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Properties

Label 16.0.21359948160...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 5^{8}\cdot 13^{10}\cdot 31^{4}$
Root discriminant $104.86$
Ramified primes $2, 5, 13, 31$
Class number $505408$ (GRH)
Class group $[2, 2, 2, 2, 31588]$ (GRH)
Galois group $(C_2\times C_4).C_2^4$ (as 16T205)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![59243323961, 39039125944, 24982140462, 7498045776, 7851350618, 539705140, 1089726432, 9111128, 73676772, -599248, 2585058, -23504, 45729, -156, 358, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 358*x^14 - 156*x^13 + 45729*x^12 - 23504*x^11 + 2585058*x^10 - 599248*x^9 + 73676772*x^8 + 9111128*x^7 + 1089726432*x^6 + 539705140*x^5 + 7851350618*x^4 + 7498045776*x^3 + 24982140462*x^2 + 39039125944*x + 59243323961)
 
gp: K = bnfinit(x^16 + 358*x^14 - 156*x^13 + 45729*x^12 - 23504*x^11 + 2585058*x^10 - 599248*x^9 + 73676772*x^8 + 9111128*x^7 + 1089726432*x^6 + 539705140*x^5 + 7851350618*x^4 + 7498045776*x^3 + 24982140462*x^2 + 39039125944*x + 59243323961, 1)
 

Normalized defining polynomial

\( x^{16} + 358 x^{14} - 156 x^{13} + 45729 x^{12} - 23504 x^{11} + 2585058 x^{10} - 599248 x^{9} + 73676772 x^{8} + 9111128 x^{7} + 1089726432 x^{6} + 539705140 x^{5} + 7851350618 x^{4} + 7498045776 x^{3} + 24982140462 x^{2} + 39039125944 x + 59243323961 \)

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:  \(213599481601886546978406400000000=2^{32}\cdot 5^{8}\cdot 13^{10}\cdot 31^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $104.86$
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:  $2, 5, 13, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{403} a^{12} + \frac{85}{403} a^{11} - \frac{58}{403} a^{10} + \frac{184}{403} a^{9} + \frac{168}{403} a^{8} + \frac{5}{13} a^{7} - \frac{11}{31} a^{6} - \frac{46}{403} a^{5} + \frac{55}{403} a^{4} - \frac{94}{403} a^{3} - \frac{119}{403} a^{2} + \frac{12}{403} a - \frac{157}{403}$, $\frac{1}{403} a^{13} - \frac{29}{403} a^{11} - \frac{125}{403} a^{10} - \frac{158}{403} a^{9} - \frac{20}{403} a^{8} - \frac{19}{403} a^{7} + \frac{19}{403} a^{6} - \frac{5}{31} a^{5} + \frac{67}{403} a^{4} - \frac{189}{403} a^{3} + \frac{4}{31} a^{2} + \frac{32}{403} a + \frac{46}{403}$, $\frac{1}{403} a^{14} - \frac{6}{31} a^{11} + \frac{175}{403} a^{10} + \frac{77}{403} a^{9} + \frac{17}{403} a^{8} + \frac{81}{403} a^{7} - \frac{14}{31} a^{6} - \frac{58}{403} a^{5} + \frac{197}{403} a^{4} + \frac{147}{403} a^{3} - \frac{15}{31} a^{2} - \frac{9}{403} a - \frac{120}{403}$, $\frac{1}{4372516973578926303456222779055104456441640534312444036495229720673860519} a^{15} - \frac{4607868555051823881395803274499318151875346039881419178356115067522190}{4372516973578926303456222779055104456441640534312444036495229720673860519} a^{14} + \frac{2616344445582305080069597162455647684788525534344769511402402493817317}{4372516973578926303456222779055104456441640534312444036495229720673860519} a^{13} + \frac{4857779884912697907124412866664800962529622269371061088487701291540692}{4372516973578926303456222779055104456441640534312444036495229720673860519} a^{12} + \frac{444136740971389308551495824251217791839766333293727568360870470082892375}{4372516973578926303456222779055104456441640534312444036495229720673860519} a^{11} + \frac{834665984820097731370864250560852182819340839136878146305786931228476957}{4372516973578926303456222779055104456441640534312444036495229720673860519} a^{10} + \frac{1727894330322213909532348605233634651251672594503191099951856171130494313}{4372516973578926303456222779055104456441640534312444036495229720673860519} a^{9} - \frac{545244119445070951586731156915921166172127827546846598168726924205368701}{4372516973578926303456222779055104456441640534312444036495229720673860519} a^{8} + \frac{1361777693291255821378016889938372449975777566235147490286688716492284427}{4372516973578926303456222779055104456441640534312444036495229720673860519} a^{7} - \frac{1293835736872361168926828251795491412296499695981912557597037797056906730}{4372516973578926303456222779055104456441640534312444036495229720673860519} a^{6} + \frac{561105944781982269097209985798101690526169871015765961812384009089135594}{4372516973578926303456222779055104456441640534312444036495229720673860519} a^{5} - \frac{1050909604789774905100937009739436409306950936206025324581050479758207977}{4372516973578926303456222779055104456441640534312444036495229720673860519} a^{4} - \frac{94880572485426566497917175023910263448248273642362268074238420379551181}{4372516973578926303456222779055104456441640534312444036495229720673860519} a^{3} - \frac{330813116145652021380978451803472943591838233572914994944581974140572495}{4372516973578926303456222779055104456441640534312444036495229720673860519} a^{2} + \frac{1336573475560858545773271582958993621006733816526393108509518153964214540}{4372516973578926303456222779055104456441640534312444036495229720673860519} a + \frac{50251334444446461526353608043504049958365704499210520759049333918983897}{106646755453144543986737140952563523327844891080791317963298285870094159}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{31588}$, which has order $505408$ (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:  \( 3710.59482488 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times C_4).C_2^4$ (as 16T205):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 44 conjugacy class representatives for $(C_2\times C_4).C_2^4$
Character table for $(C_2\times C_4).C_2^4$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{2}, \sqrt{5})\), 8.8.432640000.1

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 R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
2Data not computed
5Data not computed
$13$13.8.6.4$x^{8} - 13 x^{4} + 338$$4$$2$$6$$C_8$$[\ ]_{4}^{2}$
13.8.4.2$x^{8} + 169 x^{4} - 2197 x^{2} + 57122$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
$31$31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.4.2.2$x^{4} - 31 x^{2} + 11532$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
31.4.0.1$x^{4} - 2 x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$