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Properties

Label 16.12.2767007058...0000.1
Degree $16$
Signature $[12, 2]$
Discriminant $2^{16}\cdot 3^{6}\cdot 5^{8}\cdot 7^{6}\cdot 17^{8}\cdot 13441^{2}$
Root discriminant $189.51$
Ramified primes $2, 3, 5, 7, 17, 13441$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1228

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![79671272121, 0, -102178199739, 0, 26685015734, 0, -1136668468, 0, -88640101, 0, 1920664, 0, 2326, 0, -217, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 217*x^14 + 2326*x^12 + 1920664*x^10 - 88640101*x^8 - 1136668468*x^6 + 26685015734*x^4 - 102178199739*x^2 + 79671272121)
 
gp: K = bnfinit(x^16 - 217*x^14 + 2326*x^12 + 1920664*x^10 - 88640101*x^8 - 1136668468*x^6 + 26685015734*x^4 - 102178199739*x^2 + 79671272121, 1)
 

Normalized defining polynomial

\( x^{16} - 217 x^{14} + 2326 x^{12} + 1920664 x^{10} - 88640101 x^{8} - 1136668468 x^{6} + 26685015734 x^{4} - 102178199739 x^{2} + 79671272121 \)

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:  \(2767007058000819292896502809600000000=2^{16}\cdot 3^{6}\cdot 5^{8}\cdot 7^{6}\cdot 17^{8}\cdot 13441^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $189.51$
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:  $2, 3, 5, 7, 17, 13441$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8} - \frac{1}{5} a^{6} + \frac{2}{5} a^{2} - \frac{1}{5}$, $\frac{1}{5} a^{9} - \frac{1}{5} a^{7} + \frac{2}{5} a^{3} - \frac{1}{5} a$, $\frac{1}{5} a^{10} - \frac{1}{5} a^{6} + \frac{2}{5} a^{4} + \frac{1}{5} a^{2} - \frac{1}{5}$, $\frac{1}{5} a^{11} - \frac{1}{5} a^{7} + \frac{2}{5} a^{5} + \frac{1}{5} a^{3} - \frac{1}{5} a$, $\frac{1}{170} a^{12} - \frac{1}{17} a^{10} + \frac{3}{170} a^{8} - \frac{7}{170} a^{6} - \frac{59}{170} a^{4} + \frac{6}{85} a^{2} + \frac{1}{170}$, $\frac{1}{170} a^{13} - \frac{1}{17} a^{11} + \frac{3}{170} a^{9} - \frac{7}{170} a^{7} - \frac{59}{170} a^{5} + \frac{6}{85} a^{3} + \frac{1}{170} a$, $\frac{1}{17631171211536393855742785256150323514710} a^{14} - \frac{2768999456659325954314408869881864501}{1259369372252599561124484661153594536765} a^{12} + \frac{1159646172552609246591995831205602201101}{17631171211536393855742785256150323514710} a^{10} + \frac{1178176073465881480942812202332835122583}{17631171211536393855742785256150323514710} a^{8} - \frac{5141456953468070851152334540833285236731}{17631171211536393855742785256150323514710} a^{6} - \frac{1301105718341375355384745506855323378438}{8815585605768196927871392628075161757355} a^{4} - \frac{965692967360164432791963706075599205127}{3526234242307278771148557051230064702942} a^{2} + \frac{95279650750234302881350018894668}{31232035618694034697926361162453055}$, $\frac{1}{17631171211536393855742785256150323514710} a^{15} - \frac{2768999456659325954314408869881864501}{1259369372252599561124484661153594536765} a^{13} + \frac{1159646172552609246591995831205602201101}{17631171211536393855742785256150323514710} a^{11} + \frac{1178176073465881480942812202332835122583}{17631171211536393855742785256150323514710} a^{9} - \frac{5141456953468070851152334540833285236731}{17631171211536393855742785256150323514710} a^{7} - \frac{1301105718341375355384745506855323378438}{8815585605768196927871392628075161757355} a^{5} - \frac{965692967360164432791963706075599205127}{3526234242307278771148557051230064702942} a^{3} + \frac{95279650750234302881350018894668}{31232035618694034697926361162453055} a$

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:  $13$
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:  \( 2425599547010 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1228:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 61 conjugacy class representatives for t16n1228 are not computed
Character table for t16n1228 is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), \(\Q(\sqrt{85}) \), \(\Q(\sqrt{5}) \), 4.4.151725.1, 4.4.151725.2, \(\Q(\sqrt{5}, \sqrt{17})\), 8.8.23020475625.1

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 R R R R ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$2$2.8.8.6$x^{8} + 2 x^{7} + 2 x^{6} + 16 x^{2} + 16$$2$$4$$8$$(C_8:C_2):C_2$$[2, 2, 2]^{4}$
2.8.8.6$x^{8} + 2 x^{7} + 2 x^{6} + 16 x^{2} + 16$$2$$4$$8$$(C_8:C_2):C_2$$[2, 2, 2]^{4}$
$3$3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$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}$
$7$7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$17$17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
13441Data not computed