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Properties

Label 16.0.25849857514...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 3^{12}\cdot 5^{14}\cdot 41^{6}$
Root discriminant $106.12$
Ramified primes $2, 3, 5, 41$
Class number $34240$ (GRH)
Class group $[2, 2, 2, 2, 2140]$ (GRH)
Galois group $(C_2\times Q_8).C_2^3$ (as 16T226)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![31721125, -275908750, 667377250, -230804400, 270742200, -41923450, 40409550, -4826910, 3081316, -328966, 152482, -11798, 5025, -282, 92, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 92*x^14 - 282*x^13 + 5025*x^12 - 11798*x^11 + 152482*x^10 - 328966*x^9 + 3081316*x^8 - 4826910*x^7 + 40409550*x^6 - 41923450*x^5 + 270742200*x^4 - 230804400*x^3 + 667377250*x^2 - 275908750*x + 31721125)
 
gp: K = bnfinit(x^16 - 4*x^15 + 92*x^14 - 282*x^13 + 5025*x^12 - 11798*x^11 + 152482*x^10 - 328966*x^9 + 3081316*x^8 - 4826910*x^7 + 40409550*x^6 - 41923450*x^5 + 270742200*x^4 - 230804400*x^3 + 667377250*x^2 - 275908750*x + 31721125, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 92 x^{14} - 282 x^{13} + 5025 x^{12} - 11798 x^{11} + 152482 x^{10} - 328966 x^{9} + 3081316 x^{8} - 4826910 x^{7} + 40409550 x^{6} - 41923450 x^{5} + 270742200 x^{4} - 230804400 x^{3} + 667377250 x^{2} - 275908750 x + 31721125 \)

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:  \(258498575149187174400000000000000=2^{24}\cdot 3^{12}\cdot 5^{14}\cdot 41^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $106.12$
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:  $2, 3, 5, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{5} a^{11} + \frac{1}{5} a^{10} + \frac{2}{5} a^{9} - \frac{2}{5} a^{8} + \frac{2}{5} a^{6} + \frac{2}{5} a^{5} - \frac{1}{5} a^{4} + \frac{1}{5} a^{3}$, $\frac{1}{5} a^{12} + \frac{1}{5} a^{10} + \frac{1}{5} a^{9} + \frac{2}{5} a^{8} + \frac{2}{5} a^{7} + \frac{2}{5} a^{5} + \frac{2}{5} a^{4} - \frac{1}{5} a^{3}$, $\frac{1}{5} a^{13} - \frac{1}{5} a^{8} - \frac{1}{5} a^{3}$, $\frac{1}{25} a^{14} + \frac{1}{25} a^{13} + \frac{2}{25} a^{12} - \frac{2}{25} a^{11} - \frac{2}{5} a^{10} - \frac{3}{25} a^{9} + \frac{12}{25} a^{8} + \frac{4}{25} a^{7} + \frac{1}{25} a^{6} - \frac{1}{5} a^{5} - \frac{2}{5} a^{4} - \frac{2}{5} a^{3}$, $\frac{1}{23057803679357974049200809278462843753383564559670115833885025} a^{15} - \frac{369486344389816720958926557458422577180901839835809716081008}{23057803679357974049200809278462843753383564559670115833885025} a^{14} + \frac{1272045719155306836941901087889871622188353925041803476862078}{23057803679357974049200809278462843753383564559670115833885025} a^{13} - \frac{178630390792622052575726902405188464525630575610243179911842}{4611560735871594809840161855692568750676712911934023166777005} a^{12} - \frac{395868487176907083595230175108604876618010301917937385164087}{23057803679357974049200809278462843753383564559670115833885025} a^{11} + \frac{11334419633676243651311428701617579452280548585816358145177027}{23057803679357974049200809278462843753383564559670115833885025} a^{10} + \frac{867723642113002769977431906198995383397735686639151248383784}{23057803679357974049200809278462843753383564559670115833885025} a^{9} + \frac{6263953425738379018038165080835489878872506462825968664464096}{23057803679357974049200809278462843753383564559670115833885025} a^{8} - \frac{1204667700132685804081707643650527742995767465582315126678583}{4611560735871594809840161855692568750676712911934023166777005} a^{7} + \frac{6285010265280654752066307261701860894925000136999740666974321}{23057803679357974049200809278462843753383564559670115833885025} a^{6} + \frac{761210998888249724179421057893042814391369959129127279049238}{4611560735871594809840161855692568750676712911934023166777005} a^{5} + \frac{1068038098357093119377267823037682812853013963839567779992294}{4611560735871594809840161855692568750676712911934023166777005} a^{4} - \frac{158937874033201211415158389251623998180066961707696359991515}{922312147174318961968032371138513750135342582386804633355401} a^{3} - \frac{137506495301604645675345937119561128977017106491642940834718}{922312147174318961968032371138513750135342582386804633355401} a^{2} - \frac{40518189332256014597140888402551483637133097111997147890336}{922312147174318961968032371138513750135342582386804633355401} a + \frac{142763285997492330519983921368863496640814392026402835103195}{922312147174318961968032371138513750135342582386804633355401}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2140}$, which has order $34240$ (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:  \( -1 \) (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:  \( 118606.845528 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times Q_8).C_2^3$ (as 16T226):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 23 conjugacy class representatives for $(C_2\times Q_8).C_2^3$
Character table for $(C_2\times Q_8).C_2^3$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.16400.1, \(\Q(\zeta_{15})^+\), 4.4.738000.1, 8.8.544644000000.3

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 R R R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
3Data not computed
$5$5.8.7.1$x^{8} - 5$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
5.8.7.1$x^{8} - 5$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$41$41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$