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Properties

Label 11.11.41426511213649.1
Degree $11$
Signature $[11, 0]$
Discriminant $4.143\times 10^{13}$
Root discriminant \(17.30\)
Ramified prime $23$
Class number $1$
Class group trivial
Galois group $C_{11}$ (as 11T1)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^11 - x^10 - 10*x^9 + 9*x^8 + 36*x^7 - 28*x^6 - 56*x^5 + 35*x^4 + 35*x^3 - 15*x^2 - 6*x + 1)
 
gp: K = bnfinit(y^11 - y^10 - 10*y^9 + 9*y^8 + 36*y^7 - 28*y^6 - 56*y^5 + 35*y^4 + 35*y^3 - 15*y^2 - 6*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^11 - x^10 - 10*x^9 + 9*x^8 + 36*x^7 - 28*x^6 - 56*x^5 + 35*x^4 + 35*x^3 - 15*x^2 - 6*x + 1);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^11 - x^10 - 10*x^9 + 9*x^8 + 36*x^7 - 28*x^6 - 56*x^5 + 35*x^4 + 35*x^3 - 15*x^2 - 6*x + 1)
 

\( x^{11} - x^{10} - 10x^{9} + 9x^{8} + 36x^{7} - 28x^{6} - 56x^{5} + 35x^{4} + 35x^{3} - 15x^{2} - 6x + 1 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $11$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[11, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(41426511213649\) \(\medspace = 23^{10}\) 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:  \(17.30\)
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:  $23^{10/11}\approx 17.295550764917277$
Ramified primes:   \(23\) 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:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $11$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(23\)
Dirichlet character group:    $\lbrace$$\chi_{23}(1,·)$, $\chi_{23}(2,·)$, $\chi_{23}(3,·)$, $\chi_{23}(4,·)$, $\chi_{23}(6,·)$, $\chi_{23}(8,·)$, $\chi_{23}(9,·)$, $\chi_{23}(12,·)$, $\chi_{23}(13,·)$, $\chi_{23}(16,·)$, $\chi_{23}(18,·)$$\rbrace$
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  Yes
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:  $10$
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:   $a^{2}-2$, $a$, $a^{10}-9a^{8}+28a^{6}-35a^{4}+15a^{2}-1$, $a^{6}-6a^{4}+9a^{2}-2$, $a^{4}-4a^{2}+2$, $a^{10}-a^{9}-10a^{8}+9a^{7}+36a^{6}-28a^{5}-56a^{4}+35a^{3}+35a^{2}-14a-6$, $a^{8}-8a^{6}+21a^{4}-20a^{2}+5$, $a^{9}-9a^{7}+27a^{5}-30a^{3}+9a-1$, $a^{5}-5a^{3}+5a-1$, $a^{10}-10a^{8}+35a^{6}-50a^{4}-a^{3}+25a^{2}+3a-1$ 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:  \( 1014.31330506 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{11}\cdot(2\pi)^{0}\cdot 1014.31330506 \cdot 1}{2\cdot\sqrt{41426511213649}}\cr\approx \mathstrut & 0.161373752825 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^11 - x^10 - 10*x^9 + 9*x^8 + 36*x^7 - 28*x^6 - 56*x^5 + 35*x^4 + 35*x^3 - 15*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^11 - x^10 - 10*x^9 + 9*x^8 + 36*x^7 - 28*x^6 - 56*x^5 + 35*x^4 + 35*x^3 - 15*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^11 - x^10 - 10*x^9 + 9*x^8 + 36*x^7 - 28*x^6 - 56*x^5 + 35*x^4 + 35*x^3 - 15*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^11 - x^10 - 10*x^9 + 9*x^8 + 36*x^7 - 28*x^6 - 56*x^5 + 35*x^4 + 35*x^3 - 15*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

$C_{11}$ (as 11T1):

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 cyclic group of order 11
The 11 conjugacy class representatives for $C_{11}$
Character table for $C_{11}$

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]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.11.0.1}{11} }$ ${\href{/padicField/3.11.0.1}{11} }$ ${\href{/padicField/5.11.0.1}{11} }$ ${\href{/padicField/7.11.0.1}{11} }$ ${\href{/padicField/11.11.0.1}{11} }$ ${\href{/padicField/13.11.0.1}{11} }$ ${\href{/padicField/17.11.0.1}{11} }$ ${\href{/padicField/19.11.0.1}{11} }$ R ${\href{/padicField/29.11.0.1}{11} }$ ${\href{/padicField/31.11.0.1}{11} }$ ${\href{/padicField/37.11.0.1}{11} }$ ${\href{/padicField/41.11.0.1}{11} }$ ${\href{/padicField/43.11.0.1}{11} }$ ${\href{/padicField/47.1.0.1}{1} }^{11}$ ${\href{/padicField/53.11.0.1}{11} }$ ${\href{/padicField/59.11.0.1}{11} }$

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
\(23\) Copy content Toggle raw display 23.11.10.10$x^{11} + 23$$11$$1$$10$$C_{11}$$[\ ]_{11}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.23.11t1.a.a$1$ $ 23 $ \(\Q(\zeta_{23})^+\) $C_{11}$ (as 11T1) $0$ $1$
* 1.23.11t1.a.b$1$ $ 23 $ \(\Q(\zeta_{23})^+\) $C_{11}$ (as 11T1) $0$ $1$
* 1.23.11t1.a.c$1$ $ 23 $ \(\Q(\zeta_{23})^+\) $C_{11}$ (as 11T1) $0$ $1$
* 1.23.11t1.a.d$1$ $ 23 $ \(\Q(\zeta_{23})^+\) $C_{11}$ (as 11T1) $0$ $1$
* 1.23.11t1.a.e$1$ $ 23 $ \(\Q(\zeta_{23})^+\) $C_{11}$ (as 11T1) $0$ $1$
* 1.23.11t1.a.f$1$ $ 23 $ \(\Q(\zeta_{23})^+\) $C_{11}$ (as 11T1) $0$ $1$
* 1.23.11t1.a.g$1$ $ 23 $ \(\Q(\zeta_{23})^+\) $C_{11}$ (as 11T1) $0$ $1$
* 1.23.11t1.a.h$1$ $ 23 $ \(\Q(\zeta_{23})^+\) $C_{11}$ (as 11T1) $0$ $1$
* 1.23.11t1.a.i$1$ $ 23 $ \(\Q(\zeta_{23})^+\) $C_{11}$ (as 11T1) $0$ $1$
* 1.23.11t1.a.j$1$ $ 23 $ \(\Q(\zeta_{23})^+\) $C_{11}$ (as 11T1) $0$ $1$

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 *.