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Properties

Label 16.0.13820889968...0625.2
Degree $16$
Signature $[0, 8]$
Discriminant $5^{8}\cdot 29^{12}$
Root discriminant $27.94$
Ramified primes $5, 29$
Class number $18$ (GRH)
Class group $[3, 6]$ (GRH)
Galois group $C_2^2 : C_4$ (as 16T10)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6091, 3764, 6651, 5471, 5089, 1367, 401, -387, 1, 43, 55, -15, 23, 5, 3, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 3*x^14 + 5*x^13 + 23*x^12 - 15*x^11 + 55*x^10 + 43*x^9 + x^8 - 387*x^7 + 401*x^6 + 1367*x^5 + 5089*x^4 + 5471*x^3 + 6651*x^2 + 3764*x + 6091)
 
gp: K = bnfinit(x^16 - 2*x^15 + 3*x^14 + 5*x^13 + 23*x^12 - 15*x^11 + 55*x^10 + 43*x^9 + x^8 - 387*x^7 + 401*x^6 + 1367*x^5 + 5089*x^4 + 5471*x^3 + 6651*x^2 + 3764*x + 6091, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 3 x^{14} + 5 x^{13} + 23 x^{12} - 15 x^{11} + 55 x^{10} + 43 x^{9} + x^{8} - 387 x^{7} + 401 x^{6} + 1367 x^{5} + 5089 x^{4} + 5471 x^{3} + 6651 x^{2} + 3764 x + 6091 \)

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:  \(138208899689636344140625=5^{8}\cdot 29^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.94$
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:  $5, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{13} a^{11} - \frac{5}{13} a^{10} + \frac{5}{13} a^{9} + \frac{5}{13} a^{8} - \frac{2}{13} a^{7} + \frac{3}{13} a^{6} - \frac{6}{13} a^{5} + \frac{2}{13} a^{4} - \frac{2}{13} a^{3} + \frac{6}{13} a^{2} + \frac{3}{13} a + \frac{3}{13}$, $\frac{1}{13} a^{12} + \frac{6}{13} a^{10} + \frac{4}{13} a^{9} - \frac{3}{13} a^{8} + \frac{6}{13} a^{7} - \frac{4}{13} a^{6} - \frac{2}{13} a^{5} - \frac{5}{13} a^{4} - \frac{4}{13} a^{3} - \frac{6}{13} a^{2} + \frac{5}{13} a + \frac{2}{13}$, $\frac{1}{13} a^{13} - \frac{5}{13} a^{10} + \frac{6}{13} a^{9} + \frac{2}{13} a^{8} - \frac{5}{13} a^{7} + \frac{6}{13} a^{6} + \frac{5}{13} a^{5} - \frac{3}{13} a^{4} + \frac{6}{13} a^{3} - \frac{5}{13} a^{2} - \frac{3}{13} a - \frac{5}{13}$, $\frac{1}{1014} a^{14} - \frac{1}{78} a^{13} + \frac{5}{338} a^{12} - \frac{17}{1014} a^{11} - \frac{5}{26} a^{10} + \frac{67}{1014} a^{9} - \frac{409}{1014} a^{8} - \frac{413}{1014} a^{7} + \frac{37}{78} a^{6} + \frac{11}{78} a^{5} - \frac{379}{1014} a^{4} - \frac{135}{338} a^{3} - \frac{55}{338} a^{2} - \frac{421}{1014} a - \frac{487}{1014}$, $\frac{1}{1622014511051655440868954} a^{15} + \frac{132111500394444636938}{270335751841942573478159} a^{14} + \frac{8463515978601755732272}{811007255525827720434477} a^{13} + \frac{15950487489278020605911}{811007255525827720434477} a^{12} - \frac{7327627530064560682702}{811007255525827720434477} a^{11} - \frac{273903981089437742839093}{811007255525827720434477} a^{10} - \frac{63069615469200395877260}{270335751841942573478159} a^{9} + \frac{21135354047601750448747}{270335751841942573478159} a^{8} - \frac{16447768663696501002074}{811007255525827720434477} a^{7} - \frac{31229841813502687768}{20795057833995582575243} a^{6} - \frac{389330389260046217785972}{811007255525827720434477} a^{5} + \frac{19246540352555835352003}{62385173501986747725729} a^{4} + \frac{45669978272769082418850}{270335751841942573478159} a^{3} - \frac{203289086066541584434232}{811007255525827720434477} a^{2} - \frac{330022747447519851620149}{811007255525827720434477} a - \frac{644536253995271424219973}{1622014511051655440868954}$

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

Class group and class number

$C_{3}\times C_{6}$, which has order $18$ (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:  \( 3793.72993285 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:C_4$ (as 16T10):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $C_2^2 : C_4$
Character table for $C_2^2 : C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{145}) \), \(\Q(\sqrt{29}) \), \(\Q(\sqrt{5}, \sqrt{29})\), 4.0.609725.1 x2, 4.0.121945.1 x2, 4.4.725.1 x2, 4.4.4205.1 x2, 4.0.609725.2, 4.0.24389.1, 8.0.371764575625.5, 8.8.442050625.1, 8.0.371764575625.1, 8.0.371764575625.4 x2, 8.0.14870583025.1 x2

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

Sibling fields

Degree 8 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$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}$
5.4.2.1$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}$
$29$29.8.6.1$x^{8} - 203 x^{4} + 68121$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
29.8.6.1$x^{8} - 203 x^{4} + 68121$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$