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Properties

Label 16.4.16176501932...5625.1
Degree $16$
Signature $[4, 6]$
Discriminant $5^{12}\cdot 13^{2}\cdot 29^{8}\cdot 97^{8}$
Root discriminant $244.37$
Ramified primes $5, 13, 29, 97$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group 16T1220

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-233420339, 2686818563, -4522732490, 4001565485, -2845882150, 1072701854, -456607633, 87696510, -25428140, 2331975, -460598, -5424, 2815, -910, 150, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 150*x^14 - 910*x^13 + 2815*x^12 - 5424*x^11 - 460598*x^10 + 2331975*x^9 - 25428140*x^8 + 87696510*x^7 - 456607633*x^6 + 1072701854*x^5 - 2845882150*x^4 + 4001565485*x^3 - 4522732490*x^2 + 2686818563*x - 233420339)
 
gp: K = bnfinit(x^16 - 8*x^15 + 150*x^14 - 910*x^13 + 2815*x^12 - 5424*x^11 - 460598*x^10 + 2331975*x^9 - 25428140*x^8 + 87696510*x^7 - 456607633*x^6 + 1072701854*x^5 - 2845882150*x^4 + 4001565485*x^3 - 4522732490*x^2 + 2686818563*x - 233420339, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 150 x^{14} - 910 x^{13} + 2815 x^{12} - 5424 x^{11} - 460598 x^{10} + 2331975 x^{9} - 25428140 x^{8} + 87696510 x^{7} - 456607633 x^{6} + 1072701854 x^{5} - 2845882150 x^{4} + 4001565485 x^{3} - 4522732490 x^{2} + 2686818563 x - 233420339 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(161765019327832493697466398205322265625=5^{12}\cdot 13^{2}\cdot 29^{8}\cdot 97^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $244.37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{5} a^{4} - \frac{2}{5} a^{3} - \frac{1}{5} a^{2} + \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{5} + \frac{2}{5}$, $\frac{1}{5} a^{6} + \frac{2}{5} a$, $\frac{1}{5} a^{7} + \frac{2}{5} a^{2}$, $\frac{1}{25} a^{8} + \frac{1}{25} a^{7} + \frac{2}{25} a^{6} - \frac{2}{25} a^{5} + \frac{7}{25} a^{3} + \frac{7}{25} a^{2} - \frac{11}{25} a + \frac{11}{25}$, $\frac{1}{25} a^{9} + \frac{1}{25} a^{7} + \frac{1}{25} a^{6} + \frac{2}{25} a^{5} + \frac{2}{25} a^{4} + \frac{2}{5} a^{3} + \frac{12}{25} a^{2} - \frac{3}{25} a + \frac{9}{25}$, $\frac{1}{25} a^{10} - \frac{1}{25} a^{5} - \frac{6}{25}$, $\frac{1}{25} a^{11} - \frac{1}{25} a^{6} - \frac{6}{25} a$, $\frac{1}{250} a^{12} + \frac{2}{125} a^{11} + \frac{2}{125} a^{10} - \frac{1}{50} a^{9} - \frac{8}{125} a^{7} + \frac{1}{250} a^{6} - \frac{2}{125} a^{5} - \frac{1}{25} a^{4} + \frac{3}{10} a^{3} + \frac{32}{125} a^{2} - \frac{7}{125} a + \frac{51}{250}$, $\frac{1}{250} a^{13} - \frac{1}{125} a^{11} - \frac{1}{250} a^{10} + \frac{2}{125} a^{8} + \frac{3}{50} a^{7} + \frac{1}{125} a^{6} + \frac{3}{125} a^{5} - \frac{1}{10} a^{4} - \frac{48}{125} a^{3} - \frac{12}{25} a^{2} - \frac{63}{250} a + \frac{33}{125}$, $\frac{1}{7521193316884234026384796750} a^{14} - \frac{7}{7521193316884234026384796750} a^{13} - \frac{1891467734783053141158397}{3760596658442117013192398375} a^{12} + \frac{4539522563479327538780171}{1504238663376846805276959350} a^{11} - \frac{62826993379538642033040341}{7521193316884234026384796750} a^{10} + \frac{53036758035778682318888517}{3760596658442117013192398375} a^{9} - \frac{57575114445554206815430293}{7521193316884234026384796750} a^{8} - \frac{156466897655764345932031341}{7521193316884234026384796750} a^{7} + \frac{5008474132559926689500321}{150423866337684680527695935} a^{6} - \frac{7801951579635875817184619}{7521193316884234026384796750} a^{5} + \frac{285814768686357469026019839}{7521193316884234026384796750} a^{4} - \frac{414240580159573385284933879}{3760596658442117013192398375} a^{3} - \frac{2478902571041122369036873631}{7521193316884234026384796750} a^{2} + \frac{586165603937404102130806201}{1504238663376846805276959350} a - \frac{790865351904776319457230612}{3760596658442117013192398375}$, $\frac{1}{107395119371789977662748512793250} a^{15} + \frac{3566}{53697559685894988831374256396625} a^{14} + \frac{14677586419088458713220756489}{107395119371789977662748512793250} a^{13} - \frac{4896503180361687897541439707}{4295804774871599106509940511730} a^{12} - \frac{900588181944920085388064472497}{53697559685894988831374256396625} a^{11} + \frac{659125010850649957680082412853}{107395119371789977662748512793250} a^{10} + \frac{135048144947738633279724428273}{107395119371789977662748512793250} a^{9} - \frac{524461116324857609183254289547}{53697559685894988831374256396625} a^{8} - \frac{549279751402762884847828681753}{21479023874357995532549702558650} a^{7} - \frac{5455021785400761023722814717171}{107395119371789977662748512793250} a^{6} - \frac{11491886692624953496072087978}{429580477487159910650994051173} a^{5} - \frac{7184689179067913154713184363207}{107395119371789977662748512793250} a^{4} - \frac{38847738555188825104718305809819}{107395119371789977662748512793250} a^{3} - \frac{2671631796003067763622257792363}{10739511937178997766274851279325} a^{2} - \frac{49303060463347842344968101430791}{107395119371789977662748512793250} a + \frac{189526818034952744777250275597}{492638162255917328728204187125}$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  $9$
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:  \( 2058750769770 \) (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

\(\Q(\sqrt{2813}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{14065}) \), 4.4.725.1, \(\Q(\sqrt{5}, \sqrt{2813})\), 4.4.6821525.1, 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 ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ 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$5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$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}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.0.1$x^{4} + x^{2} - x + 2$$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}$