Properties

Label 13.7.146839998625007.1
Degree $13$
Signature $[7, 3]$
Discriminant $-1.468\times 10^{14}$
Root discriminant \(12.30\)
Ramified prime $146839998625007$
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 - 2*x^11 - 3*x^10 - 4*x^9 + 7*x^8 + 6*x^7 - 4*x^6 - 2*x^5 - 5*x^4 + 5*x^3 + 3*x^2 - 4*x + 1)
 
gp: K = bnfinit(y^13 - 2*y^11 - 3*y^10 - 4*y^9 + 7*y^8 + 6*y^7 - 4*y^6 - 2*y^5 - 5*y^4 + 5*y^3 + 3*y^2 - 4*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^13 - 2*x^11 - 3*x^10 - 4*x^9 + 7*x^8 + 6*x^7 - 4*x^6 - 2*x^5 - 5*x^4 + 5*x^3 + 3*x^2 - 4*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^13 - 2*x^11 - 3*x^10 - 4*x^9 + 7*x^8 + 6*x^7 - 4*x^6 - 2*x^5 - 5*x^4 + 5*x^3 + 3*x^2 - 4*x + 1)
 

\( x^{13} - 2x^{11} - 3x^{10} - 4x^{9} + 7x^{8} + 6x^{7} - 4x^{6} - 2x^{5} - 5x^{4} + 5x^{3} + 3x^{2} - 4x + 1 \) 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:  $[7, 3]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-146839998625007\) 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:  \(12.30\)
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:  $146839998625007^{1/2}\approx 12117755.511026248$
Ramified primes:   \(146839998625007\) 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:  $\Q(\sqrt{-146839998625007}$)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$ Copy content Toggle raw display

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

Monogenic:  Yes
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:  $9$
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:   $a$, $10a^{12}+9a^{11}-12a^{10}-42a^{9}-78a^{8}+a^{7}+66a^{6}+26a^{5}+a^{4}-52a^{3}-a^{2}+31a-9$, $13a^{12}+11a^{11}-17a^{10}-54a^{9}-97a^{8}+10a^{7}+90a^{6}+25a^{5}-8a^{4}-71a^{3}+3a^{2}+44a-14$, $21a^{12}+14a^{11}-32a^{10}-85a^{9}-141a^{8}+51a^{7}+159a^{6}+27a^{5}-25a^{4}-121a^{3}+21a^{2}+75a-30$, $8a^{12}+8a^{11}-10a^{10}-35a^{9}-65a^{8}-a^{7}+60a^{6}+25a^{5}-46a^{3}-4a^{2}+28a-6$, $8a^{12}+8a^{11}-10a^{10}-35a^{9}-65a^{8}-a^{7}+60a^{6}+25a^{5}-46a^{3}-4a^{2}+28a-7$, $17a^{12}+13a^{11}-23a^{10}-70a^{9}-122a^{8}+23a^{7}+119a^{6}+32a^{5}-13a^{4}-94a^{3}+9a^{2}+57a-19$, $32a^{12}+23a^{11}-47a^{10}-130a^{9}-222a^{8}+63a^{7}+236a^{6}+45a^{5}-31a^{4}-183a^{3}+27a^{2}+114a-44$, $4a^{12}-9a^{10}-14a^{9}-15a^{8}+33a^{7}+36a^{6}-10a^{5}-17a^{4}-26a^{3}+17a^{2}+19a-12$ 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:  \( 115.283453799 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{7}\cdot(2\pi)^{3}\cdot 115.283453799 \cdot 1}{2\cdot\sqrt{146839998625007}}\cr\approx \mathstrut & 0.151030399887 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^13 - 2*x^11 - 3*x^10 - 4*x^9 + 7*x^8 + 6*x^7 - 4*x^6 - 2*x^5 - 5*x^4 + 5*x^3 + 3*x^2 - 4*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 - 2*x^11 - 3*x^10 - 4*x^9 + 7*x^8 + 6*x^7 - 4*x^6 - 2*x^5 - 5*x^4 + 5*x^3 + 3*x^2 - 4*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 - 2*x^11 - 3*x^10 - 4*x^9 + 7*x^8 + 6*x^7 - 4*x^6 - 2*x^5 - 5*x^4 + 5*x^3 + 3*x^2 - 4*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 - 2*x^11 - 3*x^10 - 4*x^9 + 7*x^8 + 6*x^7 - 4*x^6 - 2*x^5 - 5*x^4 + 5*x^3 + 3*x^2 - 4*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 $S_{13}$ are not computed
Character table for $S_{13}$ is not computed

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 ${\href{/padicField/2.13.0.1}{13} }$ ${\href{/padicField/3.13.0.1}{13} }$ ${\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.5.0.1}{5} }$ ${\href{/padicField/7.9.0.1}{9} }{,}\,{\href{/padicField/7.4.0.1}{4} }$ ${\href{/padicField/11.13.0.1}{13} }$ ${\href{/padicField/13.13.0.1}{13} }$ ${\href{/padicField/17.11.0.1}{11} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ ${\href{/padicField/19.9.0.1}{9} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ ${\href{/padicField/23.13.0.1}{13} }$ ${\href{/padicField/29.7.0.1}{7} }{,}\,{\href{/padicField/29.6.0.1}{6} }$ ${\href{/padicField/31.13.0.1}{13} }$ ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.3.0.1}{3} }$ ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ ${\href{/padicField/43.5.0.1}{5} }^{2}{,}\,{\href{/padicField/43.3.0.1}{3} }$ ${\href{/padicField/47.9.0.1}{9} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.5.0.1}{5} }$

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
\(146839998625007\) Copy content Toggle raw display $\Q_{146839998625007}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $3$$1$$3$$0$$C_3$$[\ ]^{3}$
Deg $5$$1$$5$$0$$C_5$$[\ ]^{5}$