 x^{10} + 1547 x^{9} + 7351 x^{8} - 3809 x^{7} - 19423 x^{6} + 43589 x^{5} + 106758 x^{4} - 154765 x^{3} - 194447 x^{2} + 770640 x + 1298079 \)

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:  \(12374161428874698259143289069=13^{14}\cdot 61^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $56.99$
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:  $13, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a$, $\frac{1}{39} a^{12} + \frac{5}{39} a^{11} - \frac{5}{39} a^{10} + \frac{5}{39} a^{9} - \frac{4}{13} a^{8} + \frac{17}{39} a^{7} - \frac{5}{13} a^{6} + \frac{16}{39} a^{5} - \frac{10}{39} a^{4} - \frac{6}{13} a^{3} - \frac{19}{39} a^{2} + \frac{11}{39} a - \frac{4}{13}$, $\frac{1}{234} a^{13} - \frac{1}{234} a^{12} - \frac{35}{234} a^{11} + \frac{35}{234} a^{10} + \frac{19}{39} a^{9} - \frac{14}{117} a^{8} + \frac{1}{3} a^{7} - \frac{11}{234} a^{6} + \frac{25}{117} a^{5} + \frac{7}{39} a^{4} + \frac{25}{117} a^{3} - \frac{109}{234} a^{2} + \frac{1}{6} a - \frac{5}{26}$, $\frac{1}{468} a^{14} - \frac{1}{78} a^{12} + \frac{2}{13} a^{11} + \frac{155}{468} a^{10} + \frac{1}{234} a^{9} + \frac{1}{234} a^{8} - \frac{203}{468} a^{7} + \frac{71}{156} a^{6} - \frac{1}{9} a^{5} - \frac{5}{18} a^{4} - \frac{131}{468} a^{3} - \frac{43}{117} a^{2} + \frac{14}{39} a - \frac{19}{52}$, $\frac{1}{9397765156951710177201290411200296} a^{15} - \frac{1436096238969544300565912821369}{9397765156951710177201290411200296} a^{14} - \frac{877278737316812057203794262937}{1566294192825285029533548401866716} a^{13} - \frac{13189990447723667660120490643}{120484168678868079194888338605132} a^{12} + \frac{501871649209715662834249102510055}{9397765156951710177201290411200296} a^{11} - \frac{314800166026656716982647253353635}{3132588385650570059067096803733432} a^{10} - \frac{373773656408891355417429429560531}{783147096412642514766774200933358} a^{9} + \frac{175401954957588354589160955129731}{722905012073208475169330031630792} a^{8} + \frac{438488689828895636226009727325692}{1174720644618963772150161301400037} a^{7} - \frac{3084320771656713150779956054252765}{9397765156951710177201290411200296} a^{6} + \frac{556863099802167231628831899643807}{1566294192825285029533548401866716} a^{5} + \frac{2233317998271700460695821078913835}{9397765156951710177201290411200296} a^{4} + \frac{2902332844301485769450496425561659}{9397765156951710177201290411200296} a^{3} - \frac{399808577113334386346204096428535}{2349441289237927544300322602800074} a^{2} - \frac{1396064521002605721172838840743441}{3132588385650570059067096803733432} a + \frac{92951870997262686930500994226385}{348065376183396673229677422637048}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( 83098124.0446 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1281:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 34 conjugacy class representatives for t16n1281
Character table for t16n1281 is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), 4.0.2197.1, 8.0.17960556289.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }{,}\,{\href{/LocalNumberField/2.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
$13$13.8.7.2$x^{8} - 52$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
13.8.7.2$x^{8} - 52$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$61$61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$
61.4.2.2$x^{4} - 61 x^{2} + 7442$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
61.4.3.3$x^{4} + 122$$4$$1$$3$$C_4$$[\ ]_{4}$