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Properties

Label 16.8.15231185783...5001.1
Degree $16$
Signature $[8, 4]$
Discriminant $17^{14}\cdot 67^{6}$
Root discriminant $57.73$
Ramified primes $17, 67$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1194

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1248497, 4418113, -4367181, -286790, 2302820, -1308864, 527680, 31671, -56320, 33984, -8796, 997, -70, -109, 24, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 24*x^14 - 109*x^13 - 70*x^12 + 997*x^11 - 8796*x^10 + 33984*x^9 - 56320*x^8 + 31671*x^7 + 527680*x^6 - 1308864*x^5 + 2302820*x^4 - 286790*x^3 - 4367181*x^2 + 4418113*x - 1248497)
 
gp: K = bnfinit(x^16 - 4*x^15 + 24*x^14 - 109*x^13 - 70*x^12 + 997*x^11 - 8796*x^10 + 33984*x^9 - 56320*x^8 + 31671*x^7 + 527680*x^6 - 1308864*x^5 + 2302820*x^4 - 286790*x^3 - 4367181*x^2 + 4418113*x - 1248497, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 24 x^{14} - 109 x^{13} - 70 x^{12} + 997 x^{11} - 8796 x^{10} + 33984 x^{9} - 56320 x^{8} + 31671 x^{7} + 527680 x^{6} - 1308864 x^{5} + 2302820 x^{4} - 286790 x^{3} - 4367181 x^{2} + 4418113 x - 1248497 \)

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:  \(15231185783695887615175635001=17^{14}\cdot 67^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $57.73$
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:  $17, 67$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{271} a^{14} + \frac{71}{271} a^{13} - \frac{71}{271} a^{12} - \frac{14}{271} a^{11} - \frac{36}{271} a^{10} - \frac{77}{271} a^{9} + \frac{63}{271} a^{8} - \frac{44}{271} a^{7} - \frac{36}{271} a^{5} + \frac{53}{271} a^{4} - \frac{24}{271} a^{3} - \frac{41}{271} a^{2} + \frac{105}{271} a$, $\frac{1}{464444063719306478478837218677674691473001181491} a^{15} + \frac{210431667765694510200462722130566214074073132}{464444063719306478478837218677674691473001181491} a^{14} + \frac{110901224007194568605849103873204737127396653770}{464444063719306478478837218677674691473001181491} a^{13} + \frac{76132706391773030550929073883107838477621971792}{464444063719306478478837218677674691473001181491} a^{12} - \frac{127313926400855982160434537586971701922755220956}{464444063719306478478837218677674691473001181491} a^{11} + \frac{75867397353004769111273053290430672637586253158}{464444063719306478478837218677674691473001181491} a^{10} + \frac{63722077006767516381932793469715405646857233628}{464444063719306478478837218677674691473001181491} a^{9} - \frac{210362025307830487974586701317073550489223994512}{464444063719306478478837218677674691473001181491} a^{8} + \frac{15647870853319401087552491414580668023467173675}{464444063719306478478837218677674691473001181491} a^{7} - \frac{117162194272237692409483513493147846120409601172}{464444063719306478478837218677674691473001181491} a^{6} - \frac{44071127589690279117918983642770115496953995364}{464444063719306478478837218677674691473001181491} a^{5} - \frac{165809199637924685441831440907102710646210977692}{464444063719306478478837218677674691473001181491} a^{4} + \frac{197970301261943355208899265426808782070401252054}{464444063719306478478837218677674691473001181491} a^{3} + \frac{229407833504487778421554649375185059152849834752}{464444063719306478478837218677674691473001181491} a^{2} - \frac{181108743482934459044146976591472999224298439058}{464444063719306478478837218677674691473001181491} a - \frac{546283951218398348794698338580208721205994094}{1713815733281573721324122578146401075546129821}$

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:  \( 55015087.6883 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1194:

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 t16n1194
Character table for t16n1194 is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, \(\Q(\zeta_{17})^+\)

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$17$17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$
67Data not computed