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Properties

Label 15.5.8565893077528823.1
Degree $15$
Signature $[5, 5]$
Discriminant $-8.566\times 10^{15}$
Root discriminant \(11.54\)
Ramified primes $23,191$
Class number $1$
Class group trivial
Galois group $C_5\wr S_3$ (as 15T32)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 6*x^13 + 9*x^11 - 18*x^10 + x^9 + 32*x^8 - 17*x^7 - 19*x^6 + 24*x^5 + 3*x^4 - 15*x^3 + 3*x^2 + 4*x - 1)
 
gp: K = bnfinit(y^15 - 6*y^13 + 9*y^11 - 18*y^10 + y^9 + 32*y^8 - 17*y^7 - 19*y^6 + 24*y^5 + 3*y^4 - 15*y^3 + 3*y^2 + 4*y - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - 6*x^13 + 9*x^11 - 18*x^10 + x^9 + 32*x^8 - 17*x^7 - 19*x^6 + 24*x^5 + 3*x^4 - 15*x^3 + 3*x^2 + 4*x - 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^15 - 6*x^13 + 9*x^11 - 18*x^10 + x^9 + 32*x^8 - 17*x^7 - 19*x^6 + 24*x^5 + 3*x^4 - 15*x^3 + 3*x^2 + 4*x - 1)
 

\( x^{15} - 6 x^{13} + 9 x^{11} - 18 x^{10} + x^{9} + 32 x^{8} - 17 x^{7} - 19 x^{6} + 24 x^{5} + 3 x^{4} + \cdots - 1 \) Copy content Toggle raw display

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

Invariants

Degree:  $15$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[5, 5]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-8565893077528823\) \(\medspace = -\,23^{5}\cdot 191^{4}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(11.54\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $23^{1/2}191^{4/5}\approx 320.39860152359097$
Ramified primes:   \(23\), \(191\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q(\sqrt{-23}) \)
$\card{ \Aut(K/\Q) }$:  $5$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{70253969}a^{14}-\frac{20341715}{70253969}a^{13}+\frac{29826569}{70253969}a^{12}-\frac{32662361}{70253969}a^{11}+\frac{3665471}{70253969}a^{10}-\frac{24043703}{70253969}a^{9}+\frac{6808710}{70253969}a^{8}+\frac{14022021}{70253969}a^{7}+\frac{2281596}{70253969}a^{6}+\frac{22947435}{70253969}a^{5}+\frac{19824924}{70253969}a^{4}+\frac{3012647}{70253969}a^{3}-\frac{10018858}{70253969}a^{2}+\frac{31833807}{70253969}a-\frac{2514324}{70253969}$ Copy content Toggle raw display

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

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

Trivial group, which has order $1$

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 
oscar: class_group(K)
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $9$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:   $\frac{7609013}{70253969}a^{14}+\frac{9194900}{70253969}a^{13}-\frac{58826149}{70253969}a^{12}-\frac{66598332}{70253969}a^{11}+\frac{142066968}{70253969}a^{10}+\frac{3755544}{70253969}a^{9}-\frac{242980254}{70253969}a^{8}+\frac{442434353}{70253969}a^{7}+\frac{305599127}{70253969}a^{6}-\frac{689003472}{70253969}a^{5}+\frac{46774654}{70253969}a^{4}+\frac{453912246}{70253969}a^{3}-\frac{266211595}{70253969}a^{2}-\frac{171650655}{70253969}a+\frac{107089837}{70253969}$, $\frac{12326426}{70253969}a^{14}-\frac{14061450}{70253969}a^{13}-\frac{42069538}{70253969}a^{12}+\frac{75375941}{70253969}a^{11}-\frac{67284417}{70253969}a^{10}-\frac{304965737}{70253969}a^{9}+\frac{474285587}{70253969}a^{8}-\frac{220465614}{70253969}a^{7}-\frac{355376091}{70253969}a^{6}+\frac{730032657}{70253969}a^{5}-\frac{33826379}{70253969}a^{4}-\frac{375164088}{70253969}a^{3}+\frac{281474677}{70253969}a^{2}+\frac{56053399}{70253969}a-\frac{160555643}{70253969}$, $\frac{31215888}{70253969}a^{14}-\frac{29440847}{70253969}a^{13}-\frac{157352401}{70253969}a^{12}+\frac{149178330}{70253969}a^{11}+\frac{119496142}{70253969}a^{10}-\frac{680649843}{70253969}a^{9}+\frac{736876718}{70253969}a^{8}+\frac{281405707}{70253969}a^{7}-\frac{866543200}{70253969}a^{6}+\frac{409843759}{70253969}a^{5}+\frac{358366909}{70253969}a^{4}-\frac{495123430}{70253969}a^{3}+\frac{77470264}{70253969}a^{2}+\frac{222166882}{70253969}a-\frac{105768478}{70253969}$, $\frac{44594302}{70253969}a^{14}+\frac{5121714}{70253969}a^{13}-\frac{255911361}{70253969}a^{12}-\frac{24841032}{70253969}a^{11}+\frac{335137446}{70253969}a^{10}-\frac{784918229}{70253969}a^{9}+\frac{24904886}{70253969}a^{8}+\frac{1237976291}{70253969}a^{7}-\frac{604871607}{70253969}a^{6}-\frac{578033933}{70253969}a^{5}+\frac{805229885}{70253969}a^{4}+\frac{225218694}{70253969}a^{3}-\frac{491301104}{70253969}a^{2}-\frac{22288137}{70253969}a+\frac{108202462}{70253969}$, $\frac{4796110}{70253969}a^{14}+\frac{21989898}{70253969}a^{13}-\frac{47355024}{70253969}a^{12}-\frac{116647448}{70253969}a^{11}+\frac{140693033}{70253969}a^{10}+\frac{25400650}{70253969}a^{9}-\frac{479022325}{70253969}a^{8}+\frac{613567029}{70253969}a^{7}+\frac{237942027}{70253969}a^{6}-\frac{742817860}{70253969}a^{5}+\frac{302823257}{70253969}a^{4}+\frac{274122754}{70253969}a^{3}-\frac{289135295}{70253969}a^{2}-\frac{84736883}{70253969}a+\frac{119299210}{70253969}$, $\frac{15450992}{70253969}a^{14}+\frac{1638036}{70253969}a^{13}-\frac{80236558}{70253969}a^{12}-\frac{4899062}{70253969}a^{11}+\frac{66495820}{70253969}a^{10}-\frac{289869516}{70253969}a^{9}+\frac{79893022}{70253969}a^{8}+\frac{307930678}{70253969}a^{7}-\frac{239085092}{70253969}a^{6}+\frac{23443591}{70253969}a^{5}+\frac{201125801}{70253969}a^{4}-\frac{68560382}{70253969}a^{3}-\frac{186814086}{70253969}a^{2}+\frac{53174519}{70253969}a+\frac{29006305}{70253969}$, $\frac{11954488}{70253969}a^{14}+\frac{52302672}{70253969}a^{13}-\frac{83007377}{70253969}a^{12}-\frac{289241735}{70253969}a^{11}+\frac{164047106}{70253969}a^{10}+\frac{115142543}{70253969}a^{9}-\frac{974207199}{70253969}a^{8}+\frac{814285938}{70253969}a^{7}+\frac{964887823}{70253969}a^{6}-\frac{1175717664}{70253969}a^{5}+\frac{109923180}{70253969}a^{4}+\frac{922313018}{70253969}a^{3}-\frac{367827938}{70253969}a^{2}-\frac{245058997}{70253969}a+\frac{142518786}{70253969}$, $\frac{27123610}{70253969}a^{14}-\frac{4532177}{70253969}a^{13}-\frac{156031704}{70253969}a^{12}+\frac{24948575}{70253969}a^{11}+\frac{199100270}{70253969}a^{10}-\frac{517059731}{70253969}a^{9}+\frac{186332800}{70253969}a^{8}+\frac{731675503}{70253969}a^{7}-\frac{547401005}{70253969}a^{6}-\frac{149550096}{70253969}a^{5}+\frac{541431237}{70253969}a^{4}-\frac{74635548}{70253969}a^{3}-\frac{256913581}{70253969}a^{2}+\frac{71380926}{70253969}a+\frac{21483761}{70253969}$, $\frac{39989981}{70253969}a^{14}+\frac{9360189}{70253969}a^{13}-\frac{230140221}{70253969}a^{12}-\frac{64855497}{70253969}a^{11}+\frac{300502187}{70253969}a^{10}-\frac{587294417}{70253969}a^{9}-\frac{32944154}{70253969}a^{8}+\frac{1063639294}{70253969}a^{7}-\frac{237485570}{70253969}a^{6}-\frac{672540491}{70253969}a^{5}+\frac{419445446}{70253969}a^{4}+\frac{256026243}{70253969}a^{3}-\frac{333124590}{70253969}a^{2}-\frac{54970445}{70253969}a+\frac{132968770}{70253969}$ Copy content Toggle raw display
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 87.8024975924 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{5}\cdot(2\pi)^{5}\cdot 87.8024975924 \cdot 1}{2\cdot\sqrt{8565893077528823}}\cr\approx \mathstrut & 0.148641428275 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^15 - 6*x^13 + 9*x^11 - 18*x^10 + x^9 + 32*x^8 - 17*x^7 - 19*x^6 + 24*x^5 + 3*x^4 - 15*x^3 + 3*x^2 + 4*x - 1)
 
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
 
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
 
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^15 - 6*x^13 + 9*x^11 - 18*x^10 + x^9 + 32*x^8 - 17*x^7 - 19*x^6 + 24*x^5 + 3*x^4 - 15*x^3 + 3*x^2 + 4*x - 1, 1);
 
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^15 - 6*x^13 + 9*x^11 - 18*x^10 + x^9 + 32*x^8 - 17*x^7 - 19*x^6 + 24*x^5 + 3*x^4 - 15*x^3 + 3*x^2 + 4*x - 1);
 
OK := Integers(K); DK := Discriminant(OK);
 
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
 
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
 
hK := #clK; wK := #TorsionSubgroup(UK);
 
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^15 - 6*x^13 + 9*x^11 - 18*x^10 + x^9 + 32*x^8 - 17*x^7 - 19*x^6 + 24*x^5 + 3*x^4 - 15*x^3 + 3*x^2 + 4*x - 1);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

$C_5\wr S_3$ (as 15T32):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A solvable group of order 750
The 65 conjugacy class representatives for $C_5\wr S_3$
Character table for $C_5\wr S_3$

Intermediate fields

3.1.23.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Sibling fields

Degree 15 siblings: data not computed
Degree 30 siblings: data not computed
Minimal sibling: This field is its own minimal sibling

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $15$ $15$ ${\href{/padicField/5.10.0.1}{10} }{,}\,{\href{/padicField/5.5.0.1}{5} }$ ${\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.5.0.1}{5} }$ ${\href{/padicField/11.10.0.1}{10} }{,}\,{\href{/padicField/11.5.0.1}{5} }$ $15$ ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.5.0.1}{5} }$ ${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.5.0.1}{5} }$ R $15$ ${\href{/padicField/31.3.0.1}{3} }^{5}$ ${\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.5.0.1}{5} }$ ${\href{/padicField/41.3.0.1}{3} }^{5}$ ${\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.5.0.1}{5} }$ $15$ ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.5.0.1}{5} }$ ${\href{/padicField/59.5.0.1}{5} }^{3}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
 
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(23\) Copy content Toggle raw display 23.5.0.1$x^{5} + 3 x + 18$$1$$5$$0$$C_5$$[\ ]^{5}$
23.10.5.2$x^{10} + 115 x^{8} + 5296 x^{6} + 36 x^{5} + 120980 x^{4} - 8280 x^{3} + 1383344 x^{2} + 95328 x + 6509876$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
\(191\) Copy content Toggle raw display 191.5.4.2$x^{5} + 382$$5$$1$$4$$C_5$$[\ ]_{5}$
191.10.0.1$x^{10} + 113 x^{5} + 47 x^{4} + 173 x^{3} + 74 x^{2} + 156 x + 19$$1$$10$$0$$C_{10}$$[\ ]^{10}$