Properties

Label 16.0.27487790694...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{46}\cdot 5^{8}$
Root discriminant $16.40$
Ramified primes $2, 5$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^2\wr C_2$ (as 16T39)

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Show commands for: SageMath / Pari/GP / Magma

sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 8*x^14 - 24*x^13 - 4*x^12 - 48*x^11 + 88*x^10 + 104*x^9 + 110*x^8 - 80*x^7 - 168*x^6 - 104*x^5 + 132*x^4 + 64*x^3 - 24*x^2 - 40*x + 25)
 
gp: K = bnfinit(x^16 + 8*x^14 - 24*x^13 - 4*x^12 - 48*x^11 + 88*x^10 + 104*x^9 + 110*x^8 - 80*x^7 - 168*x^6 - 104*x^5 + 132*x^4 + 64*x^3 - 24*x^2 - 40*x + 25, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![25, -40, -24, 64, 132, -104, -168, -80, 110, 104, 88, -48, -4, -24, 8, 0, 1]);
 

Normalized defining polynomial

\( x^{16} + 8 x^{14} - 24 x^{13} - 4 x^{12} - 48 x^{11} + 88 x^{10} + 104 x^{9} + 110 x^{8} - 80 x^{7} - 168 x^{6} - 104 x^{5} + 132 x^{4} + 64 x^{3} - 24 x^{2} - 40 x + 25 \)

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:  \(27487790694400000000=2^{46}\cdot 5^{8}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $16.40$
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:  $2, 5$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $8$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{8} - \frac{1}{4} a^{4} + \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{9} - \frac{1}{4} a^{5} + \frac{1}{4} a$, $\frac{1}{12} a^{14} + \frac{1}{12} a^{12} + \frac{1}{12} a^{10} - \frac{1}{4} a^{8} + \frac{1}{6} a^{7} - \frac{1}{12} a^{6} + \frac{1}{6} a^{5} - \frac{5}{12} a^{4} + \frac{1}{6} a^{3} - \frac{1}{12} a^{2} + \frac{1}{6} a - \frac{5}{12}$, $\frac{1}{6614056414740} a^{15} + \frac{231244981}{220468547158} a^{14} + \frac{243755630699}{3307028207370} a^{13} + \frac{213708359467}{2204685471580} a^{12} - \frac{19084645717}{3307028207370} a^{11} - \frac{546624089141}{2204685471580} a^{10} - \frac{261106588149}{2204685471580} a^{9} + \frac{11171929136}{53339164635} a^{8} - \frac{277736353643}{1322811282948} a^{7} - \frac{71036461466}{330702820737} a^{6} + \frac{452939450188}{1653514103685} a^{5} - \frac{1839548074399}{6614056414740} a^{4} + \frac{25090413383}{53339164635} a^{3} + \frac{2312673461849}{6614056414740} a^{2} - \frac{2344406818859}{6614056414740} a + \frac{36394799533}{220468547158}$

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

Class group and class number

$C_{2}$, which has order $2$ (assuming GRH)

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:  \( -\frac{270159}{44076079} a^{15} - \frac{429415}{44076079} a^{14} - \frac{1114447}{44076079} a^{13} + \frac{10334055}{88152158} a^{12} + \frac{17663345}{44076079} a^{11} + \frac{5969315}{44076079} a^{10} - \frac{68446287}{44076079} a^{9} - \frac{2633915}{1421809} a^{8} - \frac{20301767}{44076079} a^{7} + \frac{207652163}{44076079} a^{6} + \frac{232414643}{44076079} a^{5} + \frac{28293737}{88152158} a^{4} - \frac{5465965}{1421809} a^{3} - \frac{39499227}{44076079} a^{2} - \frac{1120645}{44076079} a + \frac{9335578}{44076079} \) (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 (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 3495.44381092 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Galois group

$C_2^2\wr C_2$ (as 16T39):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_2^2\wr C_2$
Character table for $C_2^2\wr C_2$

Intermediate fields

\(\Q(\sqrt{-10}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{10}) \), 4.0.320.1, 4.0.1280.1, 4.0.512.1, 4.0.12800.1, 4.2.25600.2 x2, \(\Q(i, \sqrt{10})\), 4.0.2560.1 x2, 8.0.2621440000.4, 8.0.163840000.2, 8.0.2621440000.8, 8.0.26214400.2, 8.0.2621440000.6, 8.0.104857600.1, 8.0.655360000.1

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

Sibling fields

Galois closure: Deg 32
Degree 8 siblings: 8.0.26214400.2, 8.0.104857600.2, 8.0.26214400.1, 8.0.104857600.1, 8.0.655360000.1, 8.0.655360000.2, 8.0.2621440000.7, 8.0.2621440000.6
Degree 16 siblings: 16.0.429496729600000000.4, 16.4.6871947673600000000.8, 16.0.43980465111040000.1, 16.0.27487790694400000000.3, 16.0.6871947673600000000.13, 16.0.27487790694400000000.1

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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

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
$5$5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{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}$