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Properties

Label 16.0.58432555897...0000.4
Degree $16$
Signature $[0, 8]$
Discriminant $2^{48}\cdot 3^{12}\cdot 5^{8}$
Root discriminant $40.78$
Ramified primes $2, 3, 5$
Class number $72$ (GRH)
Class group $[2, 6, 6]$ (GRH)
Galois group $D_8$ (as 16T7)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2562890625, 0, -683437500, 0, 91125000, 0, -8100000, 0, 579375, 0, -36000, 0, 1800, 0, -60, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 60*x^14 + 1800*x^12 - 36000*x^10 + 579375*x^8 - 8100000*x^6 + 91125000*x^4 - 683437500*x^2 + 2562890625)
 
gp: K = bnfinit(x^16 - 60*x^14 + 1800*x^12 - 36000*x^10 + 579375*x^8 - 8100000*x^6 + 91125000*x^4 - 683437500*x^2 + 2562890625, 1)
 

Normalized defining polynomial

\( x^{16} - 60 x^{14} + 1800 x^{12} - 36000 x^{10} + 579375 x^{8} - 8100000 x^{6} + 91125000 x^{4} - 683437500 x^{2} + 2562890625 \)

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:  \(58432555897690521600000000=2^{48}\cdot 3^{12}\cdot 5^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $40.78$
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 Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $\frac{1}{5} a^{2}$, $\frac{1}{5} a^{3}$, $\frac{1}{75} a^{4}$, $\frac{1}{75} a^{5}$, $\frac{1}{375} a^{6}$, $\frac{1}{375} a^{7}$, $\frac{1}{5625} a^{8}$, $\frac{1}{5625} a^{9}$, $\frac{1}{84375} a^{10} + \frac{1}{15} a^{2}$, $\frac{1}{253125} a^{11} - \frac{1}{16875} a^{9} - \frac{1}{1125} a^{7} + \frac{1}{225} a^{5} + \frac{4}{45} a^{3}$, $\frac{1}{3796875} a^{12} - \frac{1}{253125} a^{10} - \frac{1}{16875} a^{8} + \frac{4}{3375} a^{6} + \frac{4}{675} a^{4} - \frac{1}{15} a^{2}$, $\frac{1}{3796875} a^{13} + \frac{1}{16875} a^{9} + \frac{1}{3375} a^{7} - \frac{2}{675} a^{5} + \frac{1}{45} a^{3}$, $\frac{1}{1309921875} a^{14} - \frac{1}{17465625} a^{12} - \frac{11}{1940625} a^{10} - \frac{2}{1164375} a^{8} - \frac{98}{232875} a^{6} + \frac{71}{15525} a^{4} + \frac{8}{345} a^{2} + \frac{3}{23}$, $\frac{1}{3929765625} a^{15} - \frac{28}{261984375} a^{13} - \frac{2}{3493125} a^{11} + \frac{274}{3493125} a^{9} - \frac{374}{698625} a^{7} - \frac{7}{15525} a^{5} + \frac{20}{207} a^{3} + \frac{1}{23} a$

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

Class group and class number

$C_{2}\times C_{6}\times C_{6}$, which has order $72$ (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:  \( -\frac{19}{436640625} a^{14} + \frac{193}{87328125} a^{12} - \frac{304}{5821875} a^{10} + \frac{337}{388125} a^{8} - \frac{22}{1725} a^{6} + \frac{2623}{15525} a^{4} - \frac{502}{345} a^{2} + \frac{128}{23} \) (order $24$)
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:  \( 313644.780941 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times Q_8$ (as 16T7):

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 $D_8$
Character table for $D_8$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), \(\Q(\zeta_{8})\), \(\Q(i, \sqrt{6})\), \(\Q(\zeta_{12})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\sqrt{-2}, \sqrt{-3})\), \(\Q(\sqrt{-2}, \sqrt{3})\), \(\Q(\zeta_{24})\), 8.8.7644119040000.1, 8.0.7644119040000.1

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

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
$3$3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
$5$5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$