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Properties

Label 16.0.73870292887...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{58}\cdot 3^{8}\cdot 5^{8}$
Root discriminant $47.78$
Ramified primes $2, 3, 5$
Class number $100$ (GRH)
Class group $[2, 50]$ (GRH)
Galois group $(C_2\times D_4):C_4$ (as 16T120)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![625, 0, 6000, 0, 17700, 0, 22800, 0, 14390, 0, 4560, 0, 708, 0, 48, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 48*x^14 + 708*x^12 + 4560*x^10 + 14390*x^8 + 22800*x^6 + 17700*x^4 + 6000*x^2 + 625)
 
gp: K = bnfinit(x^16 + 48*x^14 + 708*x^12 + 4560*x^10 + 14390*x^8 + 22800*x^6 + 17700*x^4 + 6000*x^2 + 625, 1)
 

Normalized defining polynomial

\( x^{16} + 48 x^{14} + 708 x^{12} + 4560 x^{10} + 14390 x^{8} + 22800 x^{6} + 17700 x^{4} + 6000 x^{2} + 625 \)

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:  \(738702928879445606400000000=2^{58}\cdot 3^{8}\cdot 5^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $47.78$
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 not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{4} + \frac{1}{4} a^{2} - \frac{1}{4}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{40} a^{8} - \frac{1}{20} a^{6} + \frac{1}{5} a^{4} - \frac{1}{4} a^{2} + \frac{3}{8}$, $\frac{1}{80} a^{9} - \frac{1}{80} a^{8} + \frac{1}{10} a^{7} - \frac{1}{10} a^{6} - \frac{1}{40} a^{5} + \frac{1}{40} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} + \frac{1}{16} a - \frac{1}{16}$, $\frac{1}{80} a^{10} - \frac{1}{80} a^{8} + \frac{3}{40} a^{6} + \frac{9}{40} a^{4} - \frac{7}{16} a^{2} - \frac{1}{16}$, $\frac{1}{80} a^{11} - \frac{1}{80} a^{8} - \frac{3}{40} a^{7} - \frac{1}{10} a^{6} - \frac{1}{20} a^{5} + \frac{1}{40} a^{4} - \frac{3}{16} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a - \frac{1}{16}$, $\frac{1}{800} a^{12} - \frac{1}{400} a^{10} - \frac{7}{800} a^{8} - \frac{3}{40} a^{6} + \frac{19}{160} a^{4} + \frac{3}{16} a^{2} + \frac{15}{32}$, $\frac{1}{1600} a^{13} - \frac{1}{1600} a^{12} - \frac{1}{800} a^{11} + \frac{1}{800} a^{10} - \frac{7}{1600} a^{9} - \frac{13}{1600} a^{8} - \frac{3}{80} a^{7} - \frac{1}{16} a^{6} + \frac{19}{320} a^{5} + \frac{69}{320} a^{4} - \frac{13}{32} a^{3} + \frac{13}{32} a^{2} + \frac{15}{64} a + \frac{29}{64}$, $\frac{1}{152000} a^{14} - \frac{3}{8000} a^{12} + \frac{43}{152000} a^{10} - \frac{371}{30400} a^{8} + \frac{223}{30400} a^{6} + \frac{249}{1216} a^{4} + \frac{67}{320} a^{2} - \frac{9}{1216}$, $\frac{1}{152000} a^{15} + \frac{1}{4000} a^{13} - \frac{1}{1600} a^{12} - \frac{147}{152000} a^{11} + \frac{1}{800} a^{10} - \frac{31}{7600} a^{9} + \frac{7}{1600} a^{8} + \frac{2123}{30400} a^{7} + \frac{3}{80} a^{6} - \frac{33}{3040} a^{5} - \frac{19}{320} a^{4} - \frac{63}{320} a^{3} + \frac{13}{32} a^{2} - \frac{35}{76} a - \frac{15}{64}$

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

Class group and class number

$C_{2}\times C_{50}$, which has order $100$ (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:  \( -1 \) (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:  \( 84223.0358383 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times D_4):C_4$ (as 16T120):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 22 conjugacy class representatives for $(C_2\times D_4):C_4$
Character table for $(C_2\times D_4):C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{6}) \), 4.4.92160.1, \(\Q(\sqrt{2}, \sqrt{3})\), 4.4.92160.2, 8.8.33973862400.1

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 R R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$