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Properties

Label 13.1.3518110599125.1
Degree $13$
Signature $[1, 6]$
Discriminant $3.518\times 10^{12}$
Root discriminant \(9.23\)
Ramified primes $5,5351,5259743$
Class number $1$
Class group trivial
Galois group $S_{13}$ (as 13T9)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^13 + x^11 - 2*x^10 + x^9 - x^8 - 4*x^7 + 4*x^6 - 5*x^5 + x^4 + 2*x^3 - 4*x^2 + 2*x - 1)
 
gp: K = bnfinit(y^13 + y^11 - 2*y^10 + y^9 - y^8 - 4*y^7 + 4*y^6 - 5*y^5 + y^4 + 2*y^3 - 4*y^2 + 2*y - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^13 + x^11 - 2*x^10 + x^9 - x^8 - 4*x^7 + 4*x^6 - 5*x^5 + x^4 + 2*x^3 - 4*x^2 + 2*x - 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^13 + x^11 - 2*x^10 + x^9 - x^8 - 4*x^7 + 4*x^6 - 5*x^5 + x^4 + 2*x^3 - 4*x^2 + 2*x - 1)
 

\( x^{13} + x^{11} - 2x^{10} + x^{9} - x^{8} - 4x^{7} + 4x^{6} - 5x^{5} + x^{4} + 2x^{3} - 4x^{2} + 2x - 1 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $13$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[1, 6]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(3518110599125\) \(\medspace = 5^{3}\cdot 5351\cdot 5259743\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(9.23\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $5^{3/4}5351^{1/2}5259743^{1/2}\approx 560953.9892720619$
Ramified primes:   \(5\), \(5351\), \(5259743\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  $\Q(\sqrt{140724423965}$)
$\card{ \Aut(K/\Q) }$:  $1$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{793}a^{12}-\frac{290}{793}a^{11}+\frac{43}{793}a^{10}+\frac{216}{793}a^{9}+\frac{8}{793}a^{8}+\frac{58}{793}a^{7}-\frac{171}{793}a^{6}-\frac{365}{793}a^{5}+\frac{376}{793}a^{4}+\frac{395}{793}a^{3}-\frac{356}{793}a^{2}+\frac{146}{793}a-\frac{309}{793}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

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

Class group and class number

Trivial group, which has order $1$

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 
oscar: class_group(K)
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $6$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:   $a$, $\frac{293}{793}a^{12}-\frac{119}{793}a^{11}-\frac{89}{793}a^{10}-\frac{945}{793}a^{9}-\frac{35}{793}a^{8}+\frac{341}{793}a^{7}-\frac{937}{793}a^{6}+\frac{2489}{793}a^{5}-\frac{852}{793}a^{4}+\frac{750}{793}a^{3}+\frac{1954}{793}a^{2}-\frac{1630}{793}a+\frac{658}{793}$, $\frac{217}{793}a^{12}+\frac{510}{793}a^{11}+\frac{608}{793}a^{10}+\frac{85}{793}a^{9}-\frac{643}{793}a^{8}-\frac{895}{793}a^{7}-\frac{1422}{793}a^{6}-\frac{1491}{793}a^{5}-\frac{87}{793}a^{4}-\frac{722}{793}a^{3}+\frac{462}{793}a^{2}+\frac{755}{793}a-\frac{441}{793}$, $\frac{148}{793}a^{12}-\frac{98}{793}a^{11}+\frac{20}{793}a^{10}-\frac{545}{793}a^{9}+\frac{391}{793}a^{8}-\frac{139}{793}a^{7}+\frac{68}{793}a^{6}+\frac{697}{793}a^{5}+\frac{138}{793}a^{4}-\frac{222}{793}a^{3}-\frac{350}{793}a^{2}-\frac{596}{793}a-\frac{531}{793}$, $\frac{181}{793}a^{12}+\frac{641}{793}a^{11}+\frac{646}{793}a^{10}+\frac{239}{793}a^{9}-\frac{931}{793}a^{8}-\frac{604}{793}a^{7}-\frac{817}{793}a^{6}-\frac{1832}{793}a^{5}-\frac{142}{793}a^{4}-\frac{1461}{793}a^{3}-\frac{203}{793}a^{2}+\frac{257}{793}a-\frac{1212}{793}$, $\frac{376}{793}a^{12}+\frac{394}{793}a^{11}+\frac{308}{793}a^{10}-\frac{463}{793}a^{9}-\frac{164}{793}a^{8}+\frac{397}{793}a^{7}-\frac{1649}{793}a^{6}-\frac{844}{793}a^{5}-\frac{571}{793}a^{4}-\frac{1357}{793}a^{3}+\frac{161}{793}a^{2}-\frac{614}{793}a-\frac{406}{793}$ Copy content Toggle raw display
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 6.17827465513 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{6}\cdot 6.17827465513 \cdot 1}{2\cdot\sqrt{3518110599125}}\cr\approx \mathstrut & 0.202671031080 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^13 + x^11 - 2*x^10 + x^9 - x^8 - 4*x^7 + 4*x^6 - 5*x^5 + x^4 + 2*x^3 - 4*x^2 + 2*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))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^13 + x^11 - 2*x^10 + x^9 - x^8 - 4*x^7 + 4*x^6 - 5*x^5 + x^4 + 2*x^3 - 4*x^2 + 2*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)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^13 + x^11 - 2*x^10 + x^9 - x^8 - 4*x^7 + 4*x^6 - 5*x^5 + x^4 + 2*x^3 - 4*x^2 + 2*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)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^13 + x^11 - 2*x^10 + x^9 - x^8 - 4*x^7 + 4*x^6 - 5*x^5 + x^4 + 2*x^3 - 4*x^2 + 2*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):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
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 $\Q$.
sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
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 ${\href{/padicField/2.7.0.1}{7} }{,}\,{\href{/padicField/2.6.0.1}{6} }$ ${\href{/padicField/3.9.0.1}{9} }{,}\,{\href{/padicField/3.4.0.1}{4} }$ R ${\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.3.0.1}{3} }$ ${\href{/padicField/11.13.0.1}{13} }$ ${\href{/padicField/13.11.0.1}{11} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ ${\href{/padicField/17.13.0.1}{13} }$ ${\href{/padicField/19.9.0.1}{9} }{,}\,{\href{/padicField/19.4.0.1}{4} }$ ${\href{/padicField/23.7.0.1}{7} }{,}\,{\href{/padicField/23.6.0.1}{6} }$ ${\href{/padicField/29.11.0.1}{11} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ ${\href{/padicField/31.7.0.1}{7} }{,}\,{\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ ${\href{/padicField/37.7.0.1}{7} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ ${\href{/padicField/41.9.0.1}{9} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.5.0.1}{5} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ ${\href{/padicField/47.11.0.1}{11} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.3.0.1}{3} }$ ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.3.0.1}{3} }^{3}$

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

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ 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 5.4.3.3$x^{4} + 10$$4$$1$$3$$C_4$$[\ ]_{4}$
5.9.0.1$x^{9} + 2 x^{3} + x + 3$$1$$9$$0$$C_9$$[\ ]^{9}$
\(5351\) Copy content Toggle raw display Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $3$$1$$3$$0$$C_3$$[\ ]^{3}$
Deg $8$$1$$8$$0$$C_8$$[\ ]^{8}$
\(5259743\) Copy content Toggle raw display $\Q_{5259743}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{5259743}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $7$$1$$7$$0$$C_7$$[\ ]^{7}$