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Properties

Label 16.12.2588240309...5625.1
Degree $16$
Signature $[12, 2]$
Discriminant $5^{8}\cdot 13^{2}\cdot 29^{8}\cdot 97^{8}$
Root discriminant $163.42$
Ramified primes $5, 13, 29, 97$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1220

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-167749, 1110642, 5607282, 9483892, -1078209, -13679900, 2828733, 3307352, -699575, -285532, 57458, 8554, -1456, 36, -21, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 21*x^14 + 36*x^13 - 1456*x^12 + 8554*x^11 + 57458*x^10 - 285532*x^9 - 699575*x^8 + 3307352*x^7 + 2828733*x^6 - 13679900*x^5 - 1078209*x^4 + 9483892*x^3 + 5607282*x^2 + 1110642*x - 167749)
 
gp: K = bnfinit(x^16 - 4*x^15 - 21*x^14 + 36*x^13 - 1456*x^12 + 8554*x^11 + 57458*x^10 - 285532*x^9 - 699575*x^8 + 3307352*x^7 + 2828733*x^6 - 13679900*x^5 - 1078209*x^4 + 9483892*x^3 + 5607282*x^2 + 1110642*x - 167749, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 21 x^{14} + 36 x^{13} - 1456 x^{12} + 8554 x^{11} + 57458 x^{10} - 285532 x^{9} - 699575 x^{8} + 3307352 x^{7} + 2828733 x^{6} - 13679900 x^{5} - 1078209 x^{4} + 9483892 x^{3} + 5607282 x^{2} + 1110642 x - 167749 \)

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:  \(258824030924531989915946237128515625=5^{8}\cdot 13^{2}\cdot 29^{8}\cdot 97^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $163.42$
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:  $5, 13, 29, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{413171396} a^{14} - \frac{12140696}{103292849} a^{13} + \frac{42005649}{206585698} a^{12} + \frac{6053545}{206585698} a^{11} - \frac{19197151}{206585698} a^{10} + \frac{4085953}{103292849} a^{9} - \frac{19027083}{103292849} a^{8} - \frac{21707511}{103292849} a^{7} - \frac{187127717}{413171396} a^{6} - \frac{33397800}{103292849} a^{5} - \frac{45781423}{103292849} a^{4} - \frac{16985387}{206585698} a^{3} + \frac{110185265}{413171396} a^{2} + \frac{100979617}{206585698} a + \frac{57391233}{413171396}$, $\frac{1}{197553093861911582288317045331644441652152} a^{15} + \frac{117840420965540117919384477987903}{197553093861911582288317045331644441652152} a^{14} - \frac{58055400194080851701779072155684382385}{1073658118814736860262592637671980661153} a^{13} - \frac{5811274184485515785055663847298126305983}{49388273465477895572079261332911110413038} a^{12} - \frac{26351212824334184608658242834661226948}{1073658118814736860262592637671980661153} a^{11} - \frac{8623101673195206161895973829651763779}{82382441143416005958430794550310442724} a^{10} - \frac{8347822171551050448069120245898458577495}{49388273465477895572079261332911110413038} a^{9} - \frac{2825079202036695210467286156695583861126}{24694136732738947786039630666455555206519} a^{8} + \frac{3816629051668809561794197021446833796367}{17959372169264689298937913211967676513832} a^{7} + \frac{39115563412373440831872904935588470808123}{197553093861911582288317045331644441652152} a^{6} - \frac{39773834776732919672403359354078479652711}{98776546930955791144158522665822220826076} a^{5} + \frac{229000304593198428450280017177986711117}{906206852577576065542738740053414869964} a^{4} - \frac{19112802707138180115969549615301866194087}{197553093861911582288317045331644441652152} a^{3} - \frac{90053818471774537994881922772942340139705}{197553093861911582288317045331644441652152} a^{2} + \frac{38979996653375045681580588545038711010711}{197553093861911582288317045331644441652152} a + \frac{37634034211965531488867119347601799834919}{197553093861911582288317045331644441652152}$

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:  \( 775367962622 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1220:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 55 conjugacy class representatives for t16n1220 are not computed
Character table for t16n1220 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{14065}) \), \(\Q(\sqrt{2813}) \), 4.4.725.1, 4.4.6821525.1, \(\Q(\sqrt{5}, \sqrt{2813})\), 8.8.39134423996850625.3

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$13$13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$29$29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$97$97.8.4.1$x^{8} + 432814 x^{4} - 912673 x^{2} + 46831989649$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
97.8.4.1$x^{8} + 432814 x^{4} - 912673 x^{2} + 46831989649$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$