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Properties

Label 16.0.69445211572...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 5^{8}\cdot 19^{6}\cdot 29^{3}\cdot 31^{4}$
Root discriminant $84.65$
Ramified primes $2, 5, 19, 29, 31$
Class number $13120$ (GRH)
Class group $[2, 2, 2, 2, 820]$ (GRH)
Galois group 16T1719

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![12543759, 37860, 7647038, -1382320, 1223950, -705304, 156958, -164802, 54324, -27256, 10766, -2812, 1015, -150, 58, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 58*x^14 - 150*x^13 + 1015*x^12 - 2812*x^11 + 10766*x^10 - 27256*x^9 + 54324*x^8 - 164802*x^7 + 156958*x^6 - 705304*x^5 + 1223950*x^4 - 1382320*x^3 + 7647038*x^2 + 37860*x + 12543759)
 
gp: K = bnfinit(x^16 - 6*x^15 + 58*x^14 - 150*x^13 + 1015*x^12 - 2812*x^11 + 10766*x^10 - 27256*x^9 + 54324*x^8 - 164802*x^7 + 156958*x^6 - 705304*x^5 + 1223950*x^4 - 1382320*x^3 + 7647038*x^2 + 37860*x + 12543759, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 58 x^{14} - 150 x^{13} + 1015 x^{12} - 2812 x^{11} + 10766 x^{10} - 27256 x^{9} + 54324 x^{8} - 164802 x^{7} + 156958 x^{6} - 705304 x^{5} + 1223950 x^{4} - 1382320 x^{3} + 7647038 x^{2} + 37860 x + 12543759 \)

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:  \(6944521157247548712550400000000=2^{24}\cdot 5^{8}\cdot 19^{6}\cdot 29^{3}\cdot 31^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $84.65$
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:  $2, 5, 19, 29, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{7} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{8} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{9} a^{13} + \frac{1}{9} a^{9} + \frac{2}{9} a^{8} + \frac{4}{9} a^{7} - \frac{1}{3} a^{6} - \frac{4}{9} a^{5} - \frac{1}{3} a^{4} - \frac{1}{9} a^{3} - \frac{4}{9} a^{2} - \frac{4}{9} a + \frac{1}{3}$, $\frac{1}{27} a^{14} + \frac{1}{9} a^{12} - \frac{1}{9} a^{11} + \frac{4}{27} a^{10} - \frac{4}{27} a^{9} + \frac{1}{27} a^{8} - \frac{4}{27} a^{6} - \frac{2}{9} a^{5} - \frac{7}{27} a^{4} + \frac{5}{27} a^{3} - \frac{13}{27} a^{2} + \frac{1}{9} a$, $\frac{1}{1254042745238726911729899860157176641650003149703} a^{15} - \frac{580631866090721258199394098207422538730218812}{139338082804302990192211095573019626850000349967} a^{14} - \frac{1864153830626294817456966675028213353158342041}{139338082804302990192211095573019626850000349967} a^{13} - \frac{68393532428557871822804906371292421768488394631}{418014248412908970576633286719058880550001049901} a^{12} + \frac{131105506409607438062287471842145075540948068958}{1254042745238726911729899860157176641650003149703} a^{11} + \frac{125608215219665288253403759184318169300582810506}{1254042745238726911729899860157176641650003149703} a^{10} + \frac{25629735654819704156710650886750688459931382495}{1254042745238726911729899860157176641650003149703} a^{9} - \frac{199030375134304615156902129539938838820011276213}{418014248412908970576633286719058880550001049901} a^{8} + \frac{514160803465366295491695908176611875987660956256}{1254042745238726911729899860157176641650003149703} a^{7} - \frac{140204330334421434416254129796594169931283079539}{418014248412908970576633286719058880550001049901} a^{6} - \frac{357872281655250203397564560335481116622395754884}{1254042745238726911729899860157176641650003149703} a^{5} - \frac{4097368749410325350477388304172251107228329314}{1254042745238726911729899860157176641650003149703} a^{4} + \frac{72106169147857531852341596733908297578659397640}{1254042745238726911729899860157176641650003149703} a^{3} + \frac{103342205776157180111712296858289367318103302238}{418014248412908970576633286719058880550001049901} a^{2} - \frac{119070725393499318236163617798324981217115435181}{418014248412908970576633286719058880550001049901} a + \frac{43212775390322917772557563862855906565587139735}{139338082804302990192211095573019626850000349967}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{820}$, which has order $13120$ (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:  \( 70657.2169247 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1719:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 8192
The 152 conjugacy class representatives for t16n1719 are not computed
Character table for t16n1719 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.12400.1, 4.4.14725.1, 4.4.7600.1, 8.8.55507360000.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ R R ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$19$19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.4.3.2$x^{4} - 19$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
19.4.3.2$x^{4} - 19$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
$29$$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
$31$31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
31.4.0.1$x^{4} - 2 x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$