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Properties

Label 16.8.12746025680...9401.3
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![1688997251, 1118219916, -1870281267, -719804582, 259011770, -69734673, 23881913, -3978878, 421736, 173122, -25901, 7086, -2054, -69, 7, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 7*x^14 - 69*x^13 - 2054*x^12 + 7086*x^11 - 25901*x^10 + 173122*x^9 + 421736*x^8 - 3978878*x^7 + 23881913*x^6 - 69734673*x^5 + 259011770*x^4 - 719804582*x^3 - 1870281267*x^2 + 1118219916*x + 1688997251)
 
gp: K = bnfinit(x^16 - 2*x^15 + 7*x^14 - 69*x^13 - 2054*x^12 + 7086*x^11 - 25901*x^10 + 173122*x^9 + 421736*x^8 - 3978878*x^7 + 23881913*x^6 - 69734673*x^5 + 259011770*x^4 - 719804582*x^3 - 1870281267*x^2 + 1118219916*x + 1688997251, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 7 x^{14} - 69 x^{13} - 2054 x^{12} + 7086 x^{11} - 25901 x^{10} + 173122 x^{9} + 421736 x^{8} - 3978878 x^{7} + 23881913 x^{6} - 69734673 x^{5} + 259011770 x^{4} - 719804582 x^{3} - 1870281267 x^{2} + 1118219916 x + 1688997251 \)

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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{7} + \frac{1}{4} a^{6} + \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{10} - \frac{1}{8} a^{8} + \frac{1}{4} a^{7} - \frac{1}{8} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} + \frac{1}{8} a - \frac{3}{8}$, $\frac{1}{16} a^{12} - \frac{1}{16} a^{10} - \frac{1}{16} a^{9} + \frac{1}{16} a^{8} + \frac{1}{8} a^{7} - \frac{1}{2} a^{6} + \frac{7}{16} a^{5} + \frac{3}{16} a^{4} + \frac{1}{8} a^{3} + \frac{7}{16} a^{2} + \frac{3}{8} a - \frac{3}{16}$, $\frac{1}{32} a^{13} - \frac{1}{32} a^{12} - \frac{1}{32} a^{11} + \frac{1}{16} a^{9} - \frac{15}{32} a^{8} - \frac{5}{16} a^{7} - \frac{1}{32} a^{6} - \frac{1}{8} a^{5} + \frac{15}{32} a^{4} + \frac{5}{32} a^{3} + \frac{15}{32} a^{2} - \frac{9}{32} a + \frac{3}{32}$, $\frac{1}{1472} a^{14} - \frac{9}{736} a^{13} + \frac{3}{184} a^{12} + \frac{9}{1472} a^{11} + \frac{65}{736} a^{10} + \frac{359}{1472} a^{9} + \frac{101}{1472} a^{8} + \frac{617}{1472} a^{7} - \frac{435}{1472} a^{6} + \frac{139}{1472} a^{5} - \frac{221}{736} a^{4} + \frac{85}{736} a^{3} + \frac{5}{46} a^{2} + \frac{105}{368} a + \frac{685}{1472}$, $\frac{1}{14246637699678649086359525170754873465830962340439198299491968} a^{15} + \frac{1053471404929562852762604087294381293362891349563303623451}{14246637699678649086359525170754873465830962340439198299491968} a^{14} - \frac{45322030338743656916901608974160610435870840277730701088791}{7123318849839324543179762585377436732915481170219599149745984} a^{13} - \frac{177823502274081232764444500167936109179639456704187614885275}{14246637699678649086359525170754873465830962340439198299491968} a^{12} + \frac{683065813718780854918182286561189243543233627553088827515379}{14246637699678649086359525170754873465830962340439198299491968} a^{11} - \frac{878357190756287348270027140649251831262535305524744569896871}{14246637699678649086359525170754873465830962340439198299491968} a^{10} + \frac{36206802040790167083557646629735928664394886559093107252211}{222603714057478891974367580793044897903608786569362473429562} a^{9} + \frac{430794767312570807629358341591295680613149276131186305471083}{7123318849839324543179762585377436732915481170219599149745984} a^{8} - \frac{270325645918652723251434938885460203597301796626014190622491}{7123318849839324543179762585377436732915481170219599149745984} a^{7} + \frac{204997222595232653721444368419749385641731727313855868428525}{890414856229915567897470323172179591614435146277449893718248} a^{6} + \frac{2235508086458244560453769620902502107373005611452029130610877}{14246637699678649086359525170754873465830962340439198299491968} a^{5} - \frac{1669413747728744388217222477786590664185277912957180812501501}{3561659424919662271589881292688718366457740585109799574872992} a^{4} - \frac{1555913932039969566797375805885826148270038622371002797328477}{7123318849839324543179762585377436732915481170219599149745984} a^{3} + \frac{429201350965998774106018613699311453673475509863974033109939}{1780829712459831135794940646344359183228870292554899787436496} a^{2} - \frac{1225992529778521911406401356847128389400126175541397224577619}{14246637699678649086359525170754873465830962340439198299491968} a + \frac{4615087319120070757573965802564035856493635509360463587584349}{14246637699678649086359525170754873465830962340439198299491968}$

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:  \( 25583240827900 \) (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 ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ $16$ $16$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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