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Properties

Label 16.0.22138542742...0625.3
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 41^{6}\cdot 661^{4}$
Root discriminant $68.25$
Ramified primes $5, 41, 661$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 16T852

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5616899611, 3769753589, 90662379, -236712984, 51478827, 26325565, 545230, 286175, 111358, 2417, 6344, 741, 524, 109, -33, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 33*x^14 + 109*x^13 + 524*x^12 + 741*x^11 + 6344*x^10 + 2417*x^9 + 111358*x^8 + 286175*x^7 + 545230*x^6 + 26325565*x^5 + 51478827*x^4 - 236712984*x^3 + 90662379*x^2 + 3769753589*x + 5616899611)
 
gp: K = bnfinit(x^16 - 2*x^15 - 33*x^14 + 109*x^13 + 524*x^12 + 741*x^11 + 6344*x^10 + 2417*x^9 + 111358*x^8 + 286175*x^7 + 545230*x^6 + 26325565*x^5 + 51478827*x^4 - 236712984*x^3 + 90662379*x^2 + 3769753589*x + 5616899611, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 33 x^{14} + 109 x^{13} + 524 x^{12} + 741 x^{11} + 6344 x^{10} + 2417 x^{9} + 111358 x^{8} + 286175 x^{7} + 545230 x^{6} + 26325565 x^{5} + 51478827 x^{4} - 236712984 x^{3} + 90662379 x^{2} + 3769753589 x + 5616899611 \)

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:  \(221385427428590205586181640625=5^{12}\cdot 41^{6}\cdot 661^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $68.25$
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:  $5, 41, 661$
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}{75765118594025939044622205507882449240821643195196205777552441} a^{15} + \frac{18267700540643413159315358233805268849812646809817464666819533}{75765118594025939044622205507882449240821643195196205777552441} a^{14} + \frac{30740937293000272375178730624256551096512679324986036277387794}{75765118594025939044622205507882449240821643195196205777552441} a^{13} + \frac{13331077203612127775026697514301498290077467073022261770157802}{75765118594025939044622205507882449240821643195196205777552441} a^{12} + \frac{35579702124764052166161673013235647478778719088336111460191615}{75765118594025939044622205507882449240821643195196205777552441} a^{11} - \frac{33106035801279910325020723356037564573389362823976416592851370}{75765118594025939044622205507882449240821643195196205777552441} a^{10} - \frac{24117995570962242694339216499337413582566367594774025915344916}{75765118594025939044622205507882449240821643195196205777552441} a^{9} + \frac{774631064141621854588808799408163734228813322790945786690037}{75765118594025939044622205507882449240821643195196205777552441} a^{8} - \frac{35772283191688853631393151852057352637267949482635266372370430}{75765118594025939044622205507882449240821643195196205777552441} a^{7} - \frac{14664309678238307978097430598181274192346094898748288057248358}{75765118594025939044622205507882449240821643195196205777552441} a^{6} + \frac{11225312997163152855655061824234275507466723882221745825229008}{75765118594025939044622205507882449240821643195196205777552441} a^{5} - \frac{1444768513843066662927380189165910441049186474658496092050668}{75765118594025939044622205507882449240821643195196205777552441} a^{4} + \frac{1019640935874423163766131024099638197455309249245486766096613}{75765118594025939044622205507882449240821643195196205777552441} a^{3} - \frac{21587546534117329856657891378922872550202885351908902182275900}{75765118594025939044622205507882449240821643195196205777552441} a^{2} - \frac{36197046434197460529529777317312375743797601841703372880932513}{75765118594025939044622205507882449240821643195196205777552441} a + \frac{185264079171196065861361698276751569891956965710622941250094}{75765118594025939044622205507882449240821643195196205777552441}$

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

Class group and class number

$C_{2}\times C_{2}$, 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:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{127066799206097184585371970332918257474274803664}{941237358447060535109492455619916777937088511432387911} a^{15} - \frac{682057823297135239218734566771276778166070190270}{941237358447060535109492455619916777937088511432387911} a^{14} - \frac{1884495277620760309335223341000715216680186887710}{941237358447060535109492455619916777937088511432387911} a^{13} + \frac{19920211688665459773122388582360315865920321205312}{941237358447060535109492455619916777937088511432387911} a^{12} - \frac{1243418817499823882838727098232676542938992829670}{941237358447060535109492455619916777937088511432387911} a^{11} + \frac{106146588302281212500269831476925574258686825448892}{941237358447060535109492455619916777937088511432387911} a^{10} + \frac{455923915485722785348762224937488578309631972339340}{941237358447060535109492455619916777937088511432387911} a^{9} - \frac{1294872282436999733152075438883442935539100742337885}{941237358447060535109492455619916777937088511432387911} a^{8} + \frac{17873089722416504615177798853958814412104049476132979}{941237358447060535109492455619916777937088511432387911} a^{7} - \frac{28504800189325868007946366037052140878297467928202242}{941237358447060535109492455619916777937088511432387911} a^{6} + \frac{152286802617076086216611108028688405814320609370412668}{941237358447060535109492455619916777937088511432387911} a^{5} + \frac{2770439358297438781404267356309323608717867172608348466}{941237358447060535109492455619916777937088511432387911} a^{4} - \frac{2912606783515690862617274892694854722564844847837387735}{941237358447060535109492455619916777937088511432387911} a^{3} - \frac{20066958664449566344201009431222010981030143158754010654}{941237358447060535109492455619916777937088511432387911} a^{2} + \frac{74619438643119003892437782534625606532326521637655342934}{941237358447060535109492455619916777937088511432387911} a + \frac{202135835767608252202211753942435349451186140790779333944}{941237358447060535109492455619916777937088511432387911} \) (order $10$)
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:  \( 98425491.4374 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T852:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 41 conjugacy class representatives for t16n852
Character table for t16n852 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 8.0.26265625.2

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 ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
5Data not computed
41Data not computed
661Data not computed