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Properties

Label 16.0.17076113336...9664.9
Degree $16$
Signature $[0, 8]$
Discriminant $2^{44}\cdot 7^{8}\cdot 17^{14}$
Root discriminant $212.34$
Ramified primes $2, 7, 17$
Class number $11202336$ (GRH)
Class group $[2, 2, 2, 6, 233382]$ (GRH)
Galois group $C_8\times C_2$ (as 16T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![26656439824, 0, 38080628320, 0, 13328219912, 0, 1942889200, 0, 133226688, 0, 4338264, 0, 68306, 0, 476, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 476*x^14 + 68306*x^12 + 4338264*x^10 + 133226688*x^8 + 1942889200*x^6 + 13328219912*x^4 + 38080628320*x^2 + 26656439824)
 
gp: K = bnfinit(x^16 + 476*x^14 + 68306*x^12 + 4338264*x^10 + 133226688*x^8 + 1942889200*x^6 + 13328219912*x^4 + 38080628320*x^2 + 26656439824, 1)
 

Normalized defining polynomial

\( x^{16} + 476 x^{14} + 68306 x^{12} + 4338264 x^{10} + 133226688 x^{8} + 1942889200 x^{6} + 13328219912 x^{4} + 38080628320 x^{2} + 26656439824 \)

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:  \(17076113336960240702301670118118129664=2^{44}\cdot 7^{8}\cdot 17^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $212.34$
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:  $2, 7, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1904=2^{4}\cdot 7\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{1904}(897,·)$, $\chi_{1904}(1413,·)$, $\chi_{1904}(1,·)$, $\chi_{1904}(909,·)$, $\chi_{1904}(461,·)$, $\chi_{1904}(1749,·)$, $\chi_{1904}(953,·)$, $\chi_{1904}(1177,·)$, $\chi_{1904}(349,·)$, $\chi_{1904}(1861,·)$, $\chi_{1904}(225,·)$, $\chi_{1904}(1121,·)$, $\chi_{1904}(169,·)$, $\chi_{1904}(797,·)$, $\chi_{1904}(1849,·)$, $\chi_{1904}(1301,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{7} a^{2}$, $\frac{1}{7} a^{3}$, $\frac{1}{98} a^{4}$, $\frac{1}{98} a^{5}$, $\frac{1}{686} a^{6}$, $\frac{1}{686} a^{7}$, $\frac{1}{163268} a^{8}$, $\frac{1}{163268} a^{9}$, $\frac{1}{1142876} a^{10}$, $\frac{1}{1142876} a^{11}$, $\frac{1}{1664027456} a^{12} - \frac{1}{3714347} a^{10} - \frac{1}{1061242} a^{8} + \frac{5}{35672} a^{6} - \frac{3}{637} a^{4} - \frac{1}{91} a^{2} - \frac{23}{104}$, $\frac{1}{1664027456} a^{13} - \frac{1}{3714347} a^{11} - \frac{1}{1061242} a^{9} + \frac{5}{35672} a^{7} - \frac{3}{637} a^{5} - \frac{1}{91} a^{3} - \frac{23}{104} a$, $\frac{1}{1036689105088} a^{14} - \frac{29}{148098443584} a^{12} + \frac{313}{1322307532} a^{10} - \frac{757}{377802152} a^{8} + \frac{47}{3174808} a^{6} + \frac{37}{56693} a^{4} - \frac{823}{64792} a^{2} - \frac{285}{9256}$, $\frac{1}{1036689105088} a^{15} - \frac{29}{148098443584} a^{13} + \frac{313}{1322307532} a^{11} - \frac{757}{377802152} a^{9} + \frac{47}{3174808} a^{7} + \frac{37}{56693} a^{5} - \frac{823}{64792} a^{3} - \frac{285}{9256} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{6}\times C_{233382}$, which has order $11202336$ (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:  \( 103646.40189541418 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_8$ (as 16T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 16
The 16 conjugacy class representatives for $C_8\times C_2$
Character table for $C_8\times C_2$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{34}) \), \(\Q(\sqrt{2}, \sqrt{17})\), 4.4.314432.1, 4.4.4913.1, 8.8.98867482624.1, 8.0.4132325415182139392.2, 8.0.4132325415182139392.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$2.8.22.5$x^{8} + 10 x^{4} + 16 x + 36$$4$$2$$22$$C_4\times C_2$$[3, 4]^{2}$
2.8.22.5$x^{8} + 10 x^{4} + 16 x + 36$$4$$2$$22$$C_4\times C_2$$[3, 4]^{2}$
$7$7.8.4.2$x^{8} + 49 x^{4} - 1029 x^{2} + 12005$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
7.8.4.2$x^{8} + 49 x^{4} - 1029 x^{2} + 12005$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
$17$17.8.7.1$x^{8} - 1377$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.7.1$x^{8} - 1377$$8$$1$$7$$C_8$$[\ ]_{8}$