Properties

Label 13.5.28261626739249.1
Degree $13$
Signature $[5, 4]$
Discriminant $2.826\times 10^{13}$
Root discriminant \(10.83\)
Ramified primes $2161,13078031809$
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 - 3*x^12 + 3*x^11 + 3*x^10 - 13*x^9 + 14*x^8 + 2*x^7 - 20*x^6 + 18*x^5 + x^4 - 11*x^3 + 6*x^2 + x - 1)
 
gp: K = bnfinit(y^13 - 3*y^12 + 3*y^11 + 3*y^10 - 13*y^9 + 14*y^8 + 2*y^7 - 20*y^6 + 18*y^5 + y^4 - 11*y^3 + 6*y^2 + y - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^13 - 3*x^12 + 3*x^11 + 3*x^10 - 13*x^9 + 14*x^8 + 2*x^7 - 20*x^6 + 18*x^5 + x^4 - 11*x^3 + 6*x^2 + x - 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^13 - 3*x^12 + 3*x^11 + 3*x^10 - 13*x^9 + 14*x^8 + 2*x^7 - 20*x^6 + 18*x^5 + x^4 - 11*x^3 + 6*x^2 + x - 1)
 

\( x^{13} - 3 x^{12} + 3 x^{11} + 3 x^{10} - 13 x^{9} + 14 x^{8} + 2 x^{7} - 20 x^{6} + 18 x^{5} + x^{4} - 11 x^{3} + 6 x^{2} + x - 1 \) 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:  $[5, 4]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(28261626739249\) \(\medspace = 2161\cdot 13078031809\) 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:  \(10.83\)
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:  $2161^{1/2}13078031809^{1/2}\approx 5316166.545477013$
Ramified primes:   \(2161\), \(13078031809\) 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:  $\Q(\sqrt{28261626739249}$)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{11}a^{12}+\frac{3}{11}a^{10}+\frac{1}{11}a^{9}+\frac{1}{11}a^{8}-\frac{5}{11}a^{7}-\frac{2}{11}a^{6}-\frac{4}{11}a^{5}-\frac{5}{11}a^{4}-\frac{3}{11}a^{3}+\frac{2}{11}a^{2}+\frac{1}{11}a+\frac{4}{11}$ 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:  $8$
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:   $\frac{13}{11}a^{12}-3a^{11}+\frac{28}{11}a^{10}+\frac{35}{11}a^{9}-\frac{130}{11}a^{8}+\frac{122}{11}a^{7}+\frac{29}{11}a^{6}-\frac{162}{11}a^{5}+\frac{122}{11}a^{4}+\frac{16}{11}a^{3}-\frac{62}{11}a^{2}+\frac{24}{11}a+\frac{8}{11}$, $a$, $\frac{7}{11}a^{12}-a^{11}-\frac{1}{11}a^{10}+\frac{29}{11}a^{9}-\frac{48}{11}a^{8}+\frac{9}{11}a^{7}+\frac{52}{11}a^{6}-\frac{50}{11}a^{5}-\frac{2}{11}a^{4}+\frac{23}{11}a^{3}+\frac{3}{11}a^{2}-\frac{4}{11}a+\frac{6}{11}$, $\frac{4}{11}a^{12}-a^{11}+\frac{12}{11}a^{10}+\frac{15}{11}a^{9}-\frac{62}{11}a^{8}+\frac{68}{11}a^{7}+\frac{3}{11}a^{6}-\frac{104}{11}a^{5}+\frac{112}{11}a^{4}-\frac{12}{11}a^{3}-\frac{47}{11}a^{2}+\frac{37}{11}a+\frac{5}{11}$, $\frac{5}{11}a^{12}-a^{11}+\frac{15}{11}a^{10}+\frac{5}{11}a^{9}-\frac{50}{11}a^{8}+\frac{63}{11}a^{7}-\frac{21}{11}a^{6}-\frac{64}{11}a^{5}+\frac{96}{11}a^{4}-\frac{15}{11}a^{3}-\frac{45}{11}a^{2}+\frac{38}{11}a+\frac{9}{11}$, $\frac{7}{11}a^{12}-2a^{11}+\frac{32}{11}a^{10}-\frac{4}{11}a^{9}-\frac{70}{11}a^{8}+\frac{119}{11}a^{7}-\frac{69}{11}a^{6}-\frac{61}{11}a^{5}+\frac{130}{11}a^{4}-\frac{65}{11}a^{3}-\frac{19}{11}a^{2}+\frac{40}{11}a-\frac{5}{11}$, $\frac{10}{11}a^{12}-2a^{11}+\frac{8}{11}a^{10}+\frac{43}{11}a^{9}-\frac{89}{11}a^{8}+\frac{38}{11}a^{7}+\frac{90}{11}a^{6}-\frac{117}{11}a^{5}+\frac{5}{11}a^{4}+\frac{91}{11}a^{3}-\frac{35}{11}a^{2}-\frac{23}{11}a+\frac{18}{11}$, $\frac{10}{11}a^{12}-2a^{11}+\frac{19}{11}a^{10}+\frac{21}{11}a^{9}-\frac{89}{11}a^{8}+\frac{93}{11}a^{7}-\frac{9}{11}a^{6}-\frac{95}{11}a^{5}+\frac{126}{11}a^{4}-\frac{52}{11}a^{3}-\frac{24}{11}a^{2}+\frac{54}{11}a-\frac{15}{11}$ 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:  \( 33.0726626498 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{5}\cdot(2\pi)^{4}\cdot 33.0726626498 \cdot 1}{2\cdot\sqrt{28261626739249}}\cr\approx \mathstrut & 0.155135088919 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^13 - 3*x^12 + 3*x^11 + 3*x^10 - 13*x^9 + 14*x^8 + 2*x^7 - 20*x^6 + 18*x^5 + x^4 - 11*x^3 + 6*x^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 - 3*x^12 + 3*x^11 + 3*x^10 - 13*x^9 + 14*x^8 + 2*x^7 - 20*x^6 + 18*x^5 + x^4 - 11*x^3 + 6*x^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 - 3*x^12 + 3*x^11 + 3*x^10 - 13*x^9 + 14*x^8 + 2*x^7 - 20*x^6 + 18*x^5 + x^4 - 11*x^3 + 6*x^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 - 3*x^12 + 3*x^11 + 3*x^10 - 13*x^9 + 14*x^8 + 2*x^7 - 20*x^6 + 18*x^5 + x^4 - 11*x^3 + 6*x^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 $S_{13}$ 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 ${\href{/padicField/2.13.0.1}{13} }$ ${\href{/padicField/3.13.0.1}{13} }$ ${\href{/padicField/5.13.0.1}{13} }$ ${\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ ${\href{/padicField/11.11.0.1}{11} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ ${\href{/padicField/13.5.0.1}{5} }{,}\,{\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }$ ${\href{/padicField/17.13.0.1}{13} }$ ${\href{/padicField/19.7.0.1}{7} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ ${\href{/padicField/23.7.0.1}{7} }{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}$ ${\href{/padicField/29.7.0.1}{7} }{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ ${\href{/padicField/31.13.0.1}{13} }$ ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.3.0.1}{3} }$ ${\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}$ ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ ${\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ ${\href{/padicField/59.5.0.1}{5} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.1.0.1}{1} }$

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
\(2161\) Copy content Toggle raw display $\Q_{2161}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $6$$1$$6$$0$$C_6$$[\ ]^{6}$
\(13078031809\) Copy content Toggle raw display $\Q_{13078031809}$$x$$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}$