Properties

Label 11.1.1461660310351.1
Degree $11$
Signature $[1, 5]$
Discriminant $-1.462\times 10^{12}$
Root discriminant \(12.76\)
Ramified prime $271$
Class number $1$
Class group trivial
Galois group $D_{11}$ (as 11T2)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

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

\( x^{11} + x^{9} - x^{8} + 3x^{7} + 3x^{6} - 6x^{5} + 3x^{4} + 5x^{3} - 6x^{2} + 5x + 1 \) 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:  $[1, 5]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-1461660310351\) \(\medspace = -\,271^{5}\) 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:  \(12.76\)
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:  $271^{1/2}\approx 16.46207763315433$
Ramified primes:   \(271\) 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:  \(\Q(\sqrt{-271}) \)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}-\frac{1}{3}$, $\frac{1}{3}a^{9}-\frac{1}{3}a$, $\frac{1}{741}a^{10}-\frac{101}{741}a^{9}+\frac{25}{247}a^{8}+\frac{27}{247}a^{7}-\frac{9}{247}a^{6}-\frac{6}{19}a^{5}-\frac{28}{247}a^{4}+\frac{112}{247}a^{3}+\frac{155}{741}a^{2}-\frac{100}{741}a+\frac{75}{247}$ Copy content Toggle raw display

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

Monogenic:  No
Index:  Not computed
Inessential primes:  $3$

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:  $5$
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:   $\frac{34}{741}a^{10}+\frac{8}{247}a^{9}-\frac{167}{741}a^{8}-\frac{70}{247}a^{7}-\frac{59}{247}a^{6}+\frac{5}{19}a^{5}-\frac{211}{247}a^{4}-\frac{391}{247}a^{3}+\frac{824}{741}a^{2}+\frac{184}{247}a-\frac{748}{741}$, $\frac{79}{741}a^{10}-\frac{25}{247}a^{9}-\frac{1}{247}a^{8}-\frac{90}{247}a^{7}+\frac{30}{247}a^{6}+\frac{1}{19}a^{5}-\frac{236}{247}a^{4}+\frac{203}{247}a^{3}+\frac{389}{741}a^{2}-\frac{328}{247}a+\frac{244}{247}$, $\frac{280}{741}a^{10}+\frac{125}{741}a^{9}+\frac{84}{247}a^{8}-\frac{97}{247}a^{7}+\frac{197}{247}a^{6}+\frac{30}{19}a^{5}-\frac{430}{247}a^{4}-\frac{9}{247}a^{3}+\frac{1163}{741}a^{2}-\frac{89}{741}a+\frac{252}{247}$, $\frac{66}{247}a^{10}+\frac{3}{247}a^{9}+\frac{277}{741}a^{8}-\frac{88}{247}a^{7}+\frac{194}{247}a^{6}+\frac{9}{19}a^{5}-\frac{357}{247}a^{4}+\frac{193}{247}a^{3}+\frac{103}{247}a^{2}-\frac{425}{247}a+\frac{1325}{741}$, $a$ 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:  \( 55.1714638864 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{5}\cdot 55.1714638864 \cdot 1}{2\cdot\sqrt{1461660310351}}\cr\approx \mathstrut & 0.446879695079 \end{aligned}\]

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

$D_{11}$ (as 11T2):

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 22
The 7 conjugacy class representatives for $D_{11}$
Character table for $D_{11}$

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]
 

Sibling fields

Galois closure: 22.0.578978183833808423828407471.1
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 ${\href{/padicField/2.11.0.1}{11} }$ ${\href{/padicField/3.2.0.1}{2} }^{5}{,}\,{\href{/padicField/3.1.0.1}{1} }$ ${\href{/padicField/5.11.0.1}{11} }$ ${\href{/padicField/7.11.0.1}{11} }$ ${\href{/padicField/11.11.0.1}{11} }$ ${\href{/padicField/13.2.0.1}{2} }^{5}{,}\,{\href{/padicField/13.1.0.1}{1} }$ ${\href{/padicField/17.11.0.1}{11} }$ ${\href{/padicField/19.2.0.1}{2} }^{5}{,}\,{\href{/padicField/19.1.0.1}{1} }$ ${\href{/padicField/23.2.0.1}{2} }^{5}{,}\,{\href{/padicField/23.1.0.1}{1} }$ ${\href{/padicField/29.2.0.1}{2} }^{5}{,}\,{\href{/padicField/29.1.0.1}{1} }$ ${\href{/padicField/31.11.0.1}{11} }$ ${\href{/padicField/37.11.0.1}{11} }$ ${\href{/padicField/41.11.0.1}{11} }$ ${\href{/padicField/43.2.0.1}{2} }^{5}{,}\,{\href{/padicField/43.1.0.1}{1} }$ ${\href{/padicField/47.2.0.1}{2} }^{5}{,}\,{\href{/padicField/47.1.0.1}{1} }$ ${\href{/padicField/53.11.0.1}{11} }$ ${\href{/padicField/59.2.0.1}{2} }^{5}{,}\,{\href{/padicField/59.1.0.1}{1} }$

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
\(271\) Copy content Toggle raw display $\Q_{271}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$