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Properties

Label 15.15.239...032.1
Degree $15$
Signature $[15, 0]$
Discriminant $2.398\times 10^{25}$
Root discriminant \(49.20\)
Ramified primes $2,3,37,53$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_3\wr F_5$ (as 15T56)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 24*x^13 - 27*x^12 + 167*x^11 + 341*x^10 - 181*x^9 - 864*x^8 - 357*x^7 + 621*x^6 + 583*x^5 + 16*x^4 - 170*x^3 - 81*x^2 - 15*x - 1)
 
gp: K = bnfinit(y^15 - 24*y^13 - 27*y^12 + 167*y^11 + 341*y^10 - 181*y^9 - 864*y^8 - 357*y^7 + 621*y^6 + 583*y^5 + 16*y^4 - 170*y^3 - 81*y^2 - 15*y - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - 24*x^13 - 27*x^12 + 167*x^11 + 341*x^10 - 181*x^9 - 864*x^8 - 357*x^7 + 621*x^6 + 583*x^5 + 16*x^4 - 170*x^3 - 81*x^2 - 15*x - 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^15 - 24*x^13 - 27*x^12 + 167*x^11 + 341*x^10 - 181*x^9 - 864*x^8 - 357*x^7 + 621*x^6 + 583*x^5 + 16*x^4 - 170*x^3 - 81*x^2 - 15*x - 1)
 

\( x^{15} - 24 x^{13} - 27 x^{12} + 167 x^{11} + 341 x^{10} - 181 x^{9} - 864 x^{8} - 357 x^{7} + 621 x^{6} + \cdots - 1 \) 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:  $[15, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(23980112932460739050668032\) \(\medspace = 2^{16}\cdot 3^{4}\cdot 37^{2}\cdot 53^{9}\) 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:  \(49.20\)
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:  $2^{4/3}3^{4/3}37^{2/3}53^{3/4}\approx 2377.988328757685$
Ramified primes:   \(2\), \(3\), \(37\), \(53\) 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:  \(\Q(\sqrt{53}) \)
$\card{ \Aut(K/\Q) }$:  $3$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{111}a^{14}+\frac{10}{111}a^{13}-\frac{35}{111}a^{12}-\frac{44}{111}a^{11}-\frac{17}{37}a^{10}+\frac{53}{111}a^{9}+\frac{16}{111}a^{8}-\frac{38}{111}a^{7}+\frac{40}{111}a^{6}+\frac{22}{111}a^{5}+\frac{26}{111}a^{4}+\frac{18}{37}a^{3}+\frac{1}{3}a^{2}-\frac{44}{111}a-\frac{11}{111}$ 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 $1$ (assuming GRH)

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:  $14$
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:   $a$, $a^{14}-a^{13}-23a^{12}-4a^{11}+171a^{10}+170a^{9}-351a^{8}-513a^{7}+156a^{6}+465a^{5}+118a^{4}-102a^{3}-68a^{2}-13a-1$, $\frac{5593}{111}a^{14}-\frac{2123}{111}a^{13}-\frac{133262}{111}a^{12}-\frac{100571}{111}a^{11}+\frac{322797}{37}a^{10}+\frac{1538519}{111}a^{9}-\frac{1567853}{111}a^{8}-\frac{4205759}{111}a^{7}-\frac{458375}{111}a^{6}+\frac{3552946}{111}a^{5}+\frac{1937513}{111}a^{4}-\frac{185743}{37}a^{3}-\frac{19454}{3}a^{2}-\frac{202469}{111}a-\frac{18455}{111}$, $\frac{4150}{111}a^{14}-\frac{1901}{111}a^{13}-\frac{98630}{111}a^{12}-\frac{67049}{111}a^{11}+\frac{240546}{37}a^{10}+\frac{1085861}{111}a^{9}-\frac{1232855}{111}a^{8}-\frac{3016283}{111}a^{7}-\frac{136808}{111}a^{6}+\frac{2618326}{111}a^{5}+\frac{1250312}{111}a^{4}-\frac{161471}{37}a^{3}-\frac{13214}{3}a^{2}-\frac{118664}{111}a-\frac{9131}{111}$, $\frac{5593}{111}a^{14}-\frac{2123}{111}a^{13}-\frac{133262}{111}a^{12}-\frac{100571}{111}a^{11}+\frac{322797}{37}a^{10}+\frac{1538519}{111}a^{9}-\frac{1567853}{111}a^{8}-\frac{4205759}{111}a^{7}-\frac{458375}{111}a^{6}+\frac{3552946}{111}a^{5}+\frac{1937513}{111}a^{4}-\frac{185743}{37}a^{3}-\frac{19454}{3}a^{2}-\frac{202580}{111}a-\frac{18344}{111}$, $\frac{9359}{111}a^{14}-\frac{5200}{111}a^{13}-\frac{221782}{111}a^{12}-\frac{129412}{111}a^{11}+\frac{545377}{37}a^{10}+\frac{2282572}{111}a^{9}-\frac{2971798}{111}a^{8}-\frac{6444436}{111}a^{7}+\frac{260030}{111}a^{6}+\frac{5697179}{111}a^{5}+\frac{2279851}{111}a^{4}-\frac{382061}{37}a^{3}-\frac{25927}{3}a^{2}-\frac{217102}{111}a-\frac{16147}{111}$, $\frac{3160}{111}a^{14}-\frac{2144}{111}a^{13}-\frac{74636}{111}a^{12}-\frac{34478}{111}a^{11}+\frac{185670}{37}a^{10}+\frac{701612}{111}a^{9}-\frac{1092518}{111}a^{8}-\frac{2039159}{111}a^{7}+\frac{349066}{111}a^{6}+\frac{1878043}{111}a^{5}+\frac{522053}{111}a^{4}-\frac{150838}{37}a^{3}-\frac{6983}{3}a^{2}-\frac{39251}{111}a-\frac{1571}{111}$, $a^{13}-a^{12}-23a^{11}-4a^{10}+171a^{9}+170a^{8}-351a^{7}-513a^{6}+156a^{5}+465a^{4}+118a^{3}-102a^{2}-69a-13$, $\frac{8773}{111}a^{14}-\frac{3623}{111}a^{13}-\frac{208820}{111}a^{12}-\frac{150914}{111}a^{11}+\frac{507350}{37}a^{10}+\frac{2363069}{111}a^{9}-\frac{2523632}{111}a^{8}-\frac{6504863}{111}a^{7}-\frac{533417}{111}a^{6}+\frac{5562964}{111}a^{5}+\frac{2871230}{111}a^{4}-\frac{314798}{37}a^{3}-\frac{29486}{3}a^{2}-\frac{287000}{111}a-\frac{23909}{111}$, $\frac{3710}{111}a^{14}-\frac{2194}{111}a^{13}-\frac{87448}{111}a^{12}-\frac{48688}{111}a^{11}+\frac{213838}{37}a^{10}+\frac{883054}{111}a^{9}-\frac{1142992}{111}a^{8}-\frac{2467429}{111}a^{7}+\frac{36068}{111}a^{6}+\frac{2099489}{111}a^{5}+\frac{950494}{111}a^{4}-\frac{110746}{37}a^{3}-\frac{10372}{3}a^{2}-\frac{115954}{111}a-\frac{11728}{111}$, $\frac{498}{37}a^{14}-\frac{126}{37}a^{13}-\frac{11991}{37}a^{12}-\frac{10294}{37}a^{11}+\frac{87304}{37}a^{10}+\frac{147014}{37}a^{9}-\frac{138589}{37}a^{8}-\frac{399802}{37}a^{7}-\frac{51342}{37}a^{6}+\frac{339812}{37}a^{5}+\frac{185701}{37}a^{4}-\frac{55396}{37}a^{3}-1854a^{2}-\frac{18360}{37}a-\frac{1519}{37}$, $\frac{10922}{111}a^{14}-\frac{2779}{111}a^{13}-\frac{261280}{111}a^{12}-\frac{228598}{111}a^{11}+\frac{626328}{37}a^{10}+\frac{3246640}{111}a^{9}-\frac{2779513}{111}a^{8}-\frac{8711842}{111}a^{7}-\frac{1733725}{111}a^{6}+\frac{7165796}{111}a^{5}+\frac{4576564}{111}a^{4}-\frac{311266}{37}a^{3}-\frac{43639}{3}a^{2}-\frac{488782}{111}a-\frac{45217}{111}$, $117a^{14}-66a^{13}-2771a^{12}-1596a^{11}+20445a^{10}+28373a^{9}-37221a^{8}-80190a^{7}+3494a^{6}+70918a^{5}+28314a^{4}-14253a^{3}-11974a^{2}-2718a-201$, $\frac{1654}{111}a^{14}+\frac{1444}{111}a^{13}-\frac{40907}{111}a^{12}-\frac{78659}{111}a^{11}+\frac{88691}{37}a^{10}+\frac{822038}{111}a^{9}-\frac{21155}{111}a^{8}-\frac{1986482}{111}a^{7}-\frac{1438445}{111}a^{6}+\frac{1353625}{111}a^{5}+\frac{1794917}{111}a^{4}+\frac{36543}{37}a^{3}-\frac{14264}{3}a^{2}-\frac{213080}{111}a-\frac{22745}{111}$ Copy content Toggle raw display (assuming GRH)
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:  \( 122000230.472 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{15}\cdot(2\pi)^{0}\cdot 122000230.472 \cdot 1}{2\cdot\sqrt{23980112932460739050668032}}\cr\approx \mathstrut & 0.408183061199 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^15 - 24*x^13 - 27*x^12 + 167*x^11 + 341*x^10 - 181*x^9 - 864*x^8 - 357*x^7 + 621*x^6 + 583*x^5 + 16*x^4 - 170*x^3 - 81*x^2 - 15*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 - 24*x^13 - 27*x^12 + 167*x^11 + 341*x^10 - 181*x^9 - 864*x^8 - 357*x^7 + 621*x^6 + 583*x^5 + 16*x^4 - 170*x^3 - 81*x^2 - 15*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 - 24*x^13 - 27*x^12 + 167*x^11 + 341*x^10 - 181*x^9 - 864*x^8 - 357*x^7 + 621*x^6 + 583*x^5 + 16*x^4 - 170*x^3 - 81*x^2 - 15*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 - 24*x^13 - 27*x^12 + 167*x^11 + 341*x^10 - 181*x^9 - 864*x^8 - 357*x^7 + 621*x^6 + 583*x^5 + 16*x^4 - 170*x^3 - 81*x^2 - 15*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

$C_3\wr F_5$ (as 15T56):

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 4860
The 63 conjugacy class representatives for $C_3\wr F_5$
Character table for $C_3\wr F_5$

Intermediate fields

5.5.2382032.1

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 sibling: data not computed
Degree 30 siblings: data not computed
Degree 45 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 R R ${\href{/padicField/5.12.0.1}{12} }{,}\,{\href{/padicField/5.3.0.1}{3} }$ $15$ $15$ ${\href{/padicField/13.5.0.1}{5} }^{3}$ ${\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{6}$ ${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.3.0.1}{3} }$ ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{3}$ ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{3}$ ${\href{/padicField/31.4.0.1}{4} }^{3}{,}\,{\href{/padicField/31.3.0.1}{3} }$ R ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.3.0.1}{3} }$ $15$ ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{3}$ R $15$

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
\(2\) Copy content Toggle raw display 2.3.0.1$x^{3} + x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.12.16.11$x^{12} + 4 x^{11} + 4 x^{10} + 2 x^{9} + 8 x^{8} + 8 x^{7} + 6 x^{6} + 8 x^{5} + 4 x^{4} + 20 x^{3} + 4 x^{2} + 28$$6$$2$$16$$C_3\times (C_3 : C_4)$$[2]_{3}^{6}$
\(3\) Copy content Toggle raw display 3.3.4.3$x^{3} + 6 x^{2} + 12$$3$$1$$4$$C_3$$[2]$
3.12.0.1$x^{12} + x^{6} + x^{5} + x^{4} + x^{2} + 2$$1$$12$$0$$C_{12}$$[\ ]^{12}$
\(37\) Copy content Toggle raw display 37.2.0.1$x^{2} + 33 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.0.1$x^{2} + 33 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.0.1$x^{2} + 33 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.0.1$x^{2} + 33 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.0.1$x^{2} + 33 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.0.1$x^{2} + 33 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
37.3.2.1$x^{3} + 37$$3$$1$$2$$C_3$$[\ ]_{3}$
\(53\) Copy content Toggle raw display 53.3.0.1$x^{3} + 3 x + 51$$1$$3$$0$$C_3$$[\ ]^{3}$
53.4.3.2$x^{4} + 53$$4$$1$$3$$C_4$$[\ ]_{4}$
53.4.3.2$x^{4} + 53$$4$$1$$3$$C_4$$[\ ]_{4}$
53.4.3.2$x^{4} + 53$$4$$1$$3$$C_4$$[\ ]_{4}$