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Properties

Label 16.2.65203029999...8687.9
Degree $16$
Signature $[2, 7]$
Discriminant $-\,17^{12}\cdot 47^{9}$
Root discriminant $73.01$
Ramified primes $17, 47$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1251

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![822209, -344185, -246861, -270587, 19354, -121028, 36105, -31872, 16001, -9793, 5292, -2002, 803, -166, 50, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 50*x^14 - 166*x^13 + 803*x^12 - 2002*x^11 + 5292*x^10 - 9793*x^9 + 16001*x^8 - 31872*x^7 + 36105*x^6 - 121028*x^5 + 19354*x^4 - 270587*x^3 - 246861*x^2 - 344185*x + 822209)
 
gp: K = bnfinit(x^16 - 4*x^15 + 50*x^14 - 166*x^13 + 803*x^12 - 2002*x^11 + 5292*x^10 - 9793*x^9 + 16001*x^8 - 31872*x^7 + 36105*x^6 - 121028*x^5 + 19354*x^4 - 270587*x^3 - 246861*x^2 - 344185*x + 822209, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 50 x^{14} - 166 x^{13} + 803 x^{12} - 2002 x^{11} + 5292 x^{10} - 9793 x^{9} + 16001 x^{8} - 31872 x^{7} + 36105 x^{6} - 121028 x^{5} + 19354 x^{4} - 270587 x^{3} - 246861 x^{2} - 344185 x + 822209 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 7]$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{173087950532561024818880562687774138115124467} a^{15} + \frac{23384625545515679773748516922602789980544478}{173087950532561024818880562687774138115124467} a^{14} - \frac{45915673920320431134392583991016849673920750}{173087950532561024818880562687774138115124467} a^{13} - \frac{37596014376186989047867386674121023566294917}{173087950532561024818880562687774138115124467} a^{12} - \frac{24865422432631796087039860052827384478661277}{173087950532561024818880562687774138115124467} a^{11} + \frac{22322465555899985286373176777945949573246164}{173087950532561024818880562687774138115124467} a^{10} + \frac{43624683436201093158387177144152490704418088}{173087950532561024818880562687774138115124467} a^{9} + \frac{63970258295275613758973976515304166458815374}{173087950532561024818880562687774138115124467} a^{8} + \frac{15126895437313329776033916381855071286440868}{173087950532561024818880562687774138115124467} a^{7} - \frac{16880826403818578381465899717798449058600859}{173087950532561024818880562687774138115124467} a^{6} + \frac{22052797660627008792341949866931002758808345}{173087950532561024818880562687774138115124467} a^{5} + \frac{75476408928399029980629940517194314221585912}{173087950532561024818880562687774138115124467} a^{4} + \frac{23284022669204880024313306188437294847910072}{173087950532561024818880562687774138115124467} a^{3} - \frac{61409231931618349989814383533921224373379782}{173087950532561024818880562687774138115124467} a^{2} - \frac{9885848109645715783809920351303756247269825}{173087950532561024818880562687774138115124467} a - \frac{31075243768922851586658065065937647576986168}{173087950532561024818880562687774138115124467}$

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:  $8$
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:  \( 94680095.5605 \) (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