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Properties

Label 15.1.656...000.1
Degree $15$
Signature $(1, 7)$
Discriminant $-6.564\times 10^{46}$
Root discriminant \(1321.74\)
Ramified primes $2,5,47,109$
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 - 5*x^14 + 245*x^13 - 715*x^12 + 22565*x^11 - 21480*x^10 + 992875*x^9 + 1050450*x^8 + 22166375*x^7 + 73659575*x^6 + 276602050*x^5 + 1171874500*x^4 + 1993898625*x^3 + 3823779000*x^2 - 1444686000*x + 13303150250)
 
Copy content gp:K = bnfinit(y^15 - 5*y^14 + 245*y^13 - 715*y^12 + 22565*y^11 - 21480*y^10 + 992875*y^9 + 1050450*y^8 + 22166375*y^7 + 73659575*y^6 + 276602050*y^5 + 1171874500*y^4 + 1993898625*y^3 + 3823779000*y^2 - 1444686000*y + 13303150250, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - 5*x^14 + 245*x^13 - 715*x^12 + 22565*x^11 - 21480*x^10 + 992875*x^9 + 1050450*x^8 + 22166375*x^7 + 73659575*x^6 + 276602050*x^5 + 1171874500*x^4 + 1993898625*x^3 + 3823779000*x^2 - 1444686000*x + 13303150250);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^15 - 5*x^14 + 245*x^13 - 715*x^12 + 22565*x^11 - 21480*x^10 + 992875*x^9 + 1050450*x^8 + 22166375*x^7 + 73659575*x^6 + 276602050*x^5 + 1171874500*x^4 + 1993898625*x^3 + 3823779000*x^2 - 1444686000*x + 13303150250)
 

\( x^{15} - 5 x^{14} + 245 x^{13} - 715 x^{12} + 22565 x^{11} - 21480 x^{10} + 992875 x^{9} + \cdots + 13303150250 \) 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:  $(1, 7)$
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:   \(-65643945043273354879634974613689843750000000000\) \(\medspace = -\,2^{10}\cdot 5^{17}\cdot 47^{13}\cdot 109^{5}\) 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:  \(1321.74\)
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:  $2^{2/3}5^{71/60}47^{9/10}109^{1/2}\approx 3559.61058896397$
Ramified primes:   \(2\), \(5\), \(47\), \(109\) 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:  \(\Q(\sqrt{-25615}) \)
$\Aut(K/\Q)$:   $C_1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{5}a^{5}$, $\frac{1}{5}a^{6}$, $\frac{1}{5}a^{7}$, $\frac{1}{5}a^{8}$, $\frac{1}{25}a^{9}-\frac{2}{5}a^{4}$, $\frac{1}{1175}a^{10}-\frac{1}{235}a^{9}+\frac{2}{235}a^{8}-\frac{2}{235}a^{7}+\frac{1}{235}a^{6}-\frac{19}{235}a^{5}$, $\frac{1}{1175}a^{11}-\frac{3}{235}a^{9}+\frac{8}{235}a^{8}-\frac{9}{235}a^{7}-\frac{14}{235}a^{6}-\frac{1}{235}a^{5}$, $\frac{1}{5875}a^{12}-\frac{7}{1175}a^{9}-\frac{73}{1175}a^{8}-\frac{44}{1175}a^{7}-\frac{16}{235}a^{6}-\frac{2}{47}a^{5}-\frac{2}{5}a^{3}+\frac{2}{5}a^{2}$, $\frac{1}{5875}a^{13}-\frac{14}{1175}a^{9}+\frac{26}{1175}a^{8}+\frac{17}{235}a^{7}-\frac{3}{235}a^{6}+\frac{8}{235}a^{5}-\frac{1}{5}a^{4}+\frac{2}{5}a^{3}$, $\frac{1}{23\cdots 25}a^{14}-\frac{99\cdots 64}{23\cdots 25}a^{13}-\frac{18\cdots 67}{46\cdots 25}a^{12}-\frac{36\cdots 66}{46\cdots 25}a^{11}-\frac{12\cdots 07}{46\cdots 25}a^{10}+\frac{34\cdots 57}{46\cdots 25}a^{9}-\frac{74\cdots 97}{98\cdots 75}a^{8}-\frac{14\cdots 18}{19\cdots 35}a^{7}+\frac{10\cdots 57}{19\cdots 35}a^{6}-\frac{18\cdots 91}{19\cdots 35}a^{5}-\frac{54\cdots 04}{19\cdots 35}a^{4}-\frac{68\cdots 94}{42\cdots 05}a^{3}-\frac{29\cdots 33}{84\cdots 61}a^{2}+\frac{10\cdots 23}{84\cdots 61}a-\frac{99\cdots 89}{84\cdots 61}$ 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:  $7$
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr = \mathstrut &\frac{2^{1}\cdot(2\pi)^{7}\cdot R \cdot h}{2\cdot\sqrt{65643945043273354879634974613689843750000000000}}\cr\mathstrut & \text{ some values not computed } \end{aligned}\]

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 - 5*x^14 + 245*x^13 - 715*x^12 + 22565*x^11 - 21480*x^10 + 992875*x^9 + 1050450*x^8 + 22166375*x^7 + 73659575*x^6 + 276602050*x^5 + 1171874500*x^4 + 1993898625*x^3 + 3823779000*x^2 - 1444686000*x + 13303150250) 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 - 5*x^14 + 245*x^13 - 715*x^12 + 22565*x^11 - 21480*x^10 + 992875*x^9 + 1050450*x^8 + 22166375*x^7 + 73659575*x^6 + 276602050*x^5 + 1171874500*x^4 + 1993898625*x^3 + 3823779000*x^2 - 1444686000*x + 13303150250, 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 - 5*x^14 + 245*x^13 - 715*x^12 + 22565*x^11 - 21480*x^10 + 992875*x^9 + 1050450*x^8 + 22166375*x^7 + 73659575*x^6 + 276602050*x^5 + 1171874500*x^4 + 1993898625*x^3 + 3823779000*x^2 - 1444686000*x + 13303150250); 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 - 5*x^14 + 245*x^13 - 715*x^12 + 22565*x^11 - 21480*x^10 + 992875*x^9 + 1050450*x^8 + 22166375*x^7 + 73659575*x^6 + 276602050*x^5 + 1171874500*x^4 + 1993898625*x^3 + 3823779000*x^2 - 1444686000*x + 13303150250); 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

3.1.512300.2, 5.1.15249003125.1

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 ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.3.0.1}{3} }$ R ${\href{/padicField/7.12.0.1}{12} }{,}\,{\href{/padicField/7.3.0.1}{3} }$ $15$ ${\href{/padicField/13.4.0.1}{4} }^{3}{,}\,{\href{/padicField/13.1.0.1}{1} }^{3}$ ${\href{/padicField/17.4.0.1}{4} }^{3}{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }$ ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.3.0.1}{3} }$ ${\href{/padicField/29.2.0.1}{2} }^{7}{,}\,{\href{/padicField/29.1.0.1}{1} }$ ${\href{/padicField/31.10.0.1}{10} }{,}\,{\href{/padicField/31.5.0.1}{5} }$ ${\href{/padicField/37.4.0.1}{4} }^{3}{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ $15$ ${\href{/padicField/43.4.0.1}{4} }^{3}{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ R ${\href{/padicField/53.4.0.1}{4} }^{3}{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.2.0.1}{2} }^{7}{,}\,{\href{/padicField/59.1.0.1}{1} }$

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 2.1.3.2a1.1$x^{3} + 2$$3$$1$$2$$S_3$$$[\ ]_{3}^{2}$$
2.4.3.8a1.2$x^{12} + 3 x^{9} + 3 x^{8} + 3 x^{6} + 6 x^{5} + 3 x^{4} + x^{3} + 3 x^{2} + 3 x + 3$$3$$4$$8$$C_3 : C_4$$$[\ ]_{3}^{4}$$
\(5\) Copy content Toggle raw display 5.1.15.17a1.1$x^{15} + 5 x^{3} + 5$$15$$1$$17$$F_5 \times S_3$$$[\frac{5}{4}]_{12}^{2}$$
\(47\) Copy content Toggle raw display 47.1.5.4a1.1$x^{5} + 47$$5$$1$$4$$F_5$$$[\ ]_{5}^{4}$$
47.1.10.9a1.2$x^{10} + 235$$10$$1$$9$$F_{5}\times C_2$$$[\ ]_{10}^{4}$$
\(109\) Copy content Toggle raw display $\Q_{109}$$x + 103$$1$$1$$0$Trivial$$[\ ]$$
109.1.2.1a1.2$x^{2} + 654$$2$$1$$1$$C_2$$$[\ ]_{2}$$
109.2.1.0a1.1$x^{2} + 108 x + 6$$1$$2$$0$$C_2$$$[\ ]^{2}$$
109.2.1.0a1.1$x^{2} + 108 x + 6$$1$$2$$0$$C_2$$$[\ ]^{2}$$
109.2.2.2a1.2$x^{4} + 216 x^{3} + 11676 x^{2} + 1296 x + 145$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
109.2.2.2a1.2$x^{4} + 216 x^{3} + 11676 x^{2} + 1296 x + 145$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$

Spectrum of ring of integers

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