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Properties

Label 16.0.57704698464...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 5^{8}\cdot 29^{6}\cdot 41^{8}\cdot 83^{4}$
Root discriminant $305.55$
Ramified primes $2, 5, 29, 41, 83$
Class number $32$ (GRH)
Class group $[2, 2, 2, 4]$ (GRH)
Galois group $(C_2^2\times D_4).C_2^3$ (as 16T600)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![112783038862723321, 0, 5324472099277797, 0, 114344495249640, 0, 1468743133947, 0, 14212500309, 0, 86861493, 0, 408630, 0, 993, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 993*x^14 + 408630*x^12 + 86861493*x^10 + 14212500309*x^8 + 1468743133947*x^6 + 114344495249640*x^4 + 5324472099277797*x^2 + 112783038862723321)
 
gp: K = bnfinit(x^16 + 993*x^14 + 408630*x^12 + 86861493*x^10 + 14212500309*x^8 + 1468743133947*x^6 + 114344495249640*x^4 + 5324472099277797*x^2 + 112783038862723321, 1)
 

Normalized defining polynomial

\( x^{16} + 993 x^{14} + 408630 x^{12} + 86861493 x^{10} + 14212500309 x^{8} + 1468743133947 x^{6} + 114344495249640 x^{4} + 5324472099277797 x^{2} + 112783038862723321 \)

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:  \(5770469846478308612464977355801600000000=2^{16}\cdot 5^{8}\cdot 29^{6}\cdot 41^{8}\cdot 83^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $305.55$
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:  $2, 5, 29, 41, 83$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{205} a^{8} + \frac{9}{205} a^{6} - \frac{58}{205} a^{4} + \frac{1}{5} a^{2} + \frac{1}{5}$, $\frac{1}{205} a^{9} + \frac{9}{205} a^{7} - \frac{58}{205} a^{5} + \frac{1}{5} a^{3} + \frac{1}{5} a$, $\frac{1}{17015} a^{10} - \frac{3}{17015} a^{8} - \frac{5291}{17015} a^{6} - \frac{3568}{17015} a^{4} + \frac{169}{415} a^{2} + \frac{1}{5}$, $\frac{1}{17015} a^{11} - \frac{3}{17015} a^{9} - \frac{5291}{17015} a^{7} - \frac{3568}{17015} a^{5} + \frac{169}{415} a^{3} + \frac{1}{5} a$, $\frac{1}{37578427205} a^{12} + \frac{228994}{37578427205} a^{10} - \frac{12138231}{37578427205} a^{8} - \frac{251348736}{916547005} a^{6} + \frac{56571954}{183309401} a^{4} - \frac{30676}{269335} a^{2} + \frac{518}{3245}$, $\frac{1}{37578427205} a^{13} + \frac{228994}{37578427205} a^{11} - \frac{12138231}{37578427205} a^{9} - \frac{251348736}{916547005} a^{7} + \frac{56571954}{183309401} a^{5} - \frac{30676}{269335} a^{3} + \frac{518}{3245} a$, $\frac{1}{7559382807295245745067355255952592530654595} a^{14} + \frac{64618059558735830068506520308878}{7559382807295245745067355255952592530654595} a^{12} - \frac{104453912454048949245795840386471541976}{7559382807295245745067355255952592530654595} a^{10} + \frac{328234077682063417425037402325304800813}{184375190421835262074813542828112012942795} a^{8} - \frac{1936925241127419609063125136533138022993}{4496955863947201514019842508002732022995} a^{6} + \frac{3293012136790303345961244802583037056}{10836038226378798828963475922898149453} a^{4} - \frac{152557571842224651399557958543189721}{652773387131252941503823850776996955} a^{2} - \frac{427661859627435600854770176424}{1322916670141463296085246994593}$, $\frac{1}{7559382807295245745067355255952592530654595} a^{15} + \frac{64618059558735830068506520308878}{7559382807295245745067355255952592530654595} a^{13} - \frac{104453912454048949245795840386471541976}{7559382807295245745067355255952592530654595} a^{11} + \frac{328234077682063417425037402325304800813}{184375190421835262074813542828112012942795} a^{9} - \frac{1936925241127419609063125136533138022993}{4496955863947201514019842508002732022995} a^{7} + \frac{3293012136790303345961244802583037056}{10836038226378798828963475922898149453} a^{5} - \frac{152557571842224651399557958543189721}{652773387131252941503823850776996955} a^{3} - \frac{427661859627435600854770176424}{1322916670141463296085246994593} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{4}$, which has order $32$ (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:  \( 1099162288620 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2^2\times D_4).C_2^3$ (as 16T600):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 40 conjugacy class representatives for $(C_2^2\times D_4).C_2^3$
Character table for $(C_2^2\times D_4).C_2^3$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.29725.2, 4.4.725.1, 4.0.1025.1, 8.0.883575625.2

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$2.8.8.6$x^{8} + 2 x^{7} + 2 x^{6} + 16 x^{2} + 16$$2$$4$$8$$(C_8:C_2):C_2$$[2, 2, 2]^{4}$
2.8.8.6$x^{8} + 2 x^{7} + 2 x^{6} + 16 x^{2} + 16$$2$$4$$8$$(C_8:C_2):C_2$$[2, 2, 2]^{4}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$29$29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.4.2.2$x^{4} - 29 x^{2} + 2523$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
$41$41.4.3.3$x^{4} + 246$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.2.2$x^{4} - 41 x^{2} + 20172$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
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}$
$83$83.4.2.2$x^{4} - 83 x^{2} + 13778$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
83.4.2.2$x^{4} - 83 x^{2} + 13778$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
83.4.0.1$x^{4} - x + 22$$1$$4$$0$$C_4$$[\ ]^{4}$
83.4.0.1$x^{4} - x + 22$$1$$4$$0$$C_4$$[\ ]^{4}$