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Properties

Label 16.0.89161004482...000.13
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 3^{12}\cdot 5^{8}$
Root discriminant $20.39$
Ramified primes $2, 3, 5$
Class number $4$
Class group $[4]$
Galois group $C_2^2 \times D_4$ (as 16T25)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![201, -204, 66, 612, -1016, 460, 602, -900, 625, -340, 46, 116, -80, 0, 20, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 20*x^14 - 80*x^12 + 116*x^11 + 46*x^10 - 340*x^9 + 625*x^8 - 900*x^7 + 602*x^6 + 460*x^5 - 1016*x^4 + 612*x^3 + 66*x^2 - 204*x + 201)
 
gp: K = bnfinit(x^16 - 8*x^15 + 20*x^14 - 80*x^12 + 116*x^11 + 46*x^10 - 340*x^9 + 625*x^8 - 900*x^7 + 602*x^6 + 460*x^5 - 1016*x^4 + 612*x^3 + 66*x^2 - 204*x + 201, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 20 x^{14} - 80 x^{12} + 116 x^{11} + 46 x^{10} - 340 x^{9} + 625 x^{8} - 900 x^{7} + 602 x^{6} + 460 x^{5} - 1016 x^{4} + 612 x^{3} + 66 x^{2} - 204 x + 201 \)

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:  \(891610044825600000000=2^{32}\cdot 3^{12}\cdot 5^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.39$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{6} a^{10} + \frac{1}{6} a^{9} - \frac{1}{6} a^{8} - \frac{1}{3} a^{7} - \frac{1}{6} a^{6} + \frac{1}{6} a^{5} + \frac{1}{6} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{18} a^{11} - \frac{1}{18} a^{10} - \frac{1}{6} a^{9} + \frac{1}{6} a^{8} - \frac{1}{6} a^{7} - \frac{1}{2} a^{6} + \frac{5}{18} a^{5} + \frac{1}{18} a^{4} - \frac{1}{6} a^{3} - \frac{1}{6} a^{2} - \frac{1}{6} a + \frac{1}{6}$, $\frac{1}{54} a^{12} - \frac{2}{27} a^{10} - \frac{2}{9} a^{7} + \frac{7}{27} a^{6} + \frac{1}{9} a^{5} - \frac{10}{27} a^{4} - \frac{4}{9} a^{3} - \frac{1}{9} a^{2} - \frac{1}{3} a + \frac{7}{18}$, $\frac{1}{3726} a^{13} + \frac{14}{1863} a^{12} + \frac{16}{1863} a^{11} + \frac{34}{1863} a^{10} + \frac{4}{23} a^{9} - \frac{229}{1242} a^{8} - \frac{620}{1863} a^{7} + \frac{11}{81} a^{6} - \frac{187}{1863} a^{5} + \frac{397}{3726} a^{4} + \frac{220}{621} a^{3} + \frac{176}{621} a^{2} + \frac{217}{1242} a + \frac{565}{1242}$, $\frac{1}{2813130} a^{14} - \frac{7}{2813130} a^{13} + \frac{761}{312570} a^{12} - \frac{41003}{2813130} a^{11} + \frac{205751}{2813130} a^{10} - \frac{217687}{937710} a^{9} + \frac{194908}{1406565} a^{8} - \frac{180955}{562626} a^{7} + \frac{416827}{937710} a^{6} + \frac{1342841}{2813130} a^{5} - \frac{125968}{1406565} a^{4} - \frac{6721}{62514} a^{3} - \frac{176447}{468855} a^{2} + \frac{539}{93771} a + \frac{193573}{937710}$, $\frac{1}{42196950} a^{15} + \frac{907}{8439390} a^{13} - \frac{7381}{843939} a^{12} + \frac{21007}{2813130} a^{11} + \frac{4018}{21098475} a^{10} - \frac{4934513}{21098475} a^{9} + \frac{94477}{7032825} a^{8} + \frac{16948711}{42196950} a^{7} - \frac{4745756}{21098475} a^{6} - \frac{131434}{7032825} a^{5} - \frac{4452016}{21098475} a^{4} - \frac{480602}{7032825} a^{3} + \frac{2432336}{7032825} a^{2} - \frac{1180729}{4688550} a + \frac{3163343}{7032825}$

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

Class group and class number

$C_{4}$, which has order $4$

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:  \( \frac{204272}{21098475} a^{15} - \frac{102136}{1406565} a^{14} + \frac{142216}{843939} a^{13} + \frac{25168}{4219695} a^{12} - \frac{10628}{17365} a^{11} + \frac{18518302}{21098475} a^{10} + \frac{5431468}{21098475} a^{9} - \frac{5745739}{2344275} a^{8} + \frac{111792092}{21098475} a^{7} - \frac{176448854}{21098475} a^{6} + \frac{15642908}{2344275} a^{5} - \frac{5839264}{21098475} a^{4} - \frac{29866988}{7032825} a^{3} + \frac{26116994}{7032825} a^{2} - \frac{2045188}{2344275} a + \frac{3131072}{7032825} \) (order $6$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 28877.951681 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times D_4$ (as 16T25):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_2^2 \times D_4$
Character table for $C_2^2 \times D_4$

Intermediate fields

\(\Q(\sqrt{6}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-30}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-5}) \), 4.2.8640.2, 4.2.34560.1, 4.2.8640.1, 4.2.34560.2, \(\Q(\sqrt{-2}, \sqrt{-5})\), \(\Q(\sqrt{6}, \sqrt{10})\), \(\Q(\sqrt{-3}, \sqrt{10})\), \(\Q(\sqrt{-2}, \sqrt{-3})\), \(\Q(\sqrt{-2}, \sqrt{15})\), \(\Q(\sqrt{-5}, \sqrt{6})\), \(\Q(\sqrt{-3}, \sqrt{-5})\), 8.0.3317760000.8, 8.4.29859840000.2, 8.4.29859840000.5, 8.0.74649600.1, 8.0.1194393600.6, 8.0.29859840000.5, 8.0.29859840000.4

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

Sibling fields

Galois closure: data not computed
Degree 16 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 R R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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

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$2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
$3$3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
$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}$