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Properties

Label 16.0.27245121110...3821.1
Degree $16$
Signature $[0, 8]$
Discriminant $13^{12}\cdot 61^{9}$
Root discriminant $69.14$
Ramified primes $13, 61$
Class number $8$ (GRH)
Class group $[2, 2, 2]$ (GRH)
Galois group 16T1281

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1756431, 1962513, -1608564, -1053416, 903300, 66572, -37310, -25235, 207, 1159, 2920, -735, 184, -82, 24, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 24*x^14 - 82*x^13 + 184*x^12 - 735*x^11 + 2920*x^10 + 1159*x^9 + 207*x^8 - 25235*x^7 - 37310*x^6 + 66572*x^5 + 903300*x^4 - 1053416*x^3 - 1608564*x^2 + 1962513*x + 1756431)
 
gp: K = bnfinit(x^16 + 24*x^14 - 82*x^13 + 184*x^12 - 735*x^11 + 2920*x^10 + 1159*x^9 + 207*x^8 - 25235*x^7 - 37310*x^6 + 66572*x^5 + 903300*x^4 - 1053416*x^3 - 1608564*x^2 + 1962513*x + 1756431, 1)
 

Normalized defining polynomial

\( x^{16} + 24 x^{14} - 82 x^{13} + 184 x^{12} - 735 x^{11} + 2920 x^{10} + 1159 x^{9} + 207 x^{8} - 25235 x^{7} - 37310 x^{6} + 66572 x^{5} + 903300 x^{4} - 1053416 x^{3} - 1608564 x^{2} + 1962513 x + 1756431 \)

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:  \(272451211105578415516403423821=13^{12}\cdot 61^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $69.14$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{39} a^{14} - \frac{2}{39} a^{13} + \frac{1}{39} a^{12} - \frac{4}{39} a^{11} - \frac{17}{39} a^{10} + \frac{5}{39} a^{9} - \frac{11}{39} a^{8} + \frac{19}{39} a^{7} - \frac{5}{13} a^{6} + \frac{3}{13} a^{5} - \frac{1}{13} a^{4} - \frac{4}{39} a^{3} + \frac{2}{13} a^{2} + \frac{5}{39} a - \frac{4}{13}$, $\frac{1}{369684496838394221115504833587864805684361453} a^{15} - \frac{727130790235575777496557544766941924038032}{123228165612798073705168277862621601894787151} a^{14} - \frac{18381128794290587641103702428720527720064747}{123228165612798073705168277862621601894787151} a^{13} - \frac{55299188980432347430423993038183121155616582}{369684496838394221115504833587864805684361453} a^{12} - \frac{122257022461092835334877709188619935213120065}{369684496838394221115504833587864805684361453} a^{11} - \frac{8081635099817687194345951565540718147223393}{123228165612798073705168277862621601894787151} a^{10} + \frac{59548954410580245953534955173459679036089854}{369684496838394221115504833587864805684361453} a^{9} + \frac{160007350939632760050431727922029525898347100}{369684496838394221115504833587864805684361453} a^{8} + \frac{37122081276933396500819010322581821416484206}{123228165612798073705168277862621601894787151} a^{7} + \frac{49727293364061193518800142253064927098559920}{369684496838394221115504833587864805684361453} a^{6} + \frac{13296045601405169376473796157265913170139850}{369684496838394221115504833587864805684361453} a^{5} + \frac{3914180693477819784006619886630800179348080}{369684496838394221115504833587864805684361453} a^{4} + \frac{23747202430100735310555543722690010479988895}{123228165612798073705168277862621601894787151} a^{3} + \frac{142700428090832661666188374961321354078973133}{369684496838394221115504833587864805684361453} a^{2} - \frac{284074248733413824312435599714826852622061}{3159696554174309582183801996477476971661209} a + \frac{15820936182740096233325678360362486798545575}{41076055204266024568389425954207200631595717}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}$, which has order $8$ (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:  \( 47034604.701 \) (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 ${\href{/LocalNumberField/2.8.0.1}{8} }{,}\,{\href{/LocalNumberField/2.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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$13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$61$61.2.1.2$x^{2} + 122$$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}$
61.4.2.1$x^{4} + 183 x^{2} + 14884$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
61.4.3.3$x^{4} + 122$$4$$1$$3$$C_4$$[\ ]_{4}$