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Properties

Label 15.1.218...000.1
Degree $15$
Signature $(1, 7)$
Discriminant $-2.181\times 10^{55}$
Root discriminant \(4889.34\)
Ramified primes $2,5,7,11,47$
Class number not computed
Class group not computed
Galois group $F_5 \times S_3$ (as 15T11)

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Show commands: Magma / Oscar / Pari/GP / SageMath

Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^15 + 145*x^13 - 130*x^12 + 8410*x^11 + 5401576*x^10 + 250650*x^9 - 786071100*x^8 + 4229116205*x^7 + 30356526600*x^6 + 9759670847621*x^5 + 605028612190*x^4 - 1890020683173160*x^3 - 2535367162720480*x^2 + 27422952298756080*x + 5898464498853712720)
 
Copy content gp:K = bnfinit(y^15 + 145*y^13 - 130*y^12 + 8410*y^11 + 5401576*y^10 + 250650*y^9 - 786071100*y^8 + 4229116205*y^7 + 30356526600*y^6 + 9759670847621*y^5 + 605028612190*y^4 - 1890020683173160*y^3 - 2535367162720480*y^2 + 27422952298756080*y + 5898464498853712720, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 + 145*x^13 - 130*x^12 + 8410*x^11 + 5401576*x^10 + 250650*x^9 - 786071100*x^8 + 4229116205*x^7 + 30356526600*x^6 + 9759670847621*x^5 + 605028612190*x^4 - 1890020683173160*x^3 - 2535367162720480*x^2 + 27422952298756080*x + 5898464498853712720);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^15 + 145*x^13 - 130*x^12 + 8410*x^11 + 5401576*x^10 + 250650*x^9 - 786071100*x^8 + 4229116205*x^7 + 30356526600*x^6 + 9759670847621*x^5 + 605028612190*x^4 - 1890020683173160*x^3 - 2535367162720480*x^2 + 27422952298756080*x + 5898464498853712720)
 

\( x^{15} + 145 x^{13} - 130 x^{12} + 8410 x^{11} + 5401576 x^{10} + 250650 x^{9} - 786071100 x^{8} + \cdots + 58\!\cdots\!20 \) 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:   \(-21814741168259795746005614009978336137984000000000000000\) \(\medspace = -\,2^{23}\cdot 5^{15}\cdot 7^{13}\cdot 11^{5}\cdot 47^{13}\) 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:  \(4889.34\)
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:  $2^{19/10}5^{23/20}7^{9/10}11^{1/2}47^{9/10}\approx 14519.282237111507$
Ramified primes:   \(2\), \(5\), \(7\), \(11\), \(47\) 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:  \(\Q(\sqrt{-36190}) \)
$\Aut(K/\Q)$:   $C_1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{14}a^{4}-\frac{3}{7}a^{3}+\frac{1}{14}a^{2}-\frac{2}{7}a+\frac{1}{7}$, $\frac{1}{14}a^{5}-\frac{1}{2}a^{3}+\frac{1}{7}a^{2}+\frac{3}{7}a-\frac{1}{7}$, $\frac{1}{28}a^{6}-\frac{1}{28}a^{5}-\frac{1}{28}a^{4}+\frac{1}{28}a^{3}-\frac{1}{7}a^{2}-\frac{1}{7}a$, $\frac{1}{196}a^{7}+\frac{1}{196}a^{6}-\frac{5}{196}a^{5}-\frac{1}{196}a^{4}+\frac{24}{49}a^{3}-\frac{4}{49}a^{2}+\frac{23}{49}a+\frac{1}{49}$, $\frac{1}{392}a^{8}-\frac{3}{196}a^{6}+\frac{1}{98}a^{5}+\frac{13}{392}a^{4}-\frac{1}{2}a^{3}+\frac{3}{49}a^{2}-\frac{18}{49}a+\frac{3}{49}$, $\frac{1}{2744}a^{9}-\frac{1}{1372}a^{8}+\frac{1}{1372}a^{7}+\frac{3}{343}a^{6}+\frac{1}{56}a^{5}+\frac{39}{1372}a^{4}+\frac{79}{686}a^{3}-\frac{59}{686}a^{2}-\frac{156}{343}a-\frac{93}{343}$, $\frac{1}{257936}a^{10}-\frac{3}{257936}a^{9}-\frac{1}{2744}a^{8}-\frac{87}{128968}a^{7}-\frac{1151}{257936}a^{6}-\frac{2323}{257936}a^{5}-\frac{205}{9212}a^{4}+\frac{10985}{64484}a^{3}-\frac{47}{686}a^{2}-\frac{965}{2303}a+\frac{7666}{16121}$, $\frac{1}{515872}a^{11}-\frac{9}{515872}a^{9}+\frac{1}{36848}a^{8}-\frac{169}{515872}a^{7}+\frac{765}{128968}a^{6}-\frac{2839}{515872}a^{5}-\frac{2385}{257936}a^{4}-\frac{1923}{64484}a^{3}+\frac{1017}{16121}a^{2}+\frac{11653}{32242}a-\frac{9}{94}$, $\frac{1}{7222208}a^{12}+\frac{5}{7222208}a^{11}+\frac{3}{7222208}a^{10}-\frac{67}{7222208}a^{9}+\frac{6669}{7222208}a^{8}+\frac{15919}{7222208}a^{7}+\frac{11087}{1031744}a^{6}-\frac{57369}{7222208}a^{5}+\frac{7795}{3611104}a^{4}-\frac{169279}{902776}a^{3}-\frac{156589}{451388}a^{2}+\frac{4222}{112847}a+\frac{156833}{451388}$, $\frac{1}{1112220032}a^{13}-\frac{17}{556110016}a^{12}+\frac{2}{8689219}a^{11}+\frac{103}{139027504}a^{10}+\frac{12755}{79444288}a^{9}-\frac{139093}{278055008}a^{8}-\frac{43173}{17378438}a^{7}+\frac{137609}{139027504}a^{6}+\frac{13470845}{1112220032}a^{5}-\frac{5996677}{556110016}a^{4}+\frac{66840341}{139027504}a^{3}+\frac{369791}{1418648}a^{2}-\frac{2050133}{6319432}a+\frac{2118419}{6319432}$, $\frac{1}{17\cdots 64}a^{14}+\frac{61\cdots 23}{21\cdots 08}a^{13}-\frac{70\cdots 33}{43\cdots 16}a^{12}+\frac{17\cdots 93}{21\cdots 08}a^{11}-\frac{51\cdots 03}{87\cdots 32}a^{10}+\frac{95\cdots 73}{27\cdots 26}a^{9}+\frac{50\cdots 05}{21\cdots 08}a^{8}-\frac{46\cdots 77}{21\cdots 08}a^{7}-\frac{14\cdots 99}{17\cdots 64}a^{6}+\frac{74\cdots 47}{21\cdots 08}a^{5}+\frac{57\cdots 81}{43\cdots 16}a^{4}-\frac{10\cdots 85}{54\cdots 52}a^{3}+\frac{28\cdots 31}{10\cdots 04}a^{2}-\frac{34\cdots 31}{99\cdots 64}a-\frac{22\cdots 65}{49\cdots 32}$ 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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr = \mathstrut &\frac{2^{1}\cdot(2\pi)^{7}\cdot R \cdot h}{2\cdot\sqrt{21814741168259795746005614009978336137984000000000000000}}\cr\mathstrut & \text{ some values not computed } \end{aligned}\]

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 + 145*x^13 - 130*x^12 + 8410*x^11 + 5401576*x^10 + 250650*x^9 - 786071100*x^8 + 4229116205*x^7 + 30356526600*x^6 + 9759670847621*x^5 + 605028612190*x^4 - 1890020683173160*x^3 - 2535367162720480*x^2 + 27422952298756080*x + 5898464498853712720) 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 + 145*x^13 - 130*x^12 + 8410*x^11 + 5401576*x^10 + 250650*x^9 - 786071100*x^8 + 4229116205*x^7 + 30356526600*x^6 + 9759670847621*x^5 + 605028612190*x^4 - 1890020683173160*x^3 - 2535367162720480*x^2 + 27422952298756080*x + 5898464498853712720, 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 + 145*x^13 - 130*x^12 + 8410*x^11 + 5401576*x^10 + 250650*x^9 - 786071100*x^8 + 4229116205*x^7 + 30356526600*x^6 + 9759670847621*x^5 + 605028612190*x^4 - 1890020683173160*x^3 - 2535367162720480*x^2 + 27422952298756080*x + 5898464498853712720); 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 + 145*x^13 - 130*x^12 + 8410*x^11 + 5401576*x^10 + 250650*x^9 - 786071100*x^8 + 4229116205*x^7 + 30356526600*x^6 + 9759670847621*x^5 + 605028612190*x^4 - 1890020683173160*x^3 - 2535367162720480*x^2 + 27422952298756080*x + 5898464498853712720); 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

3.1.28952.1, 5.1.585805704050000.5

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 R R ${\href{/padicField/13.4.0.1}{4} }^{3}{,}\,{\href{/padicField/13.1.0.1}{1} }^{3}$ ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.3.0.1}{3} }$ ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }$ ${\href{/padicField/23.4.0.1}{4} }^{3}{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ ${\href{/padicField/29.2.0.1}{2} }^{7}{,}\,{\href{/padicField/29.1.0.1}{1} }$ $15$ ${\href{/padicField/37.4.0.1}{4} }^{3}{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.5.0.1}{5} }$ ${\href{/padicField/43.4.0.1}{4} }^{3}{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ R ${\href{/padicField/53.4.0.1}{4} }^{3}{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.3.0.1}{3} }$

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$$$[\ ]_{5}^{4}$$
2.1.10.19a1.10$x^{10} + 4 x^{5} + 10$$10$$1$$19$$F_{5}\times C_2$$$[3]_{5}^{4}$$
\(5\) Copy content Toggle raw display 5.1.5.5a1.1$x^{5} + 5 x + 5$$5$$1$$5$$F_5$$$[\frac{5}{4}]_{4}$$
5.2.5.10a4.1$x^{10} + 20 x^{9} + 170 x^{8} + 800 x^{7} + 2280 x^{6} + 4064 x^{5} + 4560 x^{4} + 3200 x^{3} + 1365 x^{2} + 340 x + 47$$5$$2$$10$$F_{5}\times C_2$$$[\frac{5}{4}]_{4}^{2}$$
\(7\) Copy content Toggle raw display 7.1.5.4a1.1$x^{5} + 7$$5$$1$$4$$F_5$$$[\ ]_{5}^{4}$$
7.1.10.9a1.2$x^{10} + 21$$10$$1$$9$$F_{5}\times C_2$$$[\ ]_{10}^{4}$$
\(11\) 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$$[\ ]$$
11.1.2.1a1.1$x^{2} + 11$$2$$1$$1$$C_2$$$[\ ]_{2}$$
11.1.2.1a1.1$x^{2} + 11$$2$$1$$1$$C_2$$$[\ ]_{2}$$
11.1.2.1a1.1$x^{2} + 11$$2$$1$$1$$C_2$$$[\ ]_{2}$$
11.1.2.1a1.1$x^{2} + 11$$2$$1$$1$$C_2$$$[\ ]_{2}$$
11.1.2.1a1.1$x^{2} + 11$$2$$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)