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Properties

Label 16.0.87123092217...0625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 59^{4}\cdot 131^{4}$
Root discriminant $31.35$
Ramified primes $5, 59, 131$
Class number $16$
Class group $[2, 8]$
Galois group 16T1496

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![706541, -463678, 538071, -286710, 75351, -1969, -18995, 20221, -6853, 2822, 95, -217, 236, -70, 29, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 29*x^14 - 70*x^13 + 236*x^12 - 217*x^11 + 95*x^10 + 2822*x^9 - 6853*x^8 + 20221*x^7 - 18995*x^6 - 1969*x^5 + 75351*x^4 - 286710*x^3 + 538071*x^2 - 463678*x + 706541)
 
gp: K = bnfinit(x^16 - 4*x^15 + 29*x^14 - 70*x^13 + 236*x^12 - 217*x^11 + 95*x^10 + 2822*x^9 - 6853*x^8 + 20221*x^7 - 18995*x^6 - 1969*x^5 + 75351*x^4 - 286710*x^3 + 538071*x^2 - 463678*x + 706541, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 29 x^{14} - 70 x^{13} + 236 x^{12} - 217 x^{11} + 95 x^{10} + 2822 x^{9} - 6853 x^{8} + 20221 x^{7} - 18995 x^{6} - 1969 x^{5} + 75351 x^{4} - 286710 x^{3} + 538071 x^{2} - 463678 x + 706541 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(871230922174922119140625=5^{12}\cdot 59^{4}\cdot 131^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.35$
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:  $5, 59, 131$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{1157874282135548768936367434102829597753821} a^{15} - \frac{462169214572302429445619163615542931277391}{1157874282135548768936367434102829597753821} a^{14} - \frac{290493158316847727329181467618931789405021}{1157874282135548768936367434102829597753821} a^{13} + \frac{447674969437197057424499473929490349966745}{1157874282135548768936367434102829597753821} a^{12} - \frac{86491091775658701957219949493646992284844}{1157874282135548768936367434102829597753821} a^{11} + \frac{3217049636735471391404329149654898698402}{1157874282135548768936367434102829597753821} a^{10} + \frac{486946983933333737700476985501965325473440}{1157874282135548768936367434102829597753821} a^{9} + \frac{56949078101824637192775577067906145860482}{1157874282135548768936367434102829597753821} a^{8} - \frac{131477995603630029226172595246880944046349}{1157874282135548768936367434102829597753821} a^{7} - \frac{407583903813649929965940100203547665940496}{1157874282135548768936367434102829597753821} a^{6} - \frac{219927681155388058310918548886194882601073}{1157874282135548768936367434102829597753821} a^{5} - \frac{150719030757807424484908207124120718613442}{1157874282135548768936367434102829597753821} a^{4} - \frac{408793374814575343188807394395787763793722}{1157874282135548768936367434102829597753821} a^{3} + \frac{374336413342355046901986120622464713885503}{1157874282135548768936367434102829597753821} a^{2} - \frac{305239403082936384920679149185693517886439}{1157874282135548768936367434102829597753821} a + \frac{225463982368591645163194025435629053347949}{1157874282135548768936367434102829597753821}$

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

Class group and class number

$C_{2}\times C_{8}$, which has order $16$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $7$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 11188.0989499 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1496:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2304
The 40 conjugacy class representatives for t16n1496
Character table for t16n1496 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.966125.1, 8.0.4830625.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 12 siblings: data not computed
Degree 16 siblings: data not computed
Degree 24 siblings: data not computed
Degree 32 siblings: data not computed
Degree 36 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 ${\href{/LocalNumberField/2.12.0.1}{12} }{,}\,{\href{/LocalNumberField/2.4.0.1}{4} }$ ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ R

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
5Data not computed
$59$59.2.1.2$x^{2} + 177$$2$$1$$1$$C_2$$[\ ]_{2}$
59.2.1.2$x^{2} + 177$$2$$1$$1$$C_2$$[\ ]_{2}$
59.4.0.1$x^{4} - x + 14$$1$$4$$0$$C_4$$[\ ]^{4}$
59.4.0.1$x^{4} - x + 14$$1$$4$$0$$C_4$$[\ ]^{4}$
59.4.2.1$x^{4} + 177 x^{2} + 13924$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
131Data not computed