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Properties

Label 16.0.11007531417...000.13
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 3^{8}\cdot 5^{8}$
Root discriminant $15.49$
Ramified primes $2, 3, 5$
Class number $2$
Class group $[2]$
Galois group $C_2 \times (C_4\times C_2):C_2$ (as 16T18)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![49, -392, 1312, -2324, 2144, -488, -1114, 1368, -704, 108, 128, -168, 137, -80, 32, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 32*x^14 - 80*x^13 + 137*x^12 - 168*x^11 + 128*x^10 + 108*x^9 - 704*x^8 + 1368*x^7 - 1114*x^6 - 488*x^5 + 2144*x^4 - 2324*x^3 + 1312*x^2 - 392*x + 49)
 
gp: K = bnfinit(x^16 - 8*x^15 + 32*x^14 - 80*x^13 + 137*x^12 - 168*x^11 + 128*x^10 + 108*x^9 - 704*x^8 + 1368*x^7 - 1114*x^6 - 488*x^5 + 2144*x^4 - 2324*x^3 + 1312*x^2 - 392*x + 49, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 32 x^{14} - 80 x^{13} + 137 x^{12} - 168 x^{11} + 128 x^{10} + 108 x^{9} - 704 x^{8} + 1368 x^{7} - 1114 x^{6} - 488 x^{5} + 2144 x^{4} - 2324 x^{3} + 1312 x^{2} - 392 x + 49 \)

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:  \(11007531417600000000=2^{32}\cdot 3^{8}\cdot 5^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.49$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{21} a^{10} + \frac{10}{21} a^{9} - \frac{1}{21} a^{8} - \frac{10}{21} a^{7} - \frac{1}{3} a^{5} - \frac{1}{21} a^{3} + \frac{10}{21} a^{2} + \frac{4}{21} a + \frac{1}{3}$, $\frac{1}{21} a^{11} + \frac{4}{21} a^{9} - \frac{5}{21} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{21} a^{4} - \frac{1}{21} a^{3} + \frac{3}{7} a^{2} + \frac{3}{7} a - \frac{1}{3}$, $\frac{1}{63} a^{12} - \frac{1}{63} a^{10} - \frac{29}{63} a^{9} + \frac{1}{3} a^{8} - \frac{20}{63} a^{7} + \frac{4}{9} a^{6} + \frac{13}{63} a^{5} + \frac{20}{63} a^{4} - \frac{1}{9} a^{3} + \frac{1}{63} a^{2} - \frac{2}{21} a + \frac{1}{9}$, $\frac{1}{189} a^{13} + \frac{1}{189} a^{12} + \frac{2}{189} a^{11} + \frac{1}{63} a^{10} - \frac{44}{189} a^{9} + \frac{94}{189} a^{8} - \frac{85}{189} a^{7} + \frac{20}{189} a^{6} + \frac{4}{63} a^{5} + \frac{73}{189} a^{4} + \frac{4}{9} a^{3} - \frac{26}{189} a^{2} + \frac{34}{189} a + \frac{4}{27}$, $\frac{1}{189} a^{14} + \frac{1}{189} a^{12} + \frac{1}{189} a^{11} - \frac{2}{189} a^{10} + \frac{1}{9} a^{9} - \frac{5}{27} a^{8} + \frac{11}{63} a^{7} - \frac{8}{189} a^{6} - \frac{65}{189} a^{5} + \frac{11}{189} a^{4} + \frac{34}{189} a^{3} - \frac{19}{63} a^{2} - \frac{5}{63} a - \frac{13}{27}$, $\frac{1}{567} a^{15} - \frac{1}{567} a^{13} - \frac{1}{567} a^{12} + \frac{1}{189} a^{11} - \frac{1}{189} a^{10} + \frac{14}{81} a^{9} - \frac{137}{567} a^{8} - \frac{10}{21} a^{7} + \frac{1}{27} a^{6} - \frac{13}{567} a^{5} - \frac{121}{567} a^{4} + \frac{2}{7} a^{3} + \frac{127}{567} a^{2} + \frac{13}{189} a + \frac{19}{81}$

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

Class group and class number

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

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:  \( -\frac{208}{63} a^{15} + \frac{170}{7} a^{14} - \frac{5672}{63} a^{13} + \frac{619}{3} a^{12} - \frac{6724}{21} a^{11} + \frac{22066}{63} a^{10} - \frac{4196}{21} a^{9} - \frac{30388}{63} a^{8} + \frac{126764}{63} a^{7} - \frac{203158}{63} a^{6} + \frac{34000}{21} a^{5} + \frac{165299}{63} a^{4} - \frac{338612}{63} a^{3} + \frac{267184}{63} a^{2} - 1652 a + \frac{2362}{9} \) (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:  \( 2006.23085811 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times D_4:C_2$ (as 16T18):

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 $C_2 \times (C_4\times C_2):C_2$
Character table for $C_2 \times (C_4\times C_2):C_2$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{30}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{-3}, \sqrt{-5})\), \(\Q(\sqrt{2}, \sqrt{-5})\), \(\Q(\sqrt{-5}, \sqrt{-6})\), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\sqrt{-3}, \sqrt{-10})\), \(\Q(\sqrt{-6}, \sqrt{-10})\), \(\Q(\sqrt{2}, \sqrt{15})\), 8.0.3317760000.3, 8.0.8294400.1, 8.0.132710400.3

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 ${\href{/LocalNumberField/7.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$