Properties

Label 13.3.9197165262683.1
Degree $13$
Signature $[3, 5]$
Discriminant $-9.197\times 10^{12}$
Root discriminant \(9.94\)
Ramified primes $82163,111938041$
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^10 + 2*x^9 - x^8 - 3*x^7 - 3*x^6 + 9*x^5 - 10*x^4 + 7*x^3 + 1)
 
gp: K = bnfinit(y^13 - 2*y^10 + 2*y^9 - y^8 - 3*y^7 - 3*y^6 + 9*y^5 - 10*y^4 + 7*y^3 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^13 - 2*x^10 + 2*x^9 - x^8 - 3*x^7 - 3*x^6 + 9*x^5 - 10*x^4 + 7*x^3 + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^13 - 2*x^10 + 2*x^9 - x^8 - 3*x^7 - 3*x^6 + 9*x^5 - 10*x^4 + 7*x^3 + 1)
 

\( x^{13} - 2x^{10} + 2x^{9} - x^{8} - 3x^{7} - 3x^{6} + 9x^{5} - 10x^{4} + 7x^{3} + 1 \) 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:   \(-9197165262683\) \(\medspace = -\,82163\cdot 111938041\) 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:  \(9.94\)
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:  $82163^{1/2}111938041^{1/2}\approx 3032682.849010592$
Ramified primes:   \(82163\), \(111938041\) 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:  $\Q(\sqrt{-9197165262683}$)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7979}a^{12}-\frac{1670}{7979}a^{11}-\frac{3750}{7979}a^{10}-\frac{1017}{7979}a^{9}-\frac{1135}{7979}a^{8}-\frac{3553}{7979}a^{7}-\frac{2869}{7979}a^{6}+\frac{3827}{7979}a^{5}+\frac{98}{7979}a^{4}+\frac{3889}{7979}a^{3}+\frac{283}{7979}a^{2}-\frac{1849}{7979}a-\frac{43}{7979}$ 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:   $\frac{2597}{7979}a^{12}-\frac{4393}{7979}a^{11}-\frac{4370}{7979}a^{10}-\frac{8079}{7979}a^{9}+\frac{12614}{7979}a^{8}-\frac{3417}{7979}a^{7}-\frac{6386}{7979}a^{6}+\frac{4864}{7979}a^{5}+\frac{47052}{7979}a^{4}-\frac{49555}{7979}a^{3}+\frac{32799}{7979}a^{2}-\frac{6474}{7979}a-\frac{7944}{7979}$, $\frac{6169}{7979}a^{12}-\frac{1341}{7979}a^{11}-\frac{2629}{7979}a^{10}-\frac{10358}{7979}a^{9}+\frac{19705}{7979}a^{8}-\frac{144}{7979}a^{7}-\frac{25376}{7979}a^{6}-\frac{17056}{7979}a^{5}+\frac{69969}{7979}a^{4}-\frac{65444}{7979}a^{3}+\frac{14384}{7979}a^{2}+\frac{11468}{7979}a-\frac{9939}{7979}$, $a$, $\frac{3882}{7979}a^{12}+\frac{3987}{7979}a^{11}+\frac{4175}{7979}a^{10}-\frac{6368}{7979}a^{9}-\frac{1662}{7979}a^{8}-\frac{5034}{7979}a^{7}-\frac{6753}{7979}a^{6}-\frac{16442}{7979}a^{5}+\frac{13402}{7979}a^{4}-\frac{23107}{7979}a^{3}+\frac{21441}{7979}a^{2}+\frac{3282}{7979}a+\frac{633}{7979}$, $\frac{5929}{7979}a^{12}+\frac{509}{7979}a^{11}-\frac{4256}{7979}a^{10}-\frac{13627}{7979}a^{9}+\frac{12840}{7979}a^{8}+\frac{6802}{7979}a^{7}-\frac{23010}{7979}a^{6}-\frac{25930}{7979}a^{5}+\frac{62407}{7979}a^{4}-\frac{33345}{7979}a^{3}+\frac{2317}{7979}a^{2}+\frac{16383}{7979}a-\frac{7598}{7979}$, $\frac{2809}{7979}a^{12}+\frac{622}{7979}a^{11}-\frac{1470}{7979}a^{10}-\frac{8250}{7979}a^{9}+\frac{3385}{7979}a^{8}+\frac{1352}{7979}a^{7}-\frac{8210}{7979}a^{6}-\frac{13628}{7979}a^{5}+\frac{27933}{7979}a^{4}-\frac{7029}{7979}a^{3}+\frac{13005}{7979}a^{2}+\frac{488}{7979}a+\frac{6877}{7979}$, $\frac{1830}{7979}a^{12}-\frac{143}{7979}a^{11}-\frac{560}{7979}a^{10}-\frac{2003}{7979}a^{9}+\frac{5469}{7979}a^{8}+\frac{895}{7979}a^{7}-\frac{8067}{7979}a^{6}-\frac{2152}{7979}a^{5}+\frac{19760}{7979}a^{4}-\frac{24335}{7979}a^{3}-\frac{745}{7979}a^{2}+\frac{7405}{7979}a-\frac{6879}{7979}$ 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:  \( 13.0679687985 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{5}\cdot 13.0679687985 \cdot 1}{2\cdot\sqrt{9197165262683}}\cr\approx \mathstrut & 0.168787556802 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^13 - 2*x^10 + 2*x^9 - x^8 - 3*x^7 - 3*x^6 + 9*x^5 - 10*x^4 + 7*x^3 + 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^10 + 2*x^9 - x^8 - 3*x^7 - 3*x^6 + 9*x^5 - 10*x^4 + 7*x^3 + 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^10 + 2*x^9 - x^8 - 3*x^7 - 3*x^6 + 9*x^5 - 10*x^4 + 7*x^3 + 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^10 + 2*x^9 - x^8 - 3*x^7 - 3*x^6 + 9*x^5 - 10*x^4 + 7*x^3 + 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 ${\href{/padicField/2.9.0.1}{9} }{,}\,{\href{/padicField/2.4.0.1}{4} }$ ${\href{/padicField/3.13.0.1}{13} }$ ${\href{/padicField/5.9.0.1}{9} }{,}\,{\href{/padicField/5.4.0.1}{4} }$ ${\href{/padicField/7.13.0.1}{13} }$ ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.3.0.1}{3} }$ ${\href{/padicField/13.9.0.1}{9} }{,}\,{\href{/padicField/13.4.0.1}{4} }$ ${\href{/padicField/17.11.0.1}{11} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ ${\href{/padicField/19.9.0.1}{9} }{,}\,{\href{/padicField/19.4.0.1}{4} }$ ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.5.0.1}{5} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ ${\href{/padicField/29.7.0.1}{7} }{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}$ ${\href{/padicField/31.9.0.1}{9} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }$ ${\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.5.0.1}{5} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ ${\href{/padicField/43.13.0.1}{13} }$ ${\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.6.0.1}{6} }$ ${\href{/padicField/53.11.0.1}{11} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ ${\href{/padicField/59.5.0.1}{5} }^{2}{,}\,{\href{/padicField/59.3.0.1}{3} }$

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
\(82163\) Copy content Toggle raw display Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $11$$1$$11$$0$$C_{11}$$[\ ]^{11}$
\(111938041\) Copy content Toggle raw display $\Q_{111938041}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $5$$1$$5$$0$$C_5$$[\ ]^{5}$
Deg $5$$1$$5$$0$$C_5$$[\ ]^{5}$