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Properties

Label 16.0.12796960816...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 3^{8}\cdot 5^{12}\cdot 61^{2}\cdot 181^{2}$
Root discriminant $37.08$
Ramified primes $2, 3, 5, 61, 181$
Class number $160$ (GRH)
Class group $[2, 2, 40]$ (GRH)
Galois group 16T797

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![13921, -22708, 51225, -48694, 61418, -53874, 56189, -44976, 34425, -19892, 10008, -3774, 1232, -294, 62, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 62*x^14 - 294*x^13 + 1232*x^12 - 3774*x^11 + 10008*x^10 - 19892*x^9 + 34425*x^8 - 44976*x^7 + 56189*x^6 - 53874*x^5 + 61418*x^4 - 48694*x^3 + 51225*x^2 - 22708*x + 13921)
 
gp: K = bnfinit(x^16 - 8*x^15 + 62*x^14 - 294*x^13 + 1232*x^12 - 3774*x^11 + 10008*x^10 - 19892*x^9 + 34425*x^8 - 44976*x^7 + 56189*x^6 - 53874*x^5 + 61418*x^4 - 48694*x^3 + 51225*x^2 - 22708*x + 13921, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 62 x^{14} - 294 x^{13} + 1232 x^{12} - 3774 x^{11} + 10008 x^{10} - 19892 x^{9} + 34425 x^{8} - 44976 x^{7} + 56189 x^{6} - 53874 x^{5} + 61418 x^{4} - 48694 x^{3} + 51225 x^{2} - 22708 x + 13921 \)

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:  \(12796960816656000000000000=2^{16}\cdot 3^{8}\cdot 5^{12}\cdot 61^{2}\cdot 181^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.08$
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:  $2, 3, 5, 61, 181$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{15} a^{12} - \frac{2}{5} a^{11} - \frac{4}{15} a^{10} - \frac{1}{3} a^{9} - \frac{1}{15} a^{8} - \frac{2}{15} a^{7} + \frac{7}{15} a^{6} + \frac{7}{15} a^{5} + \frac{1}{15} a^{4} + \frac{2}{5} a^{3} + \frac{2}{5} a^{2} - \frac{1}{3} a - \frac{4}{15}$, $\frac{1}{15} a^{13} + \frac{1}{3} a^{11} + \frac{1}{15} a^{10} - \frac{1}{15} a^{9} + \frac{7}{15} a^{8} - \frac{1}{3} a^{7} + \frac{4}{15} a^{6} - \frac{2}{15} a^{5} - \frac{1}{5} a^{4} - \frac{1}{5} a^{3} + \frac{1}{15} a^{2} - \frac{4}{15} a + \frac{2}{5}$, $\frac{1}{158415} a^{14} - \frac{350}{10561} a^{13} - \frac{703}{31683} a^{12} - \frac{62714}{158415} a^{11} + \frac{17933}{52805} a^{10} - \frac{10771}{52805} a^{9} - \frac{8525}{31683} a^{8} - \frac{10012}{52805} a^{7} + \frac{10281}{52805} a^{6} - \frac{16198}{158415} a^{5} - \frac{50953}{158415} a^{4} - \frac{59039}{158415} a^{3} + \frac{20066}{158415} a^{2} + \frac{829}{2685} a + \frac{10001}{31683}$, $\frac{1}{62498669330575861435215435} a^{15} - \frac{155504356666788945646}{62498669330575861435215435} a^{14} + \frac{223467487279935177606098}{20832889776858620478405145} a^{13} + \frac{1408212459369611985801682}{62498669330575861435215435} a^{12} + \frac{746666868370512856265899}{20832889776858620478405145} a^{11} - \frac{13056437028855425247609617}{62498669330575861435215435} a^{10} + \frac{21470180677391843334052789}{62498669330575861435215435} a^{9} + \frac{10265640602686903018483241}{62498669330575861435215435} a^{8} - \frac{2057321995000524462279516}{20832889776858620478405145} a^{7} + \frac{7437208061148488762349467}{62498669330575861435215435} a^{6} - \frac{8943888909651822445809772}{20832889776858620478405145} a^{5} - \frac{2844237507749956534389169}{20832889776858620478405145} a^{4} + \frac{15629255325765874392946729}{62498669330575861435215435} a^{3} + \frac{3659239282905223802911300}{12499733866115172287043087} a^{2} + \frac{23229662022202388901958288}{62498669330575861435215435} a + \frac{975890762181140345073098}{12499733866115172287043087}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{40}$, which has order $160$ (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:  \( 3121.7160225 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T797:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 62 conjugacy class representatives for t16n797 are not computed
Character table for t16n797 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{15}) \), \(\Q(\zeta_{20})^+\), \(\Q(\zeta_{15})^+\), \(\Q(\sqrt{3}, \sqrt{5})\), \(\Q(\zeta_{60})^+\)

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 R R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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
2Data not computed
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
61Data not computed
181Data not computed