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Properties

Label 16.8.12746025680...9401.1
Degree $16$
Signature $[8, 4]$
Discriminant $31^{10}\cdot 41^{15}$
Root discriminant $278.03$
Ramified primes $31, 41$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1223

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1486289473, 388958406, 2355411513, -199051862, 548662862, -263187800, 76888174, -21748618, 1972610, -133478, -39845, 34380, -2893, 104, -54, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 54*x^14 + 104*x^13 - 2893*x^12 + 34380*x^11 - 39845*x^10 - 133478*x^9 + 1972610*x^8 - 21748618*x^7 + 76888174*x^6 - 263187800*x^5 + 548662862*x^4 - 199051862*x^3 + 2355411513*x^2 + 388958406*x - 1486289473)
 
gp: K = bnfinit(x^16 - 4*x^15 - 54*x^14 + 104*x^13 - 2893*x^12 + 34380*x^11 - 39845*x^10 - 133478*x^9 + 1972610*x^8 - 21748618*x^7 + 76888174*x^6 - 263187800*x^5 + 548662862*x^4 - 199051862*x^3 + 2355411513*x^2 + 388958406*x - 1486289473, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 54 x^{14} + 104 x^{13} - 2893 x^{12} + 34380 x^{11} - 39845 x^{10} - 133478 x^{9} + 1972610 x^{8} - 21748618 x^{7} + 76888174 x^{6} - 263187800 x^{5} + 548662862 x^{4} - 199051862 x^{3} + 2355411513 x^{2} + 388958406 x - 1486289473 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1274602568003244304847660149105336749401=31^{10}\cdot 41^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $278.03$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{4922} a^{14} + \frac{499}{4922} a^{13} - \frac{595}{2461} a^{12} - \frac{1107}{4922} a^{11} - \frac{267}{2461} a^{10} - \frac{525}{2461} a^{9} + \frac{669}{4922} a^{8} + \frac{1263}{4922} a^{7} - \frac{399}{2461} a^{6} + \frac{513}{2461} a^{5} + \frac{7}{23} a^{4} + \frac{1369}{4922} a^{3} - \frac{488}{2461} a^{2} + \frac{1206}{2461} a - \frac{142}{2461}$, $\frac{1}{127158305359792712766838754953010884669302348179278475388476532012814} a^{15} - \frac{3809514420495878503965222553222866232363999737922733481509095943}{127158305359792712766838754953010884669302348179278475388476532012814} a^{14} - \frac{2398524884210918743123934376844542953404363054375065732700665915325}{127158305359792712766838754953010884669302348179278475388476532012814} a^{13} + \frac{4337010601422568494897365316743804034281084134994275759775757972160}{63579152679896356383419377476505442334651174089639237694238266006407} a^{12} + \frac{13301774469272352220181822890353739690794017524601521081876395154611}{127158305359792712766838754953010884669302348179278475388476532012814} a^{11} + \frac{15352105996860852652362583681033533409457468313037706158393534487434}{63579152679896356383419377476505442334651174089639237694238266006407} a^{10} - \frac{1170486184480940666944533659796495743501973444748098751901386861298}{63579152679896356383419377476505442334651174089639237694238266006407} a^{9} + \frac{11064767153592412288740858923106886162886144944536030976819022790548}{63579152679896356383419377476505442334651174089639237694238266006407} a^{8} - \frac{955014243195691758447036164171834024967218897075799535342208213927}{2764310986082450277539972933761106188463094525636488595401663739409} a^{7} + \frac{9124055619542958551701896338936553481420356031204523741951912428554}{63579152679896356383419377476505442334651174089639237694238266006407} a^{6} + \frac{43698925700602090671463188521546774605814192670474086664361440008279}{127158305359792712766838754953010884669302348179278475388476532012814} a^{5} - \frac{12326340461563142543221141583859946332917324425829126050824982709174}{63579152679896356383419377476505442334651174089639237694238266006407} a^{4} + \frac{13798267801800623020789286868020791470032838302549153719081835181077}{63579152679896356383419377476505442334651174089639237694238266006407} a^{3} + \frac{20374412391579059996461603453788672804841204976368749500204565494898}{63579152679896356383419377476505442334651174089639237694238266006407} a^{2} - \frac{29660474458236856166786818336801903625986409359726921571741285861985}{127158305359792712766838754953010884669302348179278475388476532012814} a + \frac{60428375774816639819909618883216413374994848940753209677640683403109}{127158305359792712766838754953010884669302348179278475388476532012814}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $11$
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:  \( 25179804757400 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1223:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 40 conjugacy class representatives for t16n1223
Character table for t16n1223 is not computed

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.68921.1, 8.8.179859661768855001.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{12}$ $16$ R ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ $16$ $16$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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
31Data not computed
41Data not computed