Properties

Label 16.0.268...000.10
Degree $16$
Signature $[0, 8]$
Discriminant $2.687\times 10^{19}$
Root discriminant $16.38$
Ramified primes $2, 3, 5$
Class number $2$
Class group $[2]$
Galois group $C_2^2 : C_4$ (as 16T10)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 18*x^14 - 30*x^13 + 41*x^12 - 90*x^11 + 252*x^10 - 432*x^9 + 426*x^8 - 180*x^7 + 90*x^6 - 360*x^5 + 740*x^4 - 750*x^3 + 450*x^2 - 150*x + 25)
 
gp: K = bnfinit(x^16 - 6*x^15 + 18*x^14 - 30*x^13 + 41*x^12 - 90*x^11 + 252*x^10 - 432*x^9 + 426*x^8 - 180*x^7 + 90*x^6 - 360*x^5 + 740*x^4 - 750*x^3 + 450*x^2 - 150*x + 25, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![25, -150, 450, -750, 740, -360, 90, -180, 426, -432, 252, -90, 41, -30, 18, -6, 1]);
 

\( x^{16} - 6 x^{15} + 18 x^{14} - 30 x^{13} + 41 x^{12} - 90 x^{11} + 252 x^{10} - 432 x^{9} + 426 x^{8} - 180 x^{7} + 90 x^{6} - 360 x^{5} + 740 x^{4} - 750 x^{3} + 450 x^{2} - 150 x + 25 \)

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

Invariants

Degree:  $16$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 8]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(26873856000000000000\)\(\medspace = 2^{24}\cdot 3^{8}\cdot 5^{12}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $16.38$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $2, 3, 5$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $16$
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{4} + \frac{1}{3}$, $\frac{1}{3} a^{9} + \frac{1}{3} a^{5} + \frac{1}{3} a$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{6} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{7} + \frac{1}{3} a^{3}$, $\frac{1}{15} a^{12} - \frac{1}{15} a^{11} - \frac{2}{15} a^{10} + \frac{1}{15} a^{8} + \frac{1}{3} a^{7} + \frac{7}{15} a^{6} + \frac{1}{5} a^{5} + \frac{1}{15} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{15} a^{13} + \frac{2}{15} a^{11} - \frac{2}{15} a^{10} + \frac{1}{15} a^{9} + \frac{1}{15} a^{8} + \frac{2}{15} a^{7} - \frac{1}{3} a^{6} + \frac{4}{15} a^{5} + \frac{1}{15} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{2715} a^{14} + \frac{41}{2715} a^{13} + \frac{26}{2715} a^{12} - \frac{19}{2715} a^{11} + \frac{296}{2715} a^{10} + \frac{109}{905} a^{9} - \frac{353}{2715} a^{8} - \frac{538}{2715} a^{7} + \frac{947}{2715} a^{6} - \frac{6}{905} a^{5} + \frac{164}{543} a^{4} - \frac{8}{543} a^{3} - \frac{4}{543} a^{2} - \frac{190}{543} a + \frac{103}{543}$, $\frac{1}{441852675} a^{15} + \frac{6434}{441852675} a^{14} - \frac{9829697}{441852675} a^{13} - \frac{2180897}{88370535} a^{12} - \frac{7088404}{441852675} a^{11} - \frac{340291}{8033685} a^{10} + \frac{756389}{147284225} a^{9} + \frac{19716566}{147284225} a^{8} - \frac{1936549}{441852675} a^{7} - \frac{12251153}{88370535} a^{6} - \frac{836515}{5891369} a^{5} + \frac{13942702}{29456845} a^{4} + \frac{1514864}{88370535} a^{3} - \frac{1748464}{5891369} a^{2} + \frac{6448910}{17674107} a + \frac{734824}{17674107}$

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

Class group and class number

$C_{2}$, which has order $2$

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $7$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( \frac{166561}{2441175} a^{15} - \frac{278692}{813725} a^{14} + \frac{2110513}{2441175} a^{13} - \frac{177557}{162745} a^{12} + \frac{1250562}{813725} a^{11} - \frac{40679}{8877} a^{10} + \frac{10188039}{813725} a^{9} - \frac{12919204}{813725} a^{8} + \frac{8480732}{813725} a^{7} - \frac{661736}{488235} a^{6} + \frac{1193204}{162745} a^{5} - \frac{3117899}{162745} a^{4} + \frac{13363714}{488235} a^{3} - \frac{1970674}{97647} a^{2} + \frac{1052639}{97647} a - \frac{78603}{32549} \) (order $4$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 2346.64862716 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{8}\cdot 2346.64862716 \cdot 2}{4\sqrt{26873856000000000000}}\approx 0.549783908213$

Galois group

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

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
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

\(\Q(\sqrt{-5}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-1}) \), \(\Q(i, \sqrt{5})\), 4.2.3600.1 x2, 4.0.2880.1 x2, 4.0.8000.1 x2, 4.2.2000.1 x2, \(\Q(\zeta_{15})^+\), 4.0.18000.1, 8.0.207360000.5, 8.0.64000000.3, 8.0.324000000.1, 8.4.324000000.3 x2, 8.0.5184000000.7 x2

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

Sibling fields

Degree 8 siblings: 8.0.5184000000.7, 8.4.324000000.3

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
2Data not computed
$3$3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$5$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$