Properties

Label 15.3.723...000.1
Degree $15$
Signature $(3, 6)$
Discriminant $7.240\times 10^{55}$
Root discriminant \(5296.40\)
Ramified primes $2,3,5,11,29,43$
Class number not computed
Class group not computed
Galois group $F_5 \times S_3$ (as 15T11)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^15 - 570*x^13 - 20*x^12 + 129960*x^11 - 9456*x^10 - 14815280*x^9 - 12147840*x^8 + 842196240*x^7 - 1490901760*x^6 - 19175238432*x^5 + 88346055360*x^4 + 80742425600*x^3 - 5233291050240*x^2 + 5166247004160*x + 118890409442816)
 
Copy content gp:K = bnfinit(y^15 - 570*y^13 - 20*y^12 + 129960*y^11 - 9456*y^10 - 14815280*y^9 - 12147840*y^8 + 842196240*y^7 - 1490901760*y^6 - 19175238432*y^5 + 88346055360*y^4 + 80742425600*y^3 - 5233291050240*y^2 + 5166247004160*y + 118890409442816, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - 570*x^13 - 20*x^12 + 129960*x^11 - 9456*x^10 - 14815280*x^9 - 12147840*x^8 + 842196240*x^7 - 1490901760*x^6 - 19175238432*x^5 + 88346055360*x^4 + 80742425600*x^3 - 5233291050240*x^2 + 5166247004160*x + 118890409442816);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^15 - 570*x^13 - 20*x^12 + 129960*x^11 - 9456*x^10 - 14815280*x^9 - 12147840*x^8 + 842196240*x^7 - 1490901760*x^6 - 19175238432*x^5 + 88346055360*x^4 + 80742425600*x^3 - 5233291050240*x^2 + 5166247004160*x + 118890409442816)
 

\( x^{15} - 570 x^{13} - 20 x^{12} + 129960 x^{11} - 9456 x^{10} - 14815280 x^{9} - 12147840 x^{8} + \cdots + 118890409442816 \) Copy content Toggle raw display

Copy content comment:Defining polynomial
 
Copy content sage:K.defining_polynomial()
 
Copy content gp:K.pol
 
Copy content magma:DefiningPolynomial(K);
 
Copy content oscar:defining_polynomial(K)
 

Invariants

Degree:  $15$
Copy content comment:Degree over Q
 
Copy content sage:K.degree()
 
Copy content gp:poldegree(K.pol)
 
Copy content magma:Degree(K);
 
Copy content oscar:degree(K)
 
Signature:  $(3, 6)$
Copy content comment:Signature
 
Copy content sage:K.signature()
 
Copy content gp:K.sign
 
Copy content magma:Signature(K);
 
Copy content oscar:signature(K)
 
Discriminant:   \(72395177317625812066702768754666689527288000000000000000\) \(\medspace = 2^{18}\cdot 3^{13}\cdot 5^{15}\cdot 11^{5}\cdot 29^{5}\cdot 43^{13}\) Copy content Toggle raw display
Copy content comment:Discriminant
 
Copy content sage:K.disc()
 
Copy content gp:K.disc
 
Copy content magma:OK := Integers(K); Discriminant(OK);
 
Copy content oscar:OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(5296.40\)
Copy content comment:Root discriminant
 
Copy content sage:(K.disc().abs())^(1./K.degree())
 
Copy content gp:abs(K.disc)^(1/poldegree(K.pol))
 
Copy content magma:Abs(Discriminant(OK))^(1/Degree(K));
 
Copy content oscar:OK = ring_of_integers(K); (1.0 * abs(discriminant(OK)))^(1/degree(K))
 
Galois root discriminant:  $2^{7/5}3^{9/10}5^{23/20}11^{1/2}29^{1/2}43^{9/10}\approx 23805.84380156761$
Ramified primes:   \(2\), \(3\), \(5\), \(11\), \(29\), \(43\) Copy content Toggle raw display
Copy content comment:Ramified primes
 
Copy content sage:K.disc().support()
 
Copy content gp:factor(abs(K.disc))[,1]~
 
Copy content magma:PrimeDivisors(Discriminant(OK));
 
Copy content oscar:prime_divisors(discriminant(OK))
 
Discriminant root field:  \(\Q(\sqrt{205755}) \)
$\Aut(K/\Q)$:   $C_1$
Copy content comment:Automorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphism_group(K)
 
This field is not Galois over $\Q$.
This is not a CM field.
This field has no CM subfields.

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

$1$, $a$, $\frac{1}{2}a^{2}$, $\frac{1}{6}a^{3}+\frac{1}{3}$, $\frac{1}{12}a^{4}-\frac{1}{3}a$, $\frac{1}{72}a^{5}-\frac{1}{36}a^{4}-\frac{1}{36}a^{3}+\frac{1}{9}a^{2}-\frac{2}{9}a-\frac{2}{9}$, $\frac{1}{72}a^{6}+\frac{1}{18}a^{3}-\frac{4}{9}$, $\frac{1}{144}a^{7}+\frac{1}{36}a^{4}-\frac{1}{12}a^{3}-\frac{2}{9}a+\frac{1}{3}$, $\frac{1}{864}a^{8}-\frac{1}{432}a^{7}+\frac{1}{216}a^{6}+\frac{1}{216}a^{5}+\frac{7}{216}a^{4}+\frac{1}{54}a^{3}-\frac{1}{27}a^{2}+\frac{11}{27}a-\frac{4}{27}$, $\frac{1}{864}a^{9}+\frac{1}{36}a^{3}-\frac{5}{27}$, $\frac{1}{222912}a^{10}+\frac{1}{111456}a^{9}+\frac{5}{18576}a^{7}+\frac{1}{18576}a^{6}-\frac{7}{1032}a^{5}-\frac{191}{4644}a^{4}+\frac{383}{4644}a^{3}+\frac{1}{6}a^{2}+\frac{1151}{3483}a-\frac{695}{3483}$, $\frac{1}{445824}a^{11}+\frac{127}{222912}a^{9}+\frac{5}{37152}a^{8}-\frac{1}{4128}a^{7}-\frac{4}{1161}a^{6}-\frac{127}{18576}a^{5}+\frac{83}{3096}a^{4}+\frac{1}{1161}a^{3}+\frac{377}{6966}a^{2}-\frac{145}{387}a+\frac{1727}{3483}$, $\frac{1}{1337472}a^{12}+\frac{17}{334368}a^{9}-\frac{1}{12384}a^{8}-\frac{61}{18576}a^{7}+\frac{131}{27864}a^{6}-\frac{5}{1032}a^{5}-\frac{10}{1161}a^{4}+\frac{343}{41796}a^{3}-\frac{34}{387}a^{2}+\frac{290}{1161}a-\frac{4486}{10449}$, $\frac{1}{29424384}a^{13}+\frac{5}{14712192}a^{12}-\frac{5}{4904064}a^{11}+\frac{1}{919512}a^{10}+\frac{4025}{7356096}a^{9}+\frac{71}{408672}a^{8}-\frac{233}{1226016}a^{7}-\frac{238}{38313}a^{6}-\frac{277}{51084}a^{5}-\frac{2647}{114939}a^{4}-\frac{10739}{229878}a^{3}-\frac{463}{76626}a^{2}-\frac{4231}{10449}a+\frac{1100}{10449}$, $\frac{1}{85\cdots 84}a^{14}+\frac{13\cdots 37}{14\cdots 64}a^{13}-\frac{10\cdots 37}{89\cdots 04}a^{12}+\frac{38\cdots 09}{10\cdots 48}a^{11}+\frac{31\cdots 87}{20\cdots 28}a^{10}+\frac{24\cdots 47}{17\cdots 08}a^{9}+\frac{65\cdots 19}{17\cdots 08}a^{8}+\frac{28\cdots 51}{12\cdots 82}a^{7}+\frac{47\cdots 57}{19\cdots 12}a^{6}+\frac{15\cdots 39}{26\cdots 12}a^{5}+\frac{45\cdots 37}{44\cdots 52}a^{4}+\frac{24\cdots 03}{11\cdots 38}a^{3}+\frac{73\cdots 19}{16\cdots 57}a^{2}-\frac{31\cdots 43}{10\cdots 58}a-\frac{90\cdots 62}{50\cdots 29}$ Copy content Toggle raw display

Copy content comment:Integral basis
 
Copy content sage:K.integral_basis()
 
Copy content gp:K.zk
 
Copy content magma:IntegralBasis(K);
 
Copy content oscar:basis(OK)
 

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

Class group and class number

Ideal class group:  not computed
Copy content comment:Class group
 
Copy content sage:K.class_group().invariants()
 
Copy content gp:K.clgp
 
Copy content magma:ClassGroup(K);
 
Copy content oscar:class_group(K)
 
Narrow class group:  not computed
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 

Unit group

Copy content comment:Unit group
 
Copy content sage:UK = K.unit_group()
 
Copy content magma:UK, fUK := UnitGroup(K);
 
Copy content oscar:UK, fUK = unit_group(OK)
 
Rank:  $8$
Copy content comment:Unit rank
 
Copy content sage:UK.rank()
 
Copy content gp:K.fu
 
Copy content magma:UnitRank(K);
 
Copy content oscar:rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
Copy content comment:Generator for roots of unity
 
Copy content sage:UK.torsion_generator()
 
Copy content gp:K.tu[2]
 
Copy content magma:K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Copy content oscar:torsion_units_generator(OK)
 
Fundamental units:  not computed
Copy content comment:Fundamental units
 
Copy content sage:UK.fundamental_units()
 
Copy content gp:K.fu
 
Copy content magma:[K|fUK(g): g in Generators(UK)];
 
Copy content oscar:[K(fUK(a)) for a in gens(UK)]
 
Regulator:  not computed
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 
Unit signature rank:  not computed

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr = \mathstrut &\frac{2^{3}\cdot(2\pi)^{6}\cdot R \cdot h}{2\cdot\sqrt{72395177317625812066702768754666689527288000000000000000}}\cr\mathstrut & \text{ some values not computed } \end{aligned}\]

Copy content comment:Analytic class number formula
 
Copy content sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^15 - 570*x^13 - 20*x^12 + 129960*x^11 - 9456*x^10 - 14815280*x^9 - 12147840*x^8 + 842196240*x^7 - 1490901760*x^6 - 19175238432*x^5 + 88346055360*x^4 + 80742425600*x^3 - 5233291050240*x^2 + 5166247004160*x + 118890409442816) 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))))
 
Copy content gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^15 - 570*x^13 - 20*x^12 + 129960*x^11 - 9456*x^10 - 14815280*x^9 - 12147840*x^8 + 842196240*x^7 - 1490901760*x^6 - 19175238432*x^5 + 88346055360*x^4 + 80742425600*x^3 - 5233291050240*x^2 + 5166247004160*x + 118890409442816, 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)))]
 
Copy content magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - 570*x^13 - 20*x^12 + 129960*x^11 - 9456*x^10 - 14815280*x^9 - 12147840*x^8 + 842196240*x^7 - 1490901760*x^6 - 19175238432*x^5 + 88346055360*x^4 + 80742425600*x^3 - 5233291050240*x^2 + 5166247004160*x + 118890409442816); 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)));
 
Copy content oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = polynomial_ring(QQ); K, a = number_field(x^15 - 570*x^13 - 20*x^12 + 129960*x^11 - 9456*x^10 - 14815280*x^9 - 12147840*x^8 + 842196240*x^7 - 1490901760*x^6 - 19175238432*x^5 + 88346055360*x^4 + 80742425600*x^3 - 5233291050240*x^2 + 5166247004160*x + 118890409442816); 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

$S_3\times F_5$ (as 15T11):

Copy content comment:Galois group
 
Copy content sage:K.galois_group()
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:G = GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
 
A solvable group of order 120
The 15 conjugacy class representatives for $F_5 \times S_3$
Character table for $F_5 \times S_3$

Intermediate fields

3.3.164604.1, 5.1.13846144050000.2

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

Copy content comment:Intermediate fields
 
Copy content sage:K.subfields()[1:-1]
 
Copy content gp:L = nfsubfields(K); L[2..length(L)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

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 R R R ${\href{/padicField/7.4.0.1}{4} }^{3}{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ R ${\href{/padicField/13.4.0.1}{4} }^{3}{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ ${\href{/padicField/17.4.0.1}{4} }^{3}{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }$ ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.3.0.1}{3} }$ R $15$ ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.3.0.1}{3} }$ ${\href{/padicField/41.2.0.1}{2} }^{5}{,}\,{\href{/padicField/41.1.0.1}{1} }^{5}$ R ${\href{/padicField/47.4.0.1}{4} }^{3}{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ ${\href{/padicField/53.4.0.1}{4} }^{3}{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.2.0.1}{2} }^{6}{,}\,{\href{/padicField/59.1.0.1}{1} }^{3}$

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

Copy content comment:Frobenius cycle types
 
Copy content sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
Copy content gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
Copy content magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
Copy content oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_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 2.1.5.4a1.1$x^{5} + 2$$5$$1$$4$$F_5$$$[\ ]_{5}^{4}$$
2.1.10.14a1.1$x^{10} + 2 x^{5} + 2$$10$$1$$14$$F_{5}\times C_2$$$[2]_{5}^{4}$$
\(3\) Copy content Toggle raw display 3.1.5.4a1.1$x^{5} + 3$$5$$1$$4$$F_5$$$[\ ]_{5}^{4}$$
3.1.10.9a1.2$x^{10} + 6$$10$$1$$9$$F_{5}\times C_2$$$[\ ]_{10}^{4}$$
\(5\) Copy content Toggle raw display 5.3.5.15a4.1$x^{15} + 15 x^{13} + 15 x^{12} + 90 x^{11} + 180 x^{10} + 360 x^{9} + 810 x^{8} + 1215 x^{7} + 1890 x^{6} + 2673 x^{5} + 2835 x^{4} + 2840 x^{3} + 2430 x^{2} + 1230 x + 263$$5$$3$$15$$F_5\times C_3$$$[\frac{5}{4}]_{4}^{3}$$
\(11\) Copy content Toggle raw display $\Q_{11}$$x + 9$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{11}$$x + 9$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{11}$$x + 9$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{11}$$x + 9$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{11}$$x + 9$$1$$1$$0$Trivial$$[\ ]$$
11.1.2.1a1.2$x^{2} + 22$$2$$1$$1$$C_2$$$[\ ]_{2}$$
11.1.2.1a1.2$x^{2} + 22$$2$$1$$1$$C_2$$$[\ ]_{2}$$
11.1.2.1a1.2$x^{2} + 22$$2$$1$$1$$C_2$$$[\ ]_{2}$$
11.1.2.1a1.2$x^{2} + 22$$2$$1$$1$$C_2$$$[\ ]_{2}$$
11.1.2.1a1.2$x^{2} + 22$$2$$1$$1$$C_2$$$[\ ]_{2}$$
\(29\) Copy content Toggle raw display $\Q_{29}$$x + 27$$1$$1$$0$Trivial$$[\ ]$$
29.2.1.0a1.1$x^{2} + 24 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
29.1.2.1a1.2$x^{2} + 58$$2$$1$$1$$C_2$$$[\ ]_{2}$$
29.2.1.0a1.1$x^{2} + 24 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
29.2.2.2a1.2$x^{4} + 48 x^{3} + 580 x^{2} + 96 x + 33$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
29.2.2.2a1.2$x^{4} + 48 x^{3} + 580 x^{2} + 96 x + 33$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
\(43\) Copy content Toggle raw display 43.1.5.4a1.1$x^{5} + 43$$5$$1$$4$$F_5$$$[\ ]_{5}^{4}$$
43.1.10.9a1.2$x^{10} + 129$$10$$1$$9$$F_{5}\times C_2$$$[\ ]_{10}^{4}$$

Spectrum of ring of integers

(0)(0)(2)(3)(5)(7)(11)(13)(17)(19)(23)(29)(31)(37)(41)(43)(47)(53)(59)