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Properties

Label 16.0.60196204493...0625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 29^{8}\cdot 149^{4}$
Root discriminant $62.91$
Ramified primes $5, 29, 149$
Class number $24$ (GRH)
Class group $[2, 2, 6]$ (GRH)
Galois group $C_4^2:C_2^2.C_2$ (as 16T382)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![15350411, -25175452, 12891428, -3179774, 1309311, -536748, 440048, -209167, 54992, -15840, 5182, -1991, 502, -58, 11, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 11*x^14 - 58*x^13 + 502*x^12 - 1991*x^11 + 5182*x^10 - 15840*x^9 + 54992*x^8 - 209167*x^7 + 440048*x^6 - 536748*x^5 + 1309311*x^4 - 3179774*x^3 + 12891428*x^2 - 25175452*x + 15350411)
 
gp: K = bnfinit(x^16 - 5*x^15 + 11*x^14 - 58*x^13 + 502*x^12 - 1991*x^11 + 5182*x^10 - 15840*x^9 + 54992*x^8 - 209167*x^7 + 440048*x^6 - 536748*x^5 + 1309311*x^4 - 3179774*x^3 + 12891428*x^2 - 25175452*x + 15350411, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 11 x^{14} - 58 x^{13} + 502 x^{12} - 1991 x^{11} + 5182 x^{10} - 15840 x^{9} + 54992 x^{8} - 209167 x^{7} + 440048 x^{6} - 536748 x^{5} + 1309311 x^{4} - 3179774 x^{3} + 12891428 x^{2} - 25175452 x + 15350411 \)

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:  \(60196204493330351894775390625=5^{12}\cdot 29^{8}\cdot 149^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $62.91$
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:  $5, 29, 149$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8} - \frac{2}{5} a^{6} + \frac{1}{5} a^{5} - \frac{1}{5} a^{4} - \frac{1}{5} a^{3} + \frac{1}{5} a^{2} - \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{9} - \frac{2}{5} a^{7} + \frac{1}{5} a^{6} - \frac{1}{5} a^{5} - \frac{1}{5} a^{4} + \frac{1}{5} a^{3} - \frac{1}{5} a^{2} + \frac{1}{5} a$, $\frac{1}{5} a^{10} + \frac{1}{5} a^{7} + \frac{1}{5} a^{5} - \frac{1}{5} a^{4} + \frac{2}{5} a^{3} - \frac{2}{5} a^{2} - \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{5} a^{11} - \frac{2}{5} a^{6} - \frac{2}{5} a^{5} - \frac{2}{5} a^{4} - \frac{1}{5} a^{3} + \frac{2}{5} a^{2} - \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{12} - \frac{2}{5} a^{7} - \frac{2}{5} a^{6} - \frac{2}{5} a^{5} - \frac{1}{5} a^{4} + \frac{2}{5} a^{3} - \frac{2}{5} a^{2} - \frac{1}{5} a$, $\frac{1}{5} a^{13} - \frac{2}{5} a^{7} - \frac{1}{5} a^{6} + \frac{1}{5} a^{5} + \frac{1}{5} a^{3} + \frac{1}{5} a^{2} - \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{5} a^{14} - \frac{1}{5} a^{7} + \frac{2}{5} a^{6} + \frac{2}{5} a^{5} - \frac{1}{5} a^{4} - \frac{1}{5} a^{3} + \frac{2}{5}$, $\frac{1}{2394394523986715720831406865660379945316476745605} a^{15} + \frac{254645763421339048280812696944049125646005221}{15447706606365907876331657197808902873009527391} a^{14} + \frac{95388470018578157007564075557176814013797131962}{2394394523986715720831406865660379945316476745605} a^{13} + \frac{22479130707801241146593889024052706697088907439}{478878904797343144166281373132075989063295349121} a^{12} + \frac{145849460590693989934608767667605876250850594174}{2394394523986715720831406865660379945316476745605} a^{11} + \frac{189017840671385483521570912913625721517575221286}{2394394523986715720831406865660379945316476745605} a^{10} + \frac{12055131029119294670973613960575508324390853932}{2394394523986715720831406865660379945316476745605} a^{9} + \frac{90691807497430403768558211728829976218214335566}{2394394523986715720831406865660379945316476745605} a^{8} + \frac{236514875027988968232435730230817509669991037096}{478878904797343144166281373132075989063295349121} a^{7} - \frac{231968014845035647844244097347815003927055351673}{478878904797343144166281373132075989063295349121} a^{6} + \frac{835541198503240274819375908304647777159059841134}{2394394523986715720831406865660379945316476745605} a^{5} + \frac{1114488835823444894518273343458072160641997942116}{2394394523986715720831406865660379945316476745605} a^{4} + \frac{132258224565614424410197038804409775215706550604}{478878904797343144166281373132075989063295349121} a^{3} - \frac{673084698961806398262821515377305836811093792722}{2394394523986715720831406865660379945316476745605} a^{2} + \frac{865637984437356334105408358728333111631966495168}{2394394523986715720831406865660379945316476745605} a + \frac{12296666106544893379635754259583191966894114576}{30308791442869819251030466653928860067297173995}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{6}$, which has order $24$ (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:  \( 1727584.25404 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4^2:C_2^2.C_2$ (as 16T382):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 26 conjugacy class representatives for $C_4^2:C_2^2.C_2$
Character table for $C_4^2:C_2^2.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.108025.2, 4.4.725.1, 4.0.3725.1, 8.0.11669400625.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$5$5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$29$29.4.2.2$x^{4} - 29 x^{2} + 2523$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
29.4.3.4$x^{4} + 232$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.3.3$x^{4} + 58$$4$$1$$3$$C_4$$[\ ]_{4}$
$149$149.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
149.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
149.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
149.2.1.1$x^{2} - 149$$2$$1$$1$$C_2$$[\ ]_{2}$
149.2.1.1$x^{2} - 149$$2$$1$$1$$C_2$$[\ ]_{2}$
149.2.1.1$x^{2} - 149$$2$$1$$1$$C_2$$[\ ]_{2}$
149.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
149.2.1.1$x^{2} - 149$$2$$1$$1$$C_2$$[\ ]_{2}$