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Properties

Label 16.0.47103964466...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 5^{10}\cdot 29^{4}\cdot 101^{4}$
Root discriminant $40.23$
Ramified primes $2, 5, 29, 101$
Class number $40$ (GRH)
Class group $[2, 20]$ (GRH)
Galois group 16T1174

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![59105, -8840, 11511, -11294, 21611, -21798, 10610, -1132, -1051, -350, 1214, -520, 26, 0, 20, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 20*x^14 + 26*x^12 - 520*x^11 + 1214*x^10 - 350*x^9 - 1051*x^8 - 1132*x^7 + 10610*x^6 - 21798*x^5 + 21611*x^4 - 11294*x^3 + 11511*x^2 - 8840*x + 59105)
 
gp: K = bnfinit(x^16 - 8*x^15 + 20*x^14 + 26*x^12 - 520*x^11 + 1214*x^10 - 350*x^9 - 1051*x^8 - 1132*x^7 + 10610*x^6 - 21798*x^5 + 21611*x^4 - 11294*x^3 + 11511*x^2 - 8840*x + 59105, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 20 x^{14} + 26 x^{12} - 520 x^{11} + 1214 x^{10} - 350 x^{9} - 1051 x^{8} - 1132 x^{7} + 10610 x^{6} - 21798 x^{5} + 21611 x^{4} - 11294 x^{3} + 11511 x^{2} - 8840 x + 59105 \)

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:  \(47103964466995840000000000=2^{16}\cdot 5^{10}\cdot 29^{4}\cdot 101^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $40.23$
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:  $2, 5, 29, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{34416199809162565} a^{14} - \frac{7}{34416199809162565} a^{13} - \frac{2417053338747581}{34416199809162565} a^{12} + \frac{14502320032485577}{34416199809162565} a^{11} + \frac{3829122118233662}{34416199809162565} a^{10} - \frac{14418744985636006}{34416199809162565} a^{9} - \frac{115205889149902}{6883239961832513} a^{8} + \frac{436982905295162}{2024482341715445} a^{7} + \frac{4818772181889348}{34416199809162565} a^{6} + \frac{892737964561739}{6883239961832513} a^{5} - \frac{1935342824224587}{34416199809162565} a^{4} + \frac{942729530610022}{6883239961832513} a^{3} + \frac{7436809055915344}{34416199809162565} a^{2} + \frac{1314060029823953}{6883239961832513} a - \frac{2194899746992375}{6883239961832513}$, $\frac{1}{2103001889338878534325} a^{15} + \frac{6109}{420600377867775706865} a^{14} - \frac{38383796005296383951}{84120075573555141373} a^{13} - \frac{29097167857612013893}{420600377867775706865} a^{12} + \frac{400850033588775114911}{2103001889338878534325} a^{11} + \frac{7799258471289811048}{191181989939898048575} a^{10} + \frac{177386378543107687188}{2103001889338878534325} a^{9} + \frac{459726852236126602219}{2103001889338878534325} a^{8} - \frac{550667286632023846619}{2103001889338878534325} a^{7} + \frac{4196462988923688998}{123705993490522266725} a^{6} - \frac{72920239679311541812}{191181989939898048575} a^{5} - \frac{414712056746324153494}{2103001889338878534325} a^{4} + \frac{363594154687500040764}{2103001889338878534325} a^{3} + \frac{383912918307591868098}{2103001889338878534325} a^{2} - \frac{168128107633693758673}{420600377867775706865} a - \frac{100297742845800246277}{420600377867775706865}$

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

Class group and class number

$C_{2}\times C_{20}$, which has order $40$ (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:  \( 63128.2950837 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1174:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 70 conjugacy class representatives for t16n1174 are not computed
Character table for t16n1174 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.2525.1, 8.0.6863232800000.1, 8.4.924465625.2, 8.4.47332640000.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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}$
$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}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$29$29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$101$101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$