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Properties

Label 16.4.15712755557...5489.6
Degree $16$
Signature $[4, 6]$
Discriminant $23^{10}\cdot 41^{14}$
Root discriminant $182.92$
Ramified primes $23, 41$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 16T1194

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4559237, -5065413, -1756820, 4589903, -1784877, -299541, 374301, -137237, 90222, -61437, 24089, -6300, 1614, -408, 63, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 63*x^14 - 408*x^13 + 1614*x^12 - 6300*x^11 + 24089*x^10 - 61437*x^9 + 90222*x^8 - 137237*x^7 + 374301*x^6 - 299541*x^5 - 1784877*x^4 + 4589903*x^3 - 1756820*x^2 - 5065413*x + 4559237)
 
gp: K = bnfinit(x^16 - 4*x^15 + 63*x^14 - 408*x^13 + 1614*x^12 - 6300*x^11 + 24089*x^10 - 61437*x^9 + 90222*x^8 - 137237*x^7 + 374301*x^6 - 299541*x^5 - 1784877*x^4 + 4589903*x^3 - 1756820*x^2 - 5065413*x + 4559237, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 63 x^{14} - 408 x^{13} + 1614 x^{12} - 6300 x^{11} + 24089 x^{10} - 61437 x^{9} + 90222 x^{8} - 137237 x^{7} + 374301 x^{6} - 299541 x^{5} - 1784877 x^{4} + 4589903 x^{3} - 1756820 x^{2} - 5065413 x + 4559237 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1571275555715210001755383712793895489=23^{10}\cdot 41^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $182.92$
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:  $23, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{25252} a^{13} - \frac{29}{236} a^{12} + \frac{5075}{25252} a^{11} + \frac{855}{25252} a^{10} - \frac{977}{12626} a^{9} - \frac{5981}{25252} a^{8} - \frac{698}{6313} a^{7} - \frac{6383}{25252} a^{6} - \frac{851}{12626} a^{5} - \frac{8767}{25252} a^{4} - \frac{1185}{25252} a^{3} - \frac{2783}{12626} a^{2} + \frac{7243}{25252} a + \frac{519}{12626}$, $\frac{1}{25252} a^{14} - \frac{1261}{12626} a^{12} + \frac{1979}{12626} a^{11} - \frac{349}{25252} a^{10} + \frac{3863}{25252} a^{9} - \frac{1615}{25252} a^{8} + \frac{4103}{25252} a^{7} - \frac{10583}{25252} a^{6} - \frac{12405}{25252} a^{5} + \frac{961}{6313} a^{4} + \frac{4171}{25252} a^{3} - \frac{4313}{25252} a^{2} - \frac{10839}{25252} a - \frac{53}{118}$, $\frac{1}{110496337287428315738103601952466764} a^{15} + \frac{126492771351298234361295601435}{110496337287428315738103601952466764} a^{14} - \frac{77320330333409913138333436148}{27624084321857078934525900488116691} a^{13} - \frac{2462448977571929346687276893438947}{27624084321857078934525900488116691} a^{12} - \frac{57599058547984107947853555695349}{1872819276058107046408535626312996} a^{11} + \frac{13423484038486740261602390253953275}{55248168643714157869051800976233382} a^{10} + \frac{6787817308658743795434534780988008}{27624084321857078934525900488116691} a^{9} - \frac{486236889463846416476125872113690}{27624084321857078934525900488116691} a^{8} - \frac{2969466859027704533007915511278592}{27624084321857078934525900488116691} a^{7} + \frac{15737351292936893257555035978658957}{55248168643714157869051800976233382} a^{6} + \frac{38170222990662538694864103046747843}{110496337287428315738103601952466764} a^{5} + \frac{28652846757474689716379179961153135}{110496337287428315738103601952466764} a^{4} - \frac{18341131763267334107372731272911917}{55248168643714157869051800976233382} a^{3} - \frac{6062212040937860001737148335497123}{55248168643714157869051800976233382} a^{2} - \frac{44570454532419131738952180061400997}{110496337287428315738103601952466764} a - \frac{6581607316042485910647804718902961}{27624084321857078934525900488116691}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  $9$
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:  \( 151455297452 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1194:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 40 conjugacy class representatives for t16n1194
Character table for t16n1194 is not computed

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.68921.1, 8.8.54500230757132921.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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
$23$23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.4.3.2$x^{4} - 23$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
23.4.3.2$x^{4} - 23$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41Data not computed