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Properties

Label 16.0.19456426971...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 5^{12}\cdot 41^{6}$
Root discriminant $38.07$
Ramified primes $2, 5, 41$
Class number $16$ (GRH)
Class group $[4, 4]$ (GRH)
Galois group $C_2^2:D_4$ (as 16T34)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5751, -10314, 13617, 2490, -3460, -7456, 2021, 7280, -4251, -1080, 1529, -568, 230, -90, 13, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 13*x^14 - 90*x^13 + 230*x^12 - 568*x^11 + 1529*x^10 - 1080*x^9 - 4251*x^8 + 7280*x^7 + 2021*x^6 - 7456*x^5 - 3460*x^4 + 2490*x^3 + 13617*x^2 - 10314*x + 5751)
 
gp: K = bnfinit(x^16 - 2*x^15 + 13*x^14 - 90*x^13 + 230*x^12 - 568*x^11 + 1529*x^10 - 1080*x^9 - 4251*x^8 + 7280*x^7 + 2021*x^6 - 7456*x^5 - 3460*x^4 + 2490*x^3 + 13617*x^2 - 10314*x + 5751, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 13 x^{14} - 90 x^{13} + 230 x^{12} - 568 x^{11} + 1529 x^{10} - 1080 x^{9} - 4251 x^{8} + 7280 x^{7} + 2021 x^{6} - 7456 x^{5} - 3460 x^{4} + 2490 x^{3} + 13617 x^{2} - 10314 x + 5751 \)

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:  \(19456426971136000000000000=2^{24}\cdot 5^{12}\cdot 41^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.07$
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:  $2, 5, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{9} a^{10} + \frac{1}{9} a^{9} - \frac{1}{9} a^{8} - \frac{1}{9} a^{7} - \frac{2}{9} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{2}{9} a^{2}$, $\frac{1}{9} a^{11} + \frac{1}{9} a^{9} + \frac{1}{9} a^{7} + \frac{1}{9} a^{6} - \frac{4}{9} a^{5} + \frac{1}{3} a^{4} + \frac{4}{9} a^{3} + \frac{2}{9} a^{2} + \frac{1}{3} a$, $\frac{1}{18} a^{12} - \frac{1}{18} a^{10} + \frac{1}{18} a^{9} - \frac{1}{6} a^{8} - \frac{1}{6} a^{7} - \frac{1}{18} a^{6} - \frac{4}{9} a^{5} - \frac{5}{18} a^{4} + \frac{1}{9} a^{3} - \frac{4}{9} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{234} a^{13} - \frac{1}{117} a^{12} - \frac{11}{234} a^{11} - \frac{11}{234} a^{10} - \frac{23}{234} a^{9} + \frac{5}{234} a^{8} - \frac{5}{78} a^{7} - \frac{14}{117} a^{6} - \frac{5}{234} a^{5} - \frac{11}{39} a^{4} + \frac{19}{117} a^{3} - \frac{7}{78} a^{2} - \frac{1}{26} a + \frac{5}{13}$, $\frac{1}{702} a^{14} - \frac{1}{351} a^{12} - \frac{7}{702} a^{11} - \frac{1}{117} a^{10} + \frac{25}{351} a^{9} + \frac{10}{117} a^{8} - \frac{5}{78} a^{7} - \frac{8}{117} a^{6} - \frac{155}{351} a^{5} + \frac{25}{234} a^{4} - \frac{283}{702} a^{3} - \frac{23}{78} a^{2} - \frac{5}{78} a + \frac{11}{26}$, $\frac{1}{5486983160060120973396246} a^{15} + \frac{1670627999844904184699}{2743491580030060486698123} a^{14} - \frac{487305844461224641559}{5486983160060120973396246} a^{13} - \frac{11477427181665420112753}{914497193343353495566041} a^{12} + \frac{243851803441490865441623}{5486983160060120973396246} a^{11} + \frac{98607769701499776706909}{2743491580030060486698123} a^{10} - \frac{21222615741778617535865}{2743491580030060486698123} a^{9} - \frac{4890693558783279130315}{1828994386686706991132082} a^{8} + \frac{133757108823767838450481}{914497193343353495566041} a^{7} - \frac{450888727127656484348317}{5486983160060120973396246} a^{6} - \frac{11554630529387784419908}{38640726479296626573213} a^{5} + \frac{731298403860002545398472}{2743491580030060486698123} a^{4} - \frac{1489408749972705872967811}{5486983160060120973396246} a^{3} - \frac{17111482477598312910100}{70345937949488730428157} a^{2} + \frac{247495018985363692424267}{609664795562235663710694} a - \frac{40151624588132042023}{220175079654111832326}$

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

Class group and class number

$C_{4}\times C_{4}$, which has order $16$ (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:  \( 91329.0333046 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:D_4$ (as 16T34):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_2^2:D_4$
Character table for $C_2^2:D_4$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{2}) \), 4.4.328000.1, 4.4.5125.1, 4.0.65600.2, \(\Q(\sqrt{2}, \sqrt{5})\), 4.0.65600.5, 8.0.4303360000.3, 8.8.107584000000.4, 8.0.107584000000.3

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 ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$41$41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$