Properties

Label 13.3.134713017952000.1
Degree $13$
Signature $[3, 5]$
Discriminant $-1.347\times 10^{14}$
Root discriminant \(12.21\)
Ramified primes $2,5,4209781811$
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 - 5*x^12 + 15*x^11 - 34*x^10 + 56*x^9 - 79*x^8 + 85*x^7 - 78*x^6 + 53*x^5 - 25*x^4 + 6*x^3 + 4*x^2 - 2*x + 1)
 
gp: K = bnfinit(y^13 - 5*y^12 + 15*y^11 - 34*y^10 + 56*y^9 - 79*y^8 + 85*y^7 - 78*y^6 + 53*y^5 - 25*y^4 + 6*y^3 + 4*y^2 - 2*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^13 - 5*x^12 + 15*x^11 - 34*x^10 + 56*x^9 - 79*x^8 + 85*x^7 - 78*x^6 + 53*x^5 - 25*x^4 + 6*x^3 + 4*x^2 - 2*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^13 - 5*x^12 + 15*x^11 - 34*x^10 + 56*x^9 - 79*x^8 + 85*x^7 - 78*x^6 + 53*x^5 - 25*x^4 + 6*x^3 + 4*x^2 - 2*x + 1)
 

\( x^{13} - 5 x^{12} + 15 x^{11} - 34 x^{10} + 56 x^{9} - 79 x^{8} + 85 x^{7} - 78 x^{6} + 53 x^{5} - 25 x^{4} + 6 x^{3} + 4 x^{2} - 2 x + 1 \) 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:  $[3, 5]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-134713017952000\) \(\medspace = -\,2^{8}\cdot 5^{3}\cdot 4209781811\) 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:  \(12.21\)
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:  $2^{2}5^{3/4}4209781811^{1/2}\approx 867795.2892169062$
Ramified primes:   \(2\), \(5\), \(4209781811\) 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:  $\Q(\sqrt{-21048909055}$)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3}a^{10}-\frac{1}{3}a^{6}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{3}a^{11}-\frac{1}{3}a^{7}+\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{3}a^{12}-\frac{1}{3}a^{8}+\frac{1}{3}a^{6}+\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}$ 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:  $7$
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:   $a^{12}-5a^{11}+15a^{10}-34a^{9}+56a^{8}-79a^{7}+85a^{6}-78a^{5}+53a^{4}-25a^{3}+6a^{2}+4a-2$, $a^{12}-\frac{14}{3}a^{11}+13a^{10}-28a^{9}+43a^{8}-\frac{175}{3}a^{7}+59a^{6}-\frac{152}{3}a^{5}+\frac{97}{3}a^{4}-\frac{35}{3}a^{3}+\frac{8}{3}a^{2}+\frac{11}{3}a-1$, $\frac{4}{3}a^{12}-7a^{11}+20a^{10}-43a^{9}+\frac{203}{3}a^{8}-88a^{7}+\frac{274}{3}a^{6}-\frac{224}{3}a^{5}+\frac{142}{3}a^{4}-\frac{52}{3}a^{3}+\frac{5}{3}a^{2}+3a-2$, $\frac{1}{3}a^{12}-a^{11}+\frac{2}{3}a^{10}+2a^{9}-\frac{31}{3}a^{8}+21a^{7}-\frac{91}{3}a^{6}+\frac{103}{3}a^{5}-27a^{4}+\frac{46}{3}a^{3}-\frac{11}{3}a^{2}-\frac{11}{3}a+\frac{4}{3}$, $\frac{4}{3}a^{12}-7a^{11}+\frac{62}{3}a^{10}-45a^{9}+\frac{212}{3}a^{8}-92a^{7}+\frac{275}{3}a^{6}-\frac{221}{3}a^{5}+42a^{4}-\frac{35}{3}a^{3}-\frac{5}{3}a^{2}+\frac{13}{3}a-\frac{2}{3}$, $a^{12}-\frac{14}{3}a^{11}+\frac{37}{3}a^{10}-25a^{9}+35a^{8}-\frac{124}{3}a^{7}+\frac{104}{3}a^{6}-\frac{62}{3}a^{5}+\frac{14}{3}a^{4}+\frac{20}{3}a^{3}-7a^{2}+\frac{10}{3}a+\frac{2}{3}$, $\frac{2}{3}a^{12}-4a^{11}+\frac{38}{3}a^{10}-29a^{9}+\frac{148}{3}a^{8}-67a^{7}+73a^{6}-\frac{187}{3}a^{5}+\frac{118}{3}a^{4}-14a^{3}-2a^{2}+\frac{16}{3}a-\frac{5}{3}$ 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:  \( 115.220069967 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{5}\cdot 115.220069967 \cdot 1}{2\cdot\sqrt{134713017952000}}\cr\approx \mathstrut & 0.388850520749 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^13 - 5*x^12 + 15*x^11 - 34*x^10 + 56*x^9 - 79*x^8 + 85*x^7 - 78*x^6 + 53*x^5 - 25*x^4 + 6*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 - 5*x^12 + 15*x^11 - 34*x^10 + 56*x^9 - 79*x^8 + 85*x^7 - 78*x^6 + 53*x^5 - 25*x^4 + 6*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 - 5*x^12 + 15*x^11 - 34*x^10 + 56*x^9 - 79*x^8 + 85*x^7 - 78*x^6 + 53*x^5 - 25*x^4 + 6*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 - 5*x^12 + 15*x^11 - 34*x^10 + 56*x^9 - 79*x^8 + 85*x^7 - 78*x^6 + 53*x^5 - 25*x^4 + 6*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 $S_{13}$ 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 R ${\href{/padicField/3.7.0.1}{7} }{,}\,{\href{/padicField/3.6.0.1}{6} }$ R ${\href{/padicField/7.13.0.1}{13} }$ ${\href{/padicField/11.10.0.1}{10} }{,}\,{\href{/padicField/11.3.0.1}{3} }$ ${\href{/padicField/13.11.0.1}{11} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ ${\href{/padicField/19.13.0.1}{13} }$ ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.5.0.1}{5} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ ${\href{/padicField/31.9.0.1}{9} }{,}\,{\href{/padicField/31.4.0.1}{4} }$ ${\href{/padicField/37.11.0.1}{11} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ ${\href{/padicField/43.11.0.1}{11} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ ${\href{/padicField/47.9.0.1}{9} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.3.0.1}{3} }$

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

# 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
\(2\) Copy content Toggle raw display 2.4.8.2$x^{4} + 2 x^{2} + 4 x + 2$$4$$1$$8$$C_2^2$$[2, 3]$
2.9.0.1$x^{9} + x^{4} + 1$$1$$9$$0$$C_9$$[\ ]^{9}$
\(5\) Copy content Toggle raw display 5.4.3.2$x^{4} + 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.9.0.1$x^{9} + 2 x^{3} + x + 3$$1$$9$$0$$C_9$$[\ ]^{9}$
\(4209781811\) Copy content Toggle raw display Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $3$$1$$3$$0$$C_3$$[\ ]^{3}$
Deg $6$$1$$6$$0$$C_6$$[\ ]^{6}$