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Properties

Label 13.1.48551226272641.1
Degree $13$
Signature $[1, 6]$
Discriminant $4.855\times 10^{13}$
Root discriminant \(11.29\)
Ramified prime $191$
Class number $1$
Class group trivial
Galois group $D_{13}$ (as 13T2)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^13 - 2*x^12 + 4*x^10 - 5*x^9 + x^8 + 5*x^7 - 11*x^6 + 19*x^5 - 22*x^4 + 16*x^3 - 10*x^2 + 6*x - 1)
 
gp: K = bnfinit(y^13 - 2*y^12 + 4*y^10 - 5*y^9 + y^8 + 5*y^7 - 11*y^6 + 19*y^5 - 22*y^4 + 16*y^3 - 10*y^2 + 6*y - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^13 - 2*x^12 + 4*x^10 - 5*x^9 + x^8 + 5*x^7 - 11*x^6 + 19*x^5 - 22*x^4 + 16*x^3 - 10*x^2 + 6*x - 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^13 - 2*x^12 + 4*x^10 - 5*x^9 + x^8 + 5*x^7 - 11*x^6 + 19*x^5 - 22*x^4 + 16*x^3 - 10*x^2 + 6*x - 1)
 

\( x^{13} - 2x^{12} + 4x^{10} - 5x^{9} + x^{8} + 5x^{7} - 11x^{6} + 19x^{5} - 22x^{4} + 16x^{3} - 10x^{2} + 6x - 1 \) 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:   \(48551226272641\) \(\medspace = 191^{6}\) 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:  \(11.29\)
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:  $191^{1/2}\approx 13.820274961085254$
Ramified primes:   \(191\) 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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{7}a^{11}+\frac{3}{7}a^{10}+\frac{3}{7}a^{9}-\frac{3}{7}a^{8}+\frac{2}{7}a^{6}+\frac{1}{7}a^{5}-\frac{2}{7}a^{4}-\frac{3}{7}a^{3}+\frac{1}{7}a^{2}+\frac{1}{7}a-\frac{3}{7}$, $\frac{1}{287}a^{12}-\frac{8}{287}a^{11}+\frac{89}{287}a^{10}-\frac{120}{287}a^{9}-\frac{23}{287}a^{8}+\frac{16}{287}a^{7}-\frac{13}{41}a^{6}+\frac{43}{287}a^{5}+\frac{89}{287}a^{4}-\frac{64}{287}a^{3}-\frac{10}{287}a^{2}+\frac{13}{41}a+\frac{75}{287}$ 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:   $\frac{92}{287}a^{12}-\frac{29}{41}a^{11}+\frac{29}{287}a^{10}+\frac{317}{287}a^{9}-\frac{558}{287}a^{8}+\frac{324}{287}a^{7}+\frac{443}{287}a^{6}-\frac{1251}{287}a^{5}+\frac{2243}{287}a^{4}-\frac{2321}{287}a^{3}+\frac{1622}{287}a^{2}-\frac{1140}{287}a+\frac{422}{287}$, $\frac{20}{287}a^{12}-\frac{78}{287}a^{11}+\frac{17}{287}a^{10}+\frac{142}{287}a^{9}-\frac{132}{287}a^{8}+\frac{33}{287}a^{7}+\frac{66}{287}a^{6}-\frac{206}{287}a^{5}+\frac{755}{287}a^{4}-\frac{136}{41}a^{3}+\frac{456}{287}a^{2}-\frac{394}{287}a+\frac{393}{287}$, $\frac{206}{287}a^{12}-\frac{254}{287}a^{11}-\frac{157}{287}a^{10}+\frac{100}{41}a^{9}-\frac{597}{287}a^{8}-\frac{148}{287}a^{7}+\frac{975}{287}a^{6}-\frac{1802}{287}a^{5}+\frac{2631}{287}a^{4}-\frac{2442}{287}a^{3}+\frac{1343}{287}a^{2}-\frac{811}{287}a+\frac{362}{287}$, $\frac{68}{287}a^{12}-\frac{25}{41}a^{11}-\frac{16}{287}a^{10}+\frac{409}{287}a^{9}-\frac{375}{287}a^{8}-\frac{60}{287}a^{7}+\frac{577}{287}a^{6}-\frac{725}{287}a^{5}+\frac{1296}{287}a^{4}-\frac{1441}{287}a^{3}+\frac{837}{287}a^{2}-\frac{331}{287}a+\frac{262}{287}$, $\frac{87}{287}a^{12}-\frac{204}{287}a^{11}+\frac{5}{41}a^{10}+\frac{507}{287}a^{9}-\frac{607}{287}a^{8}-\frac{43}{287}a^{7}+\frac{816}{287}a^{6}-\frac{1220}{287}a^{5}+\frac{1593}{287}a^{4}-\frac{1591}{287}a^{3}+\frac{151}{41}a^{2}-\frac{201}{287}a+\frac{170}{287}$, $\frac{135}{287}a^{12}-\frac{43}{41}a^{11}+\frac{2}{287}a^{10}+\frac{774}{287}a^{9}-\frac{850}{287}a^{8}-\frac{136}{287}a^{7}+\frac{1327}{287}a^{6}-\frac{1739}{287}a^{5}+\frac{2134}{287}a^{4}-\frac{2367}{287}a^{3}+\frac{1151}{287}a^{2}+\frac{149}{287}a+\frac{39}{287}$ 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:  \( 26.2176148631 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{6}\cdot 26.2176148631 \cdot 1}{2\cdot\sqrt{48551226272641}}\cr\approx \mathstrut & 0.231511350179 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^13 - 2*x^12 + 4*x^10 - 5*x^9 + x^8 + 5*x^7 - 11*x^6 + 19*x^5 - 22*x^4 + 16*x^3 - 10*x^2 + 6*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^12 + 4*x^10 - 5*x^9 + x^8 + 5*x^7 - 11*x^6 + 19*x^5 - 22*x^4 + 16*x^3 - 10*x^2 + 6*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^12 + 4*x^10 - 5*x^9 + x^8 + 5*x^7 - 11*x^6 + 19*x^5 - 22*x^4 + 16*x^3 - 10*x^2 + 6*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^12 + 4*x^10 - 5*x^9 + x^8 + 5*x^7 - 11*x^6 + 19*x^5 - 22*x^4 + 16*x^3 - 10*x^2 + 6*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

$D_{13}$ (as 13T2):

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 solvable group of order 26
The 8 conjugacy class representatives for $D_{13}$
Character table for $D_{13}$

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

Galois closure: deg 26
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 ${\href{/padicField/2.13.0.1}{13} }$ ${\href{/padicField/3.13.0.1}{13} }$ ${\href{/padicField/5.13.0.1}{13} }$ ${\href{/padicField/7.2.0.1}{2} }^{6}{,}\,{\href{/padicField/7.1.0.1}{1} }$ ${\href{/padicField/11.2.0.1}{2} }^{6}{,}\,{\href{/padicField/11.1.0.1}{1} }$ ${\href{/padicField/13.13.0.1}{13} }$ ${\href{/padicField/17.13.0.1}{13} }$ ${\href{/padicField/19.2.0.1}{2} }^{6}{,}\,{\href{/padicField/19.1.0.1}{1} }$ ${\href{/padicField/23.13.0.1}{13} }$ ${\href{/padicField/29.2.0.1}{2} }^{6}{,}\,{\href{/padicField/29.1.0.1}{1} }$ ${\href{/padicField/31.2.0.1}{2} }^{6}{,}\,{\href{/padicField/31.1.0.1}{1} }$ ${\href{/padicField/37.2.0.1}{2} }^{6}{,}\,{\href{/padicField/37.1.0.1}{1} }$ ${\href{/padicField/41.2.0.1}{2} }^{6}{,}\,{\href{/padicField/41.1.0.1}{1} }$ ${\href{/padicField/43.13.0.1}{13} }$ ${\href{/padicField/47.2.0.1}{2} }^{6}{,}\,{\href{/padicField/47.1.0.1}{1} }$ ${\href{/padicField/53.2.0.1}{2} }^{6}{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.13.0.1}{13} }$

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
\(191\) Copy content Toggle raw display $\Q_{191}$$x + 172$$1$$1$$0$Trivial$[\ ]$
191.2.1.2$x^{2} + 191$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 191$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 191$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 191$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 191$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 191$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.191.2t1.a.a$1$ $ 191 $ \(\Q(\sqrt{-191}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.191.13t2.a.e$2$ $ 191 $ 13.1.48551226272641.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.191.13t2.a.d$2$ $ 191 $ 13.1.48551226272641.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.191.13t2.a.c$2$ $ 191 $ 13.1.48551226272641.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.191.13t2.a.b$2$ $ 191 $ 13.1.48551226272641.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.191.13t2.a.f$2$ $ 191 $ 13.1.48551226272641.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.191.13t2.a.a$2$ $ 191 $ 13.1.48551226272641.1 $D_{13}$ (as 13T2) $1$ $0$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.