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Properties

Label 16.12.2610360346...5625.1
Degree $16$
Signature $[12, 2]$
Discriminant $5^{8}\cdot 11^{4}\cdot 19\cdot 29^{10}\cdot 571$
Root discriminant $59.71$
Ramified primes $5, 11, 19, 29, 571$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1722

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-32741, 114347, 532701, -1953614, 852831, 638197, -378007, 71146, -1725, -22020, 7911, -171, -329, 160, -23, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 - 23*x^14 + 160*x^13 - 329*x^12 - 171*x^11 + 7911*x^10 - 22020*x^9 - 1725*x^8 + 71146*x^7 - 378007*x^6 + 638197*x^5 + 852831*x^4 - 1953614*x^3 + 532701*x^2 + 114347*x - 32741)
 
gp: K = bnfinit(x^16 - 5*x^15 - 23*x^14 + 160*x^13 - 329*x^12 - 171*x^11 + 7911*x^10 - 22020*x^9 - 1725*x^8 + 71146*x^7 - 378007*x^6 + 638197*x^5 + 852831*x^4 - 1953614*x^3 + 532701*x^2 + 114347*x - 32741, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} - 23 x^{14} + 160 x^{13} - 329 x^{12} - 171 x^{11} + 7911 x^{10} - 22020 x^{9} - 1725 x^{8} + 71146 x^{7} - 378007 x^{6} + 638197 x^{5} + 852831 x^{4} - 1953614 x^{3} + 532701 x^{2} + 114347 x - 32741 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(26103603462974877571097265625=5^{8}\cdot 11^{4}\cdot 19\cdot 29^{10}\cdot 571\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $59.71$
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:  $5, 11, 19, 29, 571$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{377} a^{14} - \frac{97}{377} a^{13} - \frac{175}{377} a^{12} + \frac{126}{377} a^{11} + \frac{142}{377} a^{10} - \frac{175}{377} a^{9} + \frac{54}{377} a^{8} + \frac{155}{377} a^{7} - \frac{155}{377} a^{6} + \frac{11}{377} a^{5} + \frac{59}{377} a^{4} - \frac{5}{13} a^{3} + \frac{2}{13} a^{2} - \frac{5}{13} a + \frac{1}{13}$, $\frac{1}{1754343961248584224130885462839325142665745781} a^{15} + \frac{1878933056481383993147964020888982849337941}{1754343961248584224130885462839325142665745781} a^{14} + \frac{504886871940923661670932223838473365754599220}{1754343961248584224130885462839325142665745781} a^{13} - \frac{296883814547599061161652779680998915820207774}{1754343961248584224130885462839325142665745781} a^{12} - \frac{53917998106783418675995238727903743730986304}{134949535480660324933145035603025010974288137} a^{11} + \frac{638641398432923044078370100798128319143985951}{1754343961248584224130885462839325142665745781} a^{10} - \frac{88439375746932079981837206280187373857876020}{1754343961248584224130885462839325142665745781} a^{9} - \frac{77754257783227267397621797224451129862815338}{1754343961248584224130885462839325142665745781} a^{8} + \frac{60411117295688456847043873906904094578965947}{1754343961248584224130885462839325142665745781} a^{7} + \frac{108659726185396881571810626597182631431277442}{1754343961248584224130885462839325142665745781} a^{6} - \frac{160333308984442488372568298447873615198825022}{1754343961248584224130885462839325142665745781} a^{5} - \frac{694608562395102426683607576278195719023634449}{1754343961248584224130885462839325142665745781} a^{4} - \frac{1028363218763487150282841419730264197844171}{4653432257953804308039483986311207274975453} a^{3} + \frac{28160366807770247722983200444116138751651790}{60494619353399456004513291822045694574680889} a^{2} - \frac{2497815401957836206021507380790248503182736}{60494619353399456004513291822045694574680889} a + \frac{8716088570066372792649985285756556756148953}{60494619353399456004513291822045694574680889}$

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:  $13$
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:  \( 251138060.989 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1722:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 8192
The 104 conjugacy class representatives for t16n1722 are not computed
Character table for t16n1722 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.725.1, 8.8.1844418125.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ $16$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ $16$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
5Data not computed
$11$11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.8.0.1$x^{8} - x + 2$$1$$8$$0$$C_8$$[\ ]^{8}$
$29$29.4.3.4$x^{4} + 232$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.3.4$x^{4} + 232$$4$$1$$3$$C_4$$[\ ]_{4}$
29.8.4.2$x^{8} - 24389 x^{2} + 13438339$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
571Data not computed