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Properties

Label 16.6.65203029999...8687.5
Degree $16$
Signature $[6, 5]$
Discriminant $-\,17^{12}\cdot 47^{9}$
Root discriminant $73.01$
Ramified primes $17, 47$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1251

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![74303, -152393, -486895, 641845, 375334, -956679, 549035, -82210, -45268, 30348, -9812, 1812, -96, -40, 18, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 18*x^14 - 40*x^13 - 96*x^12 + 1812*x^11 - 9812*x^10 + 30348*x^9 - 45268*x^8 - 82210*x^7 + 549035*x^6 - 956679*x^5 + 375334*x^4 + 641845*x^3 - 486895*x^2 - 152393*x + 74303)
 
gp: K = bnfinit(x^16 - 6*x^15 + 18*x^14 - 40*x^13 - 96*x^12 + 1812*x^11 - 9812*x^10 + 30348*x^9 - 45268*x^8 - 82210*x^7 + 549035*x^6 - 956679*x^5 + 375334*x^4 + 641845*x^3 - 486895*x^2 - 152393*x + 74303, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 18 x^{14} - 40 x^{13} - 96 x^{12} + 1812 x^{11} - 9812 x^{10} + 30348 x^{9} - 45268 x^{8} - 82210 x^{7} + 549035 x^{6} - 956679 x^{5} + 375334 x^{4} + 641845 x^{3} - 486895 x^{2} - 152393 x + 74303 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-652030299991134977060343848687=-\,17^{12}\cdot 47^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $73.01$
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:  $17, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{58042688641178919128367224063398277941609327} a^{15} - \frac{27934361938284794814313474326395450951848149}{58042688641178919128367224063398277941609327} a^{14} + \frac{14849959417726733882577169868182437763864679}{58042688641178919128367224063398277941609327} a^{13} + \frac{5898903718857555168011835179438371869876828}{58042688641178919128367224063398277941609327} a^{12} + \frac{14462522477114659982056180222176586136201068}{58042688641178919128367224063398277941609327} a^{11} + \frac{16782984301738953680394494313789963868751362}{58042688641178919128367224063398277941609327} a^{10} + \frac{14174573502391503460358251774430761151595538}{58042688641178919128367224063398277941609327} a^{9} + \frac{14408370922965411532133030289024986637077186}{58042688641178919128367224063398277941609327} a^{8} + \frac{19457695339126013871718248116335478473049734}{58042688641178919128367224063398277941609327} a^{7} - \frac{2208476449148146633969153180326339306114425}{58042688641178919128367224063398277941609327} a^{6} + \frac{1500447044198525256373163109501121514168440}{58042688641178919128367224063398277941609327} a^{5} + \frac{4314447360487134713482787582068548307729177}{58042688641178919128367224063398277941609327} a^{4} - \frac{561864118839288898069563500965527641193073}{1234950822152742960178026043902091020034241} a^{3} - \frac{2428711559446802702122311340444406958049587}{58042688641178919128367224063398277941609327} a^{2} + \frac{7897723263863947439890587786467517973174141}{58042688641178919128367224063398277941609327} a + \frac{19371361041449950847004234580512144827482660}{58042688641178919128367224063398277941609327}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $10$
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:  \( 470864643.569 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1251:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 34 conjugacy class representatives for t16n1251
Character table for t16n1251 is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, 8.6.2506034826287.1

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
Arithmetically equvalently 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ $16$ $16$ $16$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
17Data not computed
47Data not computed