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Properties

Label 16.0.25786656580...3321.1
Degree $16$
Signature $[0, 8]$
Discriminant $31^{12}\cdot 41^{9}$
Root discriminant $106.10$
Ramified primes $31, 41$
Class number $6$ (GRH)
Class group $[6]$ (GRH)
Galois group $C_4.D_4:C_4$ (as 16T260)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![282885323, -151471363, 45856595, -105270823, 19279393, 10670397, 351327, 153755, 440098, 65716, 7309, 5462, 1061, 41, 56, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 56*x^14 + 41*x^13 + 1061*x^12 + 5462*x^11 + 7309*x^10 + 65716*x^9 + 440098*x^8 + 153755*x^7 + 351327*x^6 + 10670397*x^5 + 19279393*x^4 - 105270823*x^3 + 45856595*x^2 - 151471363*x + 282885323)
 
gp: K = bnfinit(x^16 - x^15 + 56*x^14 + 41*x^13 + 1061*x^12 + 5462*x^11 + 7309*x^10 + 65716*x^9 + 440098*x^8 + 153755*x^7 + 351327*x^6 + 10670397*x^5 + 19279393*x^4 - 105270823*x^3 + 45856595*x^2 - 151471363*x + 282885323, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 56 x^{14} + 41 x^{13} + 1061 x^{12} + 5462 x^{11} + 7309 x^{10} + 65716 x^{9} + 440098 x^{8} + 153755 x^{7} + 351327 x^{6} + 10670397 x^{5} + 19279393 x^{4} - 105270823 x^{3} + 45856595 x^{2} - 151471363 x + 282885323 \)

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:  \(257866565806827685775538626393321=31^{12}\cdot 41^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $106.10$
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:  $31, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{5} a^{13} + \frac{1}{5} a^{11} + \frac{1}{5} a^{10} + \frac{2}{5} a^{9} - \frac{2}{5} a^{8} + \frac{1}{5} a^{7} - \frac{1}{5} a^{4} + \frac{1}{5} a^{3} - \frac{2}{5} a^{2} + \frac{2}{5} a - \frac{2}{5}$, $\frac{1}{5} a^{14} + \frac{1}{5} a^{12} + \frac{1}{5} a^{11} + \frac{2}{5} a^{10} - \frac{2}{5} a^{9} + \frac{1}{5} a^{8} - \frac{1}{5} a^{5} + \frac{1}{5} a^{4} - \frac{2}{5} a^{3} + \frac{2}{5} a^{2} - \frac{2}{5} a$, $\frac{1}{217551798584467787892555532779990214137279277053082253259096905} a^{15} - \frac{1958757593094851223933587463034007359584133705014454124781576}{217551798584467787892555532779990214137279277053082253259096905} a^{14} - \frac{2200302474595067192784590485127443715521177839254133265635019}{43510359716893557578511106555998042827455855410616450651819381} a^{13} - \frac{4884670343134591763638063774521141650548599687978199275333857}{43510359716893557578511106555998042827455855410616450651819381} a^{12} - \frac{7027669466794620148285200536963878333349326154978422250122361}{43510359716893557578511106555998042827455855410616450651819381} a^{11} - \frac{10141220946518668875784465005249019877020567528840846083240219}{43510359716893557578511106555998042827455855410616450651819381} a^{10} - \frac{88849044515007763951481072470357858559141262277703464481930314}{217551798584467787892555532779990214137279277053082253259096905} a^{9} - \frac{82053990201774083526705205760450841440228782626577918447854779}{217551798584467787892555532779990214137279277053082253259096905} a^{8} - \frac{107409547037928116436996756550078696457522884218273954271231681}{217551798584467787892555532779990214137279277053082253259096905} a^{7} - \frac{95907470252488838629083914940749494417644351437540581760954576}{217551798584467787892555532779990214137279277053082253259096905} a^{6} - \frac{12388796483152130274828963265678662669381051486087948207308043}{217551798584467787892555532779990214137279277053082253259096905} a^{5} - \frac{95102602884418953563374762395926352673456855677689289886265532}{217551798584467787892555532779990214137279277053082253259096905} a^{4} + \frac{56335534707574285612494584976318656070276977626590856832276773}{217551798584467787892555532779990214137279277053082253259096905} a^{3} - \frac{41701481279345835717235452663850952837816483587880505508869027}{217551798584467787892555532779990214137279277053082253259096905} a^{2} - \frac{3412903319834542835013097909562003412948750953748509284455317}{43510359716893557578511106555998042827455855410616450651819381} a + \frac{93934546040101718463873109748189466760443764122174418260628402}{217551798584467787892555532779990214137279277053082253259096905}$

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

Class group and class number

$C_{6}$, which has order $6$ (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:  \( 847898663.713 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4.D_4:C_4$ (as 16T260):

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 $C_4.D_4:C_4$
Character table for $C_4.D_4:C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-31}) \), 4.0.39401.1, 8.0.63649990841.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 16 sibling: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{4}$ $16$ $16$ $16$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ $16$ R ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ $16$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$31$31.4.3.1$x^{4} + 217$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
31.4.3.1$x^{4} + 217$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
31.4.3.1$x^{4} + 217$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
31.4.3.1$x^{4} + 217$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
$41$41.4.2.2$x^{4} - 41 x^{2} + 20172$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.8.7.4$x^{8} - 1912896$$8$$1$$7$$C_8$$[\ ]_{8}$