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Properties

Label 16.0.37276936195...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 5^{12}\cdot 13^{12}$
Root discriminant $45.78$
Ramified primes $2, 5, 13$
Class number $32$ (GRH)
Class group $[4, 8]$ (GRH)
Galois group $C_4^2$ (as 16T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![26861, -57018, 52255, 3244, -12245, 1288, -3846, 1998, 2251, 62, 132, -198, -12, 58, -6, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 6*x^14 + 58*x^13 - 12*x^12 - 198*x^11 + 132*x^10 + 62*x^9 + 2251*x^8 + 1998*x^7 - 3846*x^6 + 1288*x^5 - 12245*x^4 + 3244*x^3 + 52255*x^2 - 57018*x + 26861)
 
gp: K = bnfinit(x^16 - 4*x^15 - 6*x^14 + 58*x^13 - 12*x^12 - 198*x^11 + 132*x^10 + 62*x^9 + 2251*x^8 + 1998*x^7 - 3846*x^6 + 1288*x^5 - 12245*x^4 + 3244*x^3 + 52255*x^2 - 57018*x + 26861, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 6 x^{14} + 58 x^{13} - 12 x^{12} - 198 x^{11} + 132 x^{10} + 62 x^{9} + 2251 x^{8} + 1998 x^{7} - 3846 x^{6} + 1288 x^{5} - 12245 x^{4} + 3244 x^{3} + 52255 x^{2} - 57018 x + 26861 \)

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:  \(372769361959696000000000000=2^{16}\cdot 5^{12}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $45.78$
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:  $2, 5, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(260=2^{2}\cdot 5\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{260}(1,·)$, $\chi_{260}(129,·)$, $\chi_{260}(203,·)$, $\chi_{260}(207,·)$, $\chi_{260}(209,·)$, $\chi_{260}(83,·)$, $\chi_{260}(21,·)$, $\chi_{260}(27,·)$, $\chi_{260}(161,·)$, $\chi_{260}(229,·)$, $\chi_{260}(103,·)$, $\chi_{260}(109,·)$, $\chi_{260}(47,·)$, $\chi_{260}(181,·)$, $\chi_{260}(183,·)$, $\chi_{260}(187,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{10} + \frac{1}{3} a^{6} - \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{11} + \frac{1}{3} a^{7} - \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{296932804347} a^{14} + \frac{5545469502}{98977601449} a^{13} + \frac{6712698989}{296932804347} a^{12} + \frac{21286620501}{98977601449} a^{11} - \frac{18609957099}{98977601449} a^{10} + \frac{1310348334}{98977601449} a^{9} - \frac{140088565865}{296932804347} a^{8} - \frac{454891779}{1252881031} a^{7} + \frac{2183068138}{98977601449} a^{6} - \frac{5862304061}{98977601449} a^{5} - \frac{98515437958}{296932804347} a^{4} - \frac{10194823519}{98977601449} a^{3} - \frac{41998987519}{296932804347} a^{2} - \frac{14778032202}{98977601449} a + \frac{30409850521}{98977601449}$, $\frac{1}{3256414880610712963031763} a^{15} - \frac{2226210547205}{3256414880610712963031763} a^{14} + \frac{254861051596106922635261}{3256414880610712963031763} a^{13} + \frac{509429877087627067833215}{3256414880610712963031763} a^{12} + \frac{47557247130948160494667}{1085471626870237654343921} a^{11} - \frac{236324098220516516905731}{1085471626870237654343921} a^{10} - \frac{240174292163490054696098}{3256414880610712963031763} a^{9} - \frac{621952214379757974819596}{3256414880610712963031763} a^{8} + \frac{201533268055382713778337}{1085471626870237654343921} a^{7} - \frac{223360049674129525427273}{1085471626870237654343921} a^{6} - \frac{3432198857229170499473}{7771873223414589410577} a^{5} - \frac{861242292546885121133230}{3256414880610712963031763} a^{4} - \frac{1329178424818165270825771}{3256414880610712963031763} a^{3} - \frac{119593447233955067682436}{3256414880610712963031763} a^{2} + \frac{353140827145951817129235}{1085471626870237654343921} a - \frac{52151128711024331083608}{1085471626870237654343921}$

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

Class group and class number

$C_{4}\times C_{8}$, which has order $32$ (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:  \( 103848.79652840877 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4^2$ (as 16T4):

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 $C_4^2$
Character table for $C_4^2$

Intermediate fields

\(\Q(\sqrt{65}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{5}) \), 4.0.4394000.2, \(\Q(\sqrt{5}, \sqrt{13})\), 4.0.4394000.1, 4.0.54925.1, 4.0.2197.1, 4.4.338000.1, \(\Q(\zeta_{20})^+\), 8.0.19307236000000.5, 8.0.3016755625.1, 8.8.114244000000.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
5Data not computed
13Data not computed