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Properties

Label 13.3.96082590998375.1
Degree $13$
Signature $(3, 5)$
Discriminant $-9.608\times 10^{13}$
Root discriminant \(11.90\)
Ramified primes $5,768660727987$
Class number $1$
Class group trivial
Galois group $S_{13}$ (as 13T9)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^13 - 4*x^11 - 3*x^10 + 9*x^9 + 10*x^8 - 4*x^7 - 12*x^6 - 13*x^5 + 5*x^4 + 12*x^3 - x + 1)
 
Copy content gp:K = bnfinit(y^13 - 4*y^11 - 3*y^10 + 9*y^9 + 10*y^8 - 4*y^7 - 12*y^6 - 13*y^5 + 5*y^4 + 12*y^3 - y + 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^13 - 4*x^11 - 3*x^10 + 9*x^9 + 10*x^8 - 4*x^7 - 12*x^6 - 13*x^5 + 5*x^4 + 12*x^3 - x + 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^13 - 4*x^11 - 3*x^10 + 9*x^9 + 10*x^8 - 4*x^7 - 12*x^6 - 13*x^5 + 5*x^4 + 12*x^3 - x + 1)
 

\( x^{13} - 4x^{11} - 3x^{10} + 9x^{9} + 10x^{8} - 4x^{7} - 12x^{6} - 13x^{5} + 5x^{4} + 12x^{3} - x + 1 \) 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:  $13$
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:  $(3, 5)$
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:   \(-96082590998375\) \(\medspace = -\,5^{3}\cdot 768660727987\) 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:  \(11.90\)
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:  $5^{3/4}768660727987^{1/2}\approx 2931533.4207658344$
Ramified primes:   \(5\), \(768660727987\) 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:  $\Q(\sqrt{-3843303639935}$)
$\Aut(K/\Q)$:   $C_1$
Copy content comment:Autmorphisms
 
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{5}a^{11}-\frac{1}{5}a^{10}-\frac{2}{5}a^{9}-\frac{2}{5}a^{8}-\frac{1}{5}a^{7}-\frac{1}{5}a^{6}+\frac{1}{5}a^{5}+\frac{1}{5}a^{4}+\frac{2}{5}a^{3}-\frac{1}{5}a^{2}-\frac{1}{5}$, $\frac{1}{535}a^{12}+\frac{52}{535}a^{11}+\frac{5}{107}a^{10}+\frac{227}{535}a^{9}+\frac{43}{535}a^{8}+\frac{106}{535}a^{7}+\frac{158}{535}a^{6}+\frac{179}{535}a^{5}+\frac{40}{107}a^{4}+\frac{48}{107}a^{3}+\frac{187}{535}a^{2}+\frac{94}{535}a+\frac{72}{535}$ 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:  Trivial group, which has order $1$
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:  Trivial group, which has order $1$
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:   $a$, $\frac{122}{107}a^{12}-\frac{166}{535}a^{11}-\frac{2084}{535}a^{10}-\frac{1058}{535}a^{9}+\frac{4937}{535}a^{8}+\frac{3456}{535}a^{7}-\frac{2274}{535}a^{6}-\frac{4016}{535}a^{5}-\frac{5116}{535}a^{4}+\frac{4518}{535}a^{3}+\frac{3111}{535}a^{2}-\frac{409}{107}a-\frac{164}{535}$, $\frac{366}{535}a^{12}-\frac{763}{535}a^{11}-\frac{310}{107}a^{10}+\frac{1227}{535}a^{9}+\frac{5573}{535}a^{8}-\frac{794}{535}a^{7}-\frac{6907}{535}a^{6}-\frac{5641}{535}a^{5}-\frac{340}{107}a^{4}+\frac{1946}{107}a^{3}+\frac{3707}{535}a^{2}-\frac{1441}{535}a+\frac{672}{535}$, $\frac{254}{535}a^{12}-\frac{274}{535}a^{11}-\frac{1033}{535}a^{10}+\frac{92}{535}a^{9}+\frac{3111}{535}a^{8}+\frac{816}{535}a^{7}-\frac{3096}{535}a^{6}-\frac{3326}{535}a^{5}-\frac{1737}{535}a^{4}+\frac{4036}{535}a^{3}+\frac{533}{107}a^{2}-\frac{734}{535}a-\frac{66}{107}$, $\frac{41}{107}a^{12}-\frac{254}{535}a^{11}-\frac{546}{535}a^{10}+\frac{418}{535}a^{9}+\frac{1753}{535}a^{8}-\frac{1061}{535}a^{7}-\frac{1636}{535}a^{6}+\frac{636}{535}a^{5}+\frac{661}{535}a^{4}+\frac{2762}{535}a^{3}-\frac{1576}{535}a^{2}-\frac{426}{107}a+\frac{529}{535}$, $\frac{48}{107}a^{12}-\frac{467}{535}a^{11}-\frac{848}{535}a^{10}+\frac{659}{535}a^{9}+\frac{3044}{535}a^{8}-\frac{668}{535}a^{7}-\frac{3168}{535}a^{6}-\frac{2622}{535}a^{5}-\frac{1327}{535}a^{4}+\frac{4956}{535}a^{3}+\frac{582}{535}a^{2}+\frac{18}{107}a+\frac{802}{535}$, $\frac{447}{535}a^{12}-\frac{403}{535}a^{11}-\frac{1558}{535}a^{10}+\frac{33}{535}a^{9}+\frac{891}{107}a^{8}+\frac{944}{535}a^{7}-\frac{3632}{535}a^{6}-\frac{3554}{535}a^{5}-\frac{2727}{535}a^{4}+\frac{5951}{535}a^{3}+\frac{1841}{535}a^{2}-\frac{1852}{535}a+\frac{191}{535}$ Copy content Toggle raw display
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:  \( 61.9626233533 \)
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{5}\cdot 61.9626233533 \cdot 1}{2\cdot\sqrt{96082590998375}}\cr\approx \mathstrut & 0.247609199809 \end{aligned}\]

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^13 - 4*x^11 - 3*x^10 + 9*x^9 + 10*x^8 - 4*x^7 - 12*x^6 - 13*x^5 + 5*x^4 + 12*x^3 - 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))))
 
Copy content gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^13 - 4*x^11 - 3*x^10 + 9*x^9 + 10*x^8 - 4*x^7 - 12*x^6 - 13*x^5 + 5*x^4 + 12*x^3 - 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)))]
 
Copy content magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^13 - 4*x^11 - 3*x^10 + 9*x^9 + 10*x^8 - 4*x^7 - 12*x^6 - 13*x^5 + 5*x^4 + 12*x^3 - 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)));
 
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^13 - 4*x^11 - 3*x^10 + 9*x^9 + 10*x^8 - 4*x^7 - 12*x^6 - 13*x^5 + 5*x^4 + 12*x^3 - 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

$S_{13}$ (as 13T9):

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 non-solvable group of order 6227020800
The 101 conjugacy class representatives for $S_{13}$
Character table for $S_{13}$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.
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 26 sibling: 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 ${\href{/padicField/2.7.0.1}{7} }{,}\,{\href{/padicField/2.3.0.1}{3} }^{2}$ ${\href{/padicField/3.13.0.1}{13} }$ R ${\href{/padicField/7.9.0.1}{9} }{,}\,{\href{/padicField/7.4.0.1}{4} }$ ${\href{/padicField/11.9.0.1}{9} }{,}\,{\href{/padicField/11.4.0.1}{4} }$ ${\href{/padicField/13.7.0.1}{7} }{,}\,{\href{/padicField/13.6.0.1}{6} }$ ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ ${\href{/padicField/19.13.0.1}{13} }$ ${\href{/padicField/23.11.0.1}{11} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ ${\href{/padicField/29.5.0.1}{5} }{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ ${\href{/padicField/31.10.0.1}{10} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ ${\href{/padicField/37.9.0.1}{9} }{,}\,{\href{/padicField/37.4.0.1}{4} }$ ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ ${\href{/padicField/43.5.0.1}{5} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ ${\href{/padicField/47.13.0.1}{13} }$ ${\href{/padicField/53.11.0.1}{11} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ ${\href{/padicField/59.13.0.1}{13} }$

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
\(5\) Copy content Toggle raw display 5.1.4.3a1.4$x^{4} + 20$$4$$1$$3$$C_4$$$[\ ]_{4}$$
5.4.1.0a1.1$x^{4} + 4 x^{2} + 4 x + 2$$1$$4$$0$$C_4$$$[\ ]^{4}$$
5.5.1.0a1.1$x^{5} + 4 x + 3$$1$$5$$0$$C_5$$$[\ ]^{5}$$
\(768660727987\) Copy content Toggle raw display $\Q_{768660727987}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{768660727987}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{768660727987}$$x$$1$$1$$0$Trivial$$[\ ]$$
Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $6$$1$$6$$0$$C_6$$$[\ ]^{6}$$

Spectrum of ring of integers

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