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Properties

Label 15.1.212...000.1
Degree $15$
Signature $(1, 7)$
Discriminant $-2.127\times 10^{52}$
Root discriminant \(3079.77\)
Ramified primes $2,3,5,31,47$
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 + 60*x^13 - 140*x^12 + 1025*x^11 - 60893*x^10 + 209370*x^9 + 2663110*x^8 + 23996600*x^7 - 102355320*x^6 + 1329796016*x^5 - 3077398160*x^4 - 76353619120*x^3 + 288748984560*x^2 + 221707794080*x - 8821936400416)
 
Copy content gp:K = bnfinit(y^15 - 5*y^14 + 60*y^13 - 140*y^12 + 1025*y^11 - 60893*y^10 + 209370*y^9 + 2663110*y^8 + 23996600*y^7 - 102355320*y^6 + 1329796016*y^5 - 3077398160*y^4 - 76353619120*y^3 + 288748984560*y^2 + 221707794080*y - 8821936400416, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - 5*x^14 + 60*x^13 - 140*x^12 + 1025*x^11 - 60893*x^10 + 209370*x^9 + 2663110*x^8 + 23996600*x^7 - 102355320*x^6 + 1329796016*x^5 - 3077398160*x^4 - 76353619120*x^3 + 288748984560*x^2 + 221707794080*x - 8821936400416);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^15 - 5*x^14 + 60*x^13 - 140*x^12 + 1025*x^11 - 60893*x^10 + 209370*x^9 + 2663110*x^8 + 23996600*x^7 - 102355320*x^6 + 1329796016*x^5 - 3077398160*x^4 - 76353619120*x^3 + 288748984560*x^2 + 221707794080*x - 8821936400416)
 

\( x^{15} - 5 x^{14} + 60 x^{13} - 140 x^{12} + 1025 x^{11} - 60893 x^{10} + 209370 x^{9} + \cdots - 8821936400416 \) 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:   \(-21270399372578540868971003301317697792000000000000000\) \(\medspace = -\,2^{23}\cdot 3^{12}\cdot 5^{15}\cdot 31^{5}\cdot 47^{13}\) 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:  \(3079.77\)
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:  $2^{19/10}3^{4/5}5^{23/20}31^{1/2}47^{9/10}\approx 10186.815010742723$
Ramified primes:   \(2\), \(3\), \(5\), \(31\), \(47\) 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:  \(\Q(\sqrt{-14570}) \)
$\Aut(K/\Q)$:   $C_1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{12}a^{6}+\frac{1}{12}a^{5}-\frac{1}{4}a^{4}+\frac{5}{12}a^{3}-\frac{1}{2}a^{2}-\frac{1}{6}a+\frac{1}{3}$, $\frac{1}{12}a^{7}+\frac{1}{6}a^{5}+\frac{1}{6}a^{4}-\frac{5}{12}a^{3}-\frac{1}{6}a^{2}-\frac{1}{2}a-\frac{1}{3}$, $\frac{1}{24}a^{8}-\frac{1}{4}a^{5}-\frac{5}{24}a^{4}-\frac{1}{4}a^{3}-\frac{1}{2}a+\frac{1}{6}$, $\frac{1}{24}a^{9}+\frac{1}{24}a^{5}+\frac{1}{4}a^{3}-\frac{1}{3}a$, $\frac{1}{1128}a^{10}-\frac{7}{1128}a^{9}+\frac{2}{141}a^{8}+\frac{1}{282}a^{7}-\frac{13}{376}a^{6}+\frac{79}{376}a^{5}+\frac{31}{188}a^{4}-\frac{41}{564}a^{3}-\frac{97}{282}a^{2}-\frac{49}{141}a+\frac{10}{141}$, $\frac{1}{2256}a^{11}-\frac{1}{2256}a^{10}-\frac{13}{1128}a^{9}+\frac{1}{376}a^{8}-\frac{5}{752}a^{7}+\frac{1}{752}a^{6}+\frac{10}{47}a^{5}-\frac{1}{12}a^{4}-\frac{55}{141}a^{3}+\frac{25}{564}a^{2}+\frac{139}{282}a-\frac{64}{141}$, $\frac{1}{13536}a^{12}-\frac{1}{13536}a^{11}+\frac{1}{3384}a^{10}+\frac{43}{3384}a^{9}+\frac{277}{13536}a^{8}-\frac{253}{13536}a^{7}-\frac{251}{6768}a^{6}+\frac{1111}{6768}a^{5}+\frac{11}{72}a^{4}+\frac{115}{3384}a^{3}-\frac{11}{36}a^{2}+\frac{5}{36}a-\frac{19}{423}$, $\frac{1}{27072}a^{13}-\frac{1}{27072}a^{12}-\frac{1}{13536}a^{11}-\frac{1}{13536}a^{10}-\frac{563}{27072}a^{9}-\frac{349}{27072}a^{8}+\frac{281}{6768}a^{7}-\frac{29}{846}a^{6}+\frac{385}{1692}a^{5}+\frac{157}{1692}a^{4}+\frac{178}{423}a^{3}+\frac{89}{423}a^{2}-\frac{839}{1692}a-\frac{125}{564}$, $\frac{1}{54\cdots 24}a^{14}-\frac{43\cdots 49}{27\cdots 12}a^{13}+\frac{52\cdots 35}{54\cdots 24}a^{12}-\frac{11\cdots 87}{13\cdots 56}a^{11}-\frac{11\cdots 49}{54\cdots 24}a^{10}+\frac{27\cdots 91}{27\cdots 12}a^{9}+\frac{17\cdots 21}{54\cdots 24}a^{8}-\frac{34\cdots 88}{84\cdots 41}a^{7}-\frac{10\cdots 03}{13\cdots 56}a^{6}-\frac{11\cdots 27}{67\cdots 28}a^{5}-\frac{47\cdots 13}{33\cdots 64}a^{4}+\frac{75\cdots 51}{33\cdots 64}a^{3}-\frac{80\cdots 61}{16\cdots 82}a^{2}+\frac{44\cdots 09}{56\cdots 94}a+\frac{46\cdots 37}{11\cdots 88}$ 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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr = \mathstrut &\frac{2^{1}\cdot(2\pi)^{7}\cdot R \cdot h}{2\cdot\sqrt{21270399372578540868971003301317697792000000000000000}}\cr\mathstrut & \text{ some values not computed } \end{aligned}\]

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 + 60*x^13 - 140*x^12 + 1025*x^11 - 60893*x^10 + 209370*x^9 + 2663110*x^8 + 23996600*x^7 - 102355320*x^6 + 1329796016*x^5 - 3077398160*x^4 - 76353619120*x^3 + 288748984560*x^2 + 221707794080*x - 8821936400416) 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 + 60*x^13 - 140*x^12 + 1025*x^11 - 60893*x^10 + 209370*x^9 + 2663110*x^8 + 23996600*x^7 - 102355320*x^6 + 1329796016*x^5 - 3077398160*x^4 - 76353619120*x^3 + 288748984560*x^2 + 221707794080*x - 8821936400416, 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 + 60*x^13 - 140*x^12 + 1025*x^11 - 60893*x^10 + 209370*x^9 + 2663110*x^8 + 23996600*x^7 - 102355320*x^6 + 1329796016*x^5 - 3077398160*x^4 - 76353619120*x^3 + 288748984560*x^2 + 221707794080*x - 8821936400416); 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 + 60*x^13 - 140*x^12 + 1025*x^11 - 60893*x^10 + 209370*x^9 + 2663110*x^8 + 23996600*x^7 - 102355320*x^6 + 1329796016*x^5 - 3077398160*x^4 - 76353619120*x^3 + 288748984560*x^2 + 221707794080*x - 8821936400416); 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

3.1.11656.1, 5.1.19762708050000.8

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 ${\href{/padicField/7.4.0.1}{4} }^{3}{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ $15$ ${\href{/padicField/13.4.0.1}{4} }^{3}{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ ${\href{/padicField/17.4.0.1}{4} }^{3}{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ ${\href{/padicField/19.2.0.1}{2} }^{7}{,}\,{\href{/padicField/19.1.0.1}{1} }$ ${\href{/padicField/23.4.0.1}{4} }^{3}{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ ${\href{/padicField/29.2.0.1}{2} }^{7}{,}\,{\href{/padicField/29.1.0.1}{1} }$ R ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.3.0.1}{3} }$ ${\href{/padicField/41.2.0.1}{2} }^{5}{,}\,{\href{/padicField/41.1.0.1}{1} }^{5}$ ${\href{/padicField/43.4.0.1}{4} }^{3}{,}\,{\href{/padicField/43.1.0.1}{1} }^{3}$ R ${\href{/padicField/53.4.0.1}{4} }^{3}{,}\,{\href{/padicField/53.1.0.1}{1} }^{3}$ ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.3.0.1}{3} }$

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$$$[\ ]_{5}^{4}$$
2.1.10.19a1.1$x^{10} + 2$$10$$1$$19$$F_{5}\times C_2$$$[3]_{5}^{4}$$
\(3\) Copy content Toggle raw display 3.1.5.4a1.1$x^{5} + 3$$5$$1$$4$$F_5$$$[\ ]_{5}^{4}$$
3.2.5.8a1.1$x^{10} + 10 x^{9} + 50 x^{8} + 160 x^{7} + 360 x^{6} + 592 x^{5} + 720 x^{4} + 640 x^{3} + 400 x^{2} + 160 x + 35$$5$$2$$8$$F_5$$$[\ ]_{5}^{4}$$
\(5\) 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$$$[\frac{5}{4}]_{4}^{3}$$
\(31\) Copy content Toggle raw display $\Q_{31}$$x + 28$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{31}$$x + 28$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{31}$$x + 28$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{31}$$x + 28$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{31}$$x + 28$$1$$1$$0$Trivial$$[\ ]$$
31.1.2.1a1.1$x^{2} + 31$$2$$1$$1$$C_2$$$[\ ]_{2}$$
31.1.2.1a1.1$x^{2} + 31$$2$$1$$1$$C_2$$$[\ ]_{2}$$
31.1.2.1a1.1$x^{2} + 31$$2$$1$$1$$C_2$$$[\ ]_{2}$$
31.1.2.1a1.1$x^{2} + 31$$2$$1$$1$$C_2$$$[\ ]_{2}$$
31.1.2.1a1.1$x^{2} + 31$$2$$1$$1$$C_2$$$[\ ]_{2}$$
\(47\) 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$$$[\ ]_{10}^{4}$$

Spectrum of ring of integers

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