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Properties

Label 16.0.12750963979...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 5^{4}\cdot 41^{6}$
Root discriminant $24.08$
Ramified primes $2, 5, 41$
Class number $2$
Class group $[2]$
Galois group $C_2\times D_4:D_4$ (as 16T265)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 44, 968, -3584, 6313, -6960, 5304, -2988, 1456, -812, 504, -240, 73, -16, 8, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 8*x^14 - 16*x^13 + 73*x^12 - 240*x^11 + 504*x^10 - 812*x^9 + 1456*x^8 - 2988*x^7 + 5304*x^6 - 6960*x^5 + 6313*x^4 - 3584*x^3 + 968*x^2 + 44*x + 1)
 
gp: K = bnfinit(x^16 - 4*x^15 + 8*x^14 - 16*x^13 + 73*x^12 - 240*x^11 + 504*x^10 - 812*x^9 + 1456*x^8 - 2988*x^7 + 5304*x^6 - 6960*x^5 + 6313*x^4 - 3584*x^3 + 968*x^2 + 44*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 8 x^{14} - 16 x^{13} + 73 x^{12} - 240 x^{11} + 504 x^{10} - 812 x^{9} + 1456 x^{8} - 2988 x^{7} + 5304 x^{6} - 6960 x^{5} + 6313 x^{4} - 3584 x^{3} + 968 x^{2} + 44 x + 1 \)

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:  \(12750963979803688960000=2^{32}\cdot 5^{4}\cdot 41^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.08$
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 not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{8} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a + \frac{1}{8}$, $\frac{1}{56} a^{13} + \frac{3}{56} a^{12} + \frac{3}{14} a^{11} + \frac{2}{7} a^{10} - \frac{1}{7} a^{9} - \frac{5}{14} a^{8} - \frac{5}{14} a^{7} + \frac{3}{14} a^{6} - \frac{1}{14} a^{5} - \frac{1}{7} a^{4} - \frac{2}{7} a^{3} + \frac{3}{14} a^{2} + \frac{13}{56} a + \frac{15}{56}$, $\frac{1}{8176} a^{14} - \frac{27}{4088} a^{13} - \frac{75}{8176} a^{12} - \frac{405}{2044} a^{11} + \frac{473}{1022} a^{10} + \frac{669}{2044} a^{9} + \frac{15}{146} a^{8} + \frac{179}{1022} a^{7} + \frac{215}{1022} a^{6} - \frac{211}{2044} a^{5} - \frac{85}{1022} a^{4} + \frac{69}{292} a^{3} - \frac{2407}{8176} a^{2} + \frac{925}{4088} a - \frac{2563}{8176}$, $\frac{1}{4537978661104} a^{15} + \frac{89237717}{2268989330552} a^{14} + \frac{14226303747}{4537978661104} a^{13} + \frac{10869065787}{283623666319} a^{12} - \frac{6967767317}{81035333234} a^{11} + \frac{30153099685}{1134494665276} a^{10} + \frac{83918099557}{567247332638} a^{9} - \frac{138662270322}{283623666319} a^{8} - \frac{2887029281}{7770511406} a^{7} + \frac{45000792587}{1134494665276} a^{6} + \frac{181522058685}{567247332638} a^{5} + \frac{333157987055}{1134494665276} a^{4} + \frac{1108765999001}{4537978661104} a^{3} + \frac{1004907115873}{2268989330552} a^{2} + \frac{1108076732459}{4537978661104} a + \frac{490087692599}{1134494665276}$

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:  \( \frac{1659869609}{81035333234} a^{15} - \frac{205229715603}{2268989330552} a^{14} + \frac{108300599423}{567247332638} a^{13} - \frac{843781176391}{2268989330552} a^{12} + \frac{902886095031}{567247332638} a^{11} - \frac{1545799694682}{283623666319} a^{10} + \frac{6763467768563}{567247332638} a^{9} - \frac{790678904575}{40517666617} a^{8} + \frac{9679095349387}{283623666319} a^{7} - \frac{19765303355540}{283623666319} a^{6} + \frac{71946058257405}{567247332638} a^{5} - \frac{48710173084192}{283623666319} a^{4} + \frac{6570804514301}{40517666617} a^{3} - \frac{220550413452155}{2268989330552} a^{2} + \frac{16617056896575}{567247332638} a - \frac{2879364543}{31082045624} \) (order $8$)
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:  \( 77896.9161128 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times D_4:D_4$ (as 16T265):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 32 conjugacy class representatives for $C_2\times D_4:D_4$
Character table for $C_2\times D_4:D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-2}) \), 4.4.2624.1, 4.0.10496.2, \(\Q(\zeta_{8})\), 8.4.112920166400.3, 8.4.7057510400.1, 8.0.110166016.2

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{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}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$41$41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.4.3.3$x^{4} + 246$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.3.3$x^{4} + 246$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$