Properties

Label 13.1.1325925503633.1
Degree $13$
Signature $[1, 6]$
Discriminant $1.326\times 10^{12}$
Root discriminant \(8.56\)
Ramified primes $97,13669335089$
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^12 - x^11 + x^10 + 3*x^8 - x^7 - 5*x^6 + x^5 + 4*x^4 + 2*x^3 - 3*x^2 - x + 1)
 
gp: K = bnfinit(y^13 - y^12 - y^11 + y^10 + 3*y^8 - y^7 - 5*y^6 + y^5 + 4*y^4 + 2*y^3 - 3*y^2 - y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^13 - x^12 - x^11 + x^10 + 3*x^8 - x^7 - 5*x^6 + x^5 + 4*x^4 + 2*x^3 - 3*x^2 - x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^13 - x^12 - x^11 + x^10 + 3*x^8 - x^7 - 5*x^6 + x^5 + 4*x^4 + 2*x^3 - 3*x^2 - x + 1)
 

\( x^{13} - x^{12} - x^{11} + x^{10} + 3x^{8} - x^{7} - 5x^{6} + x^{5} + 4x^{4} + 2x^{3} - 3x^{2} - x + 1 \) 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:  $[1, 6]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(1325925503633\) \(\medspace = 97\cdot 13669335089\) 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:  \(8.56\)
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:  $97^{1/2}13669335089^{1/2}\approx 1151488.386234529$
Ramified primes:   \(97\), \(13669335089\) 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:  $\Q(\sqrt{1325925503633}$)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{97}a^{12}-\frac{40}{97}a^{11}+\frac{7}{97}a^{10}+\frac{19}{97}a^{9}+\frac{35}{97}a^{8}-\frac{4}{97}a^{7}-\frac{39}{97}a^{6}-\frac{36}{97}a^{5}+\frac{47}{97}a^{4}+\frac{14}{97}a^{3}+\frac{38}{97}a^{2}-\frac{30}{97}a+\frac{5}{97}$ 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:   $a$, $\frac{43}{97}a^{12}+\frac{26}{97}a^{11}-\frac{87}{97}a^{10}-\frac{56}{97}a^{9}+\frac{50}{97}a^{8}+\frac{119}{97}a^{7}+\frac{166}{97}a^{6}-\frac{190}{97}a^{5}-\frac{307}{97}a^{4}+\frac{117}{97}a^{3}+\frac{373}{97}a^{2}+\frac{68}{97}a-\frac{173}{97}$, $\frac{43}{97}a^{12}+\frac{26}{97}a^{11}-\frac{87}{97}a^{10}-\frac{56}{97}a^{9}+\frac{50}{97}a^{8}+\frac{119}{97}a^{7}+\frac{166}{97}a^{6}-\frac{190}{97}a^{5}-\frac{307}{97}a^{4}+\frac{117}{97}a^{3}+\frac{276}{97}a^{2}+\frac{68}{97}a-\frac{173}{97}$, $\frac{161}{97}a^{12}-\frac{135}{97}a^{11}-\frac{134}{97}a^{10}+\frac{52}{97}a^{9}+\frac{9}{97}a^{8}+\frac{520}{97}a^{7}-\frac{71}{97}a^{6}-\frac{655}{97}a^{5}-\frac{96}{97}a^{4}+\frac{411}{97}a^{3}+\frac{492}{97}a^{2}-\frac{174}{97}a-\frac{165}{97}$, $\frac{51}{97}a^{12}-\frac{3}{97}a^{11}-\frac{128}{97}a^{10}-\frac{1}{97}a^{9}+\frac{39}{97}a^{8}+\frac{184}{97}a^{7}+\frac{145}{97}a^{6}-\frac{381}{97}a^{5}-\frac{222}{97}a^{4}+\frac{229}{97}a^{3}+\frac{386}{97}a^{2}+\frac{22}{97}a-\frac{230}{97}$, $\frac{65}{97}a^{12}+\frac{19}{97}a^{11}-\frac{127}{97}a^{10}-\frac{26}{97}a^{9}+\frac{44}{97}a^{8}+\frac{225}{97}a^{7}+\frac{181}{97}a^{6}-\frac{303}{97}a^{5}-\frac{340}{97}a^{4}+\frac{231}{97}a^{3}+\frac{433}{97}a^{2}+\frac{87}{97}a-\frac{160}{97}$ 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:  \( 3.1288056301 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{6}\cdot 3.1288056301 \cdot 1}{2\cdot\sqrt{1325925503633}}\cr\approx \mathstrut & 0.16718535530 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^13 - x^12 - x^11 + x^10 + 3*x^8 - x^7 - 5*x^6 + x^5 + 4*x^4 + 2*x^3 - 3*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 - x^12 - x^11 + x^10 + 3*x^8 - x^7 - 5*x^6 + x^5 + 4*x^4 + 2*x^3 - 3*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 - x^12 - x^11 + x^10 + 3*x^8 - x^7 - 5*x^6 + x^5 + 4*x^4 + 2*x^3 - 3*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 - x^12 - x^11 + x^10 + 3*x^8 - x^7 - 5*x^6 + x^5 + 4*x^4 + 2*x^3 - 3*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 ${\href{/padicField/2.13.0.1}{13} }$ ${\href{/padicField/3.7.0.1}{7} }{,}\,{\href{/padicField/3.6.0.1}{6} }$ ${\href{/padicField/5.9.0.1}{9} }{,}\,{\href{/padicField/5.4.0.1}{4} }$ ${\href{/padicField/7.11.0.1}{11} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ ${\href{/padicField/11.13.0.1}{13} }$ ${\href{/padicField/13.13.0.1}{13} }$ ${\href{/padicField/17.13.0.1}{13} }$ ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.5.0.1}{5} }$ ${\href{/padicField/23.5.0.1}{5} }{,}\,{\href{/padicField/23.4.0.1}{4} }^{2}$ ${\href{/padicField/29.13.0.1}{13} }$ ${\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.5.0.1}{5} }$ ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ ${\href{/padicField/41.5.0.1}{5} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ ${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.5.0.1}{5} }$ ${\href{/padicField/47.5.0.1}{5} }{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.9.0.1}{9} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.11.0.1}{11} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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
\(97\) Copy content Toggle raw display $\Q_{97}$$x + 92$$1$$1$$0$Trivial$[\ ]$
97.2.1.2$x^{2} + 485$$2$$1$$1$$C_2$$[\ ]_{2}$
97.3.0.1$x^{3} + 9 x + 92$$1$$3$$0$$C_3$$[\ ]^{3}$
97.7.0.1$x^{7} + 5 x + 92$$1$$7$$0$$C_7$$[\ ]^{7}$
\(13669335089\) 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}$