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Properties

Label 16.12.8767895277...7713.1
Degree $16$
Signature $[12, 2]$
Discriminant $61^{4}\cdot 97^{15}$
Root discriminant $203.67$
Ramified primes $61, 97$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $(C_2\times OD_{16}).D_4$ (as 16T591)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-55934785129, 12256345154, 12277163359, -1274011149, -885218873, -141618630, 6505217, 24361457, 3091178, -1118637, -230205, 15299, 7902, 161, -138, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 138*x^14 + 161*x^13 + 7902*x^12 + 15299*x^11 - 230205*x^10 - 1118637*x^9 + 3091178*x^8 + 24361457*x^7 + 6505217*x^6 - 141618630*x^5 - 885218873*x^4 - 1274011149*x^3 + 12277163359*x^2 + 12256345154*x - 55934785129)
 
gp: K = bnfinit(x^16 - 4*x^15 - 138*x^14 + 161*x^13 + 7902*x^12 + 15299*x^11 - 230205*x^10 - 1118637*x^9 + 3091178*x^8 + 24361457*x^7 + 6505217*x^6 - 141618630*x^5 - 885218873*x^4 - 1274011149*x^3 + 12277163359*x^2 + 12256345154*x - 55934785129, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 138 x^{14} + 161 x^{13} + 7902 x^{12} + 15299 x^{11} - 230205 x^{10} - 1118637 x^{9} + 3091178 x^{8} + 24361457 x^{7} + 6505217 x^{6} - 141618630 x^{5} - 885218873 x^{4} - 1274011149 x^{3} + 12277163359 x^{2} + 12256345154 x - 55934785129 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(8767895277848913089642817986417897713=61^{4}\cdot 97^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $203.67$
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:  $61, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{180308462368977642243538889312721913645834562262109487762150013269} a^{15} + \frac{51873371498968166795656242109957561084052812128219862893263011945}{180308462368977642243538889312721913645834562262109487762150013269} a^{14} - \frac{31725540821213834019905862919750556957873047964577026442462107872}{180308462368977642243538889312721913645834562262109487762150013269} a^{13} + \frac{32484030940537372587472278631392001286344937504507016289150688813}{180308462368977642243538889312721913645834562262109487762150013269} a^{12} + \frac{75584511291611226435352393107563055183196677539096352497543451105}{180308462368977642243538889312721913645834562262109487762150013269} a^{11} - \frac{49974365904843506553352405679826099759206657758308879432847611171}{180308462368977642243538889312721913645834562262109487762150013269} a^{10} - \frac{82196689184573442284788234642449883653845981235315699498177286423}{180308462368977642243538889312721913645834562262109487762150013269} a^{9} - \frac{82865098550751521336950401709143133666167611264114595806658438499}{180308462368977642243538889312721913645834562262109487762150013269} a^{8} - \frac{50787770235764875306411562011240748016532429901850658788614804556}{180308462368977642243538889312721913645834562262109487762150013269} a^{7} - \frac{46930946981319594213145118383458099004848560074656445046960030288}{180308462368977642243538889312721913645834562262109487762150013269} a^{6} + \frac{16269249572356819038578242971945889847648437867252614796275778182}{180308462368977642243538889312721913645834562262109487762150013269} a^{5} + \frac{37821063231735781033310811034870145436990490265298753976939057861}{180308462368977642243538889312721913645834562262109487762150013269} a^{4} + \frac{82354380508127228127058712815859324126295709187915490134136653058}{180308462368977642243538889312721913645834562262109487762150013269} a^{3} + \frac{86641332574670089680396742265099259717420744578885520225703866723}{180308462368977642243538889312721913645834562262109487762150013269} a^{2} + \frac{89576641791896471987012390057040313987075828043682051240671388870}{180308462368977642243538889312721913645834562262109487762150013269} a - \frac{14302544655864455453023357006205469582630501107642174387895039931}{180308462368977642243538889312721913645834562262109487762150013269}$

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:  $13$
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:  \( 2697567588720 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times OD_{16}).D_4$ (as 16T591):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 28 conjugacy class representatives for $(C_2\times OD_{16}).D_4$
Character table for $(C_2\times OD_{16}).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{97}) \), 4.4.912673.1, 8.8.80798284478113.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ $16$

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
61Data not computed
97Data not computed