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Properties

Label 16.0.71411348623...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 5^{9}\cdot 11^{7}\cdot 31^{5}$
Root discriminant $41.29$
Ramified primes $2, 5, 11, 31$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1782

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![29161, -68728, 122166, -154396, 150974, -119942, 77587, -42224, 19375, -7002, 1853, -304, -11, 12, 9, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 9*x^14 + 12*x^13 - 11*x^12 - 304*x^11 + 1853*x^10 - 7002*x^9 + 19375*x^8 - 42224*x^7 + 77587*x^6 - 119942*x^5 + 150974*x^4 - 154396*x^3 + 122166*x^2 - 68728*x + 29161)
 
gp: K = bnfinit(x^16 - 6*x^15 + 9*x^14 + 12*x^13 - 11*x^12 - 304*x^11 + 1853*x^10 - 7002*x^9 + 19375*x^8 - 42224*x^7 + 77587*x^6 - 119942*x^5 + 150974*x^4 - 154396*x^3 + 122166*x^2 - 68728*x + 29161, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 9 x^{14} + 12 x^{13} - 11 x^{12} - 304 x^{11} + 1853 x^{10} - 7002 x^{9} + 19375 x^{8} - 42224 x^{7} + 77587 x^{6} - 119942 x^{5} + 150974 x^{4} - 154396 x^{3} + 122166 x^{2} - 68728 x + 29161 \)

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:  \(71411348623593088000000000=2^{16}\cdot 5^{9}\cdot 11^{7}\cdot 31^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.29$
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:  $2, 5, 11, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} + \frac{1}{5} a^{10} + \frac{2}{5} a^{8} - \frac{2}{5} a^{6} - \frac{2}{5} a^{4} + \frac{1}{5} a^{2} - \frac{1}{5}$, $\frac{1}{55} a^{13} - \frac{4}{55} a^{12} + \frac{1}{55} a^{11} - \frac{19}{55} a^{10} + \frac{17}{55} a^{9} + \frac{27}{55} a^{8} - \frac{7}{55} a^{7} + \frac{13}{55} a^{6} + \frac{8}{55} a^{5} - \frac{12}{55} a^{4} - \frac{9}{55} a^{3} + \frac{6}{55} a^{2} - \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{275} a^{14} + \frac{2}{275} a^{13} - \frac{12}{275} a^{12} - \frac{68}{275} a^{11} + \frac{79}{275} a^{10} - \frac{91}{275} a^{9} + \frac{67}{275} a^{8} + \frac{81}{275} a^{7} - \frac{101}{275} a^{6} + \frac{91}{275} a^{5} - \frac{103}{275} a^{4} + \frac{62}{275} a^{3} - \frac{74}{275} a^{2} + \frac{3}{25} a + \frac{3}{25}$, $\frac{1}{238152583891290282201268325} a^{15} + \frac{93102269398135367351518}{238152583891290282201268325} a^{14} + \frac{39429024546241304716898}{47630516778258056440253665} a^{13} - \frac{3405882285741348891756669}{47630516778258056440253665} a^{12} - \frac{71183765061266289873111939}{238152583891290282201268325} a^{11} - \frac{101027634721642918607341087}{238152583891290282201268325} a^{10} - \frac{69032641657588678872840999}{238152583891290282201268325} a^{9} - \frac{96646225598130750225699767}{238152583891290282201268325} a^{8} - \frac{19032277147793843733234969}{47630516778258056440253665} a^{7} + \frac{4961659682394156016227954}{47630516778258056440253665} a^{6} - \frac{22002089597703137194116137}{238152583891290282201268325} a^{5} + \frac{8438082188108656976392009}{238152583891290282201268325} a^{4} + \frac{83948355209875964047244138}{238152583891290282201268325} a^{3} - \frac{76605204908818431030434961}{238152583891290282201268325} a^{2} - \frac{5062341398321021689780869}{21650234899208207472842575} a + \frac{9709703281784830122735583}{21650234899208207472842575}$

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:  $7$
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:  \( 3675832.26532 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1782:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16384
The 130 conjugacy class representatives for t16n1782 are not computed
Character table for t16n1782 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.8525.1, 8.4.204654560000.6

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 ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
$11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.4.3.2$x^{4} - 11$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
31Data not computed