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Properties

Label 16.6.10009146154...6143.3
Degree $16$
Signature $[6, 5]$
Discriminant $-\,23^{5}\cdot 41^{15}$
Root discriminant $86.60$
Ramified primes $23, 41$
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![-439133, 666313, 62219, -501358, 94405, 161836, -22884, -44205, 780, 10183, -712, -1328, 195, 90, -15, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - 15*x^14 + 90*x^13 + 195*x^12 - 1328*x^11 - 712*x^10 + 10183*x^9 + 780*x^8 - 44205*x^7 - 22884*x^6 + 161836*x^5 + 94405*x^4 - 501358*x^3 + 62219*x^2 + 666313*x - 439133)
 
gp: K = bnfinit(x^16 - 3*x^15 - 15*x^14 + 90*x^13 + 195*x^12 - 1328*x^11 - 712*x^10 + 10183*x^9 + 780*x^8 - 44205*x^7 - 22884*x^6 + 161836*x^5 + 94405*x^4 - 501358*x^3 + 62219*x^2 + 666313*x - 439133, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} - 15 x^{14} + 90 x^{13} + 195 x^{12} - 1328 x^{11} - 712 x^{10} + 10183 x^{9} + 780 x^{8} - 44205 x^{7} - 22884 x^{6} + 161836 x^{5} + 94405 x^{4} - 501358 x^{3} + 62219 x^{2} + 666313 x - 439133 \)

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:  \(-10009146154007580091982470826143=-\,23^{5}\cdot 41^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $86.60$
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:  $23, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{92} a^{12} + \frac{11}{46} a^{11} + \frac{29}{92} a^{10} + \frac{8}{23} a^{9} + \frac{6}{23} a^{8} + \frac{31}{92} a^{7} - \frac{7}{23} a^{6} + \frac{21}{92} a^{5} - \frac{5}{23} a^{4} - \frac{9}{23} a^{3} - \frac{39}{92} a^{2} - \frac{19}{46} a + \frac{1}{92}$, $\frac{1}{184} a^{13} - \frac{1}{184} a^{12} - \frac{17}{184} a^{11} + \frac{9}{184} a^{10} + \frac{3}{23} a^{9} + \frac{31}{184} a^{8} - \frac{5}{184} a^{7} + \frac{21}{184} a^{6} - \frac{43}{184} a^{5} - \frac{9}{46} a^{4} - \frac{39}{184} a^{3} + \frac{31}{184} a^{2} + \frac{47}{184} a - \frac{1}{8}$, $\frac{1}{184} a^{14} + \frac{5}{46} a^{11} + \frac{3}{184} a^{10} + \frac{79}{184} a^{9} + \frac{45}{92} a^{8} + \frac{11}{92} a^{7} + \frac{13}{92} a^{6} - \frac{3}{8} a^{5} - \frac{67}{184} a^{4} + \frac{10}{23} a^{3} - \frac{9}{23} a^{2} + \frac{19}{46} a - \frac{5}{184}$, $\frac{1}{814568091679809942135694641105808} a^{15} + \frac{920697155898576580931847969835}{407284045839904971067847320552904} a^{14} - \frac{1947005457211365718570214913611}{814568091679809942135694641105808} a^{13} - \frac{1654323683292287374097655811959}{814568091679809942135694641105808} a^{12} + \frac{5328165171659568919677854345611}{407284045839904971067847320552904} a^{11} - \frac{77585668393194896277779026553821}{203642022919952485533923660276452} a^{10} - \frac{84889520176163263874378539371285}{203642022919952485533923660276452} a^{9} + \frac{268727833964324185652838691482773}{814568091679809942135694641105808} a^{8} + \frac{396153614979538089072570434617267}{814568091679809942135694641105808} a^{7} - \frac{82207963567253191504275052756219}{203642022919952485533923660276452} a^{6} + \frac{90563720429150524404394201868815}{407284045839904971067847320552904} a^{5} + \frac{79762814776568387362764899878869}{407284045839904971067847320552904} a^{4} - \frac{322240071928998542379876836288083}{814568091679809942135694641105808} a^{3} - \frac{131955879350839152155082986498343}{814568091679809942135694641105808} a^{2} - \frac{176872061049470922719815812705937}{407284045839904971067847320552904} a - \frac{196310458494528523198643840486995}{814568091679809942135694641105808}$

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:  \( 10483241933.8 \) (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

\(\Q(\sqrt{41}) \), 4.4.68921.1, 8.6.4479348299263.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{4}$ $16$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ R $16$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ $16$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$23$$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
41Data not computed