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Properties

Label 16.0.71776839744...0000.3
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 3^{4}\cdot 5^{10}\cdot 61^{4}$
Root discriminant $20.11$
Ramified primes $2, 3, 5, 61$
Class number $2$
Class group $[2]$
Galois group $C_2^4.C_2^4$ (as 16T459)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![45, -360, 1200, -2100, 1876, -336, -1042, 1140, -497, 32, 100, -118, 109, -72, 31, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 31*x^14 - 72*x^13 + 109*x^12 - 118*x^11 + 100*x^10 + 32*x^9 - 497*x^8 + 1140*x^7 - 1042*x^6 - 336*x^5 + 1876*x^4 - 2100*x^3 + 1200*x^2 - 360*x + 45)
 
gp: K = bnfinit(x^16 - 8*x^15 + 31*x^14 - 72*x^13 + 109*x^12 - 118*x^11 + 100*x^10 + 32*x^9 - 497*x^8 + 1140*x^7 - 1042*x^6 - 336*x^5 + 1876*x^4 - 2100*x^3 + 1200*x^2 - 360*x + 45, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 31 x^{14} - 72 x^{13} + 109 x^{12} - 118 x^{11} + 100 x^{10} + 32 x^{9} - 497 x^{8} + 1140 x^{7} - 1042 x^{6} - 336 x^{5} + 1876 x^{4} - 2100 x^{3} + 1200 x^{2} - 360 x + 45 \)

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:  \(717768397440000000000=2^{16}\cdot 3^{4}\cdot 5^{10}\cdot 61^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.11$
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:  $2, 3, 5, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{9} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{9} a^{11} - \frac{1}{9} a^{10} - \frac{1}{9} a^{9} - \frac{1}{9} a^{8} - \frac{1}{9} a^{6} + \frac{4}{9} a^{5} + \frac{1}{9} a^{4} + \frac{1}{3} a^{3}$, $\frac{1}{27} a^{12} + \frac{1}{27} a^{10} + \frac{1}{27} a^{9} + \frac{2}{27} a^{8} - \frac{4}{27} a^{7} - \frac{1}{9} a^{6} + \frac{2}{27} a^{5} + \frac{7}{27} a^{4} + \frac{1}{9} a^{3} + \frac{1}{9} a^{2} - \frac{1}{3}$, $\frac{1}{27} a^{13} + \frac{1}{27} a^{11} + \frac{1}{27} a^{10} + \frac{2}{27} a^{9} - \frac{4}{27} a^{8} - \frac{1}{9} a^{7} + \frac{2}{27} a^{6} + \frac{7}{27} a^{5} + \frac{1}{9} a^{4} + \frac{1}{9} a^{3} - \frac{1}{3} a$, $\frac{1}{81} a^{14} - \frac{1}{81} a^{13} - \frac{1}{81} a^{12} - \frac{1}{81} a^{10} + \frac{10}{81} a^{9} + \frac{2}{27} a^{8} + \frac{13}{81} a^{7} + \frac{11}{81} a^{6} + \frac{28}{81} a^{5} - \frac{14}{81} a^{4} - \frac{5}{27} a^{2} + \frac{4}{9} a + \frac{2}{9}$, $\frac{1}{243} a^{15} + \frac{4}{243} a^{13} - \frac{4}{243} a^{12} + \frac{5}{243} a^{11} + \frac{4}{81} a^{10} - \frac{29}{243} a^{9} + \frac{16}{243} a^{8} - \frac{4}{27} a^{7} - \frac{7}{81} a^{6} + \frac{23}{243} a^{5} - \frac{71}{243} a^{4} + \frac{16}{81} a^{3} + \frac{40}{81} a^{2} - \frac{1}{3} a - \frac{13}{27}$

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:  \( \frac{301}{81} a^{15} - \frac{2224}{81} a^{14} + \frac{7976}{81} a^{13} - \frac{5609}{27} a^{12} + \frac{22622}{81} a^{11} - \frac{21872}{81} a^{10} + \frac{5645}{27} a^{9} + \frac{19810}{81} a^{8} - \frac{137362}{81} a^{7} + \frac{259579}{81} a^{6} - \frac{156794}{81} a^{5} - \frac{21598}{9} a^{4} + \frac{148495}{27} a^{3} - \frac{40340}{9} a^{2} + \frac{15950}{9} a - 283 \) (order $4$)
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:  \( 15225.1298419 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^4$ (as 16T459):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 46 conjugacy class representatives for $C_2^4.C_2^4$
Character table for $C_2^4.C_2^4$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5}) \), 4.0.1525.1, 4.4.24400.1, \(\Q(i, \sqrt{5})\), 8.0.595360000.6

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
$3$3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$61$61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$