Properties

Label 11.7.1135258109569.1
Degree $11$
Signature $[7, 2]$
Discriminant $1.135\times 10^{12}$
Root discriminant \(12.47\)
Ramified primes $71,73,219034943$
Class number $1$
Class group trivial
Galois group $S_{11}$ (as 11T8)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

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

\( x^{11} - 2x^{10} - x^{9} + 5x^{8} - 3x^{7} - 2x^{6} - 3x^{5} + 3x^{4} + 11x^{3} - 5x^{2} - 4x + 1 \) 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:  $[7, 2]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(1135258109569\) \(\medspace = 71\cdot 73\cdot 219034943\) 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:  \(12.47\)
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:  $71^{1/2}73^{1/2}219034943^{1/2}\approx 1065484.9175699297$
Ramified primes:   \(71\), \(73\), \(219034943\) 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:  $\Q(\sqrt{1135258109569}$)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{53}a^{10}-\frac{14}{53}a^{9}+\frac{8}{53}a^{8}+\frac{15}{53}a^{7}-\frac{24}{53}a^{6}+\frac{21}{53}a^{5}+\frac{10}{53}a^{4}-\frac{11}{53}a^{3}-\frac{16}{53}a^{2}-\frac{25}{53}a-\frac{22}{53}$ 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:  $8$
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:   $\frac{44}{53}a^{10}-\frac{33}{53}a^{9}-\frac{72}{53}a^{8}+\frac{130}{53}a^{7}+\frac{4}{53}a^{6}-\frac{83}{53}a^{5}-\frac{196}{53}a^{4}-\frac{113}{53}a^{3}+\frac{250}{53}a^{2}+\frac{13}{53}a-\frac{120}{53}$, $a$, $\frac{105}{53}a^{10}-\frac{92}{53}a^{9}-\frac{167}{53}a^{8}+\frac{303}{53}a^{7}-\frac{29}{53}a^{6}-\frac{127}{53}a^{5}-\frac{487}{53}a^{4}-\frac{254}{53}a^{3}+\frac{652}{53}a^{2}+\frac{78}{53}a-\frac{84}{53}$, $\frac{24}{53}a^{10}+\frac{35}{53}a^{9}-\frac{73}{53}a^{8}-\frac{11}{53}a^{7}+\frac{113}{53}a^{6}-\frac{26}{53}a^{5}-\frac{131}{53}a^{4}-\frac{370}{53}a^{3}-\frac{66}{53}a^{2}+\frac{248}{53}a+\frac{55}{53}$, $\frac{34}{53}a^{10}+\frac{1}{53}a^{9}-\frac{46}{53}a^{8}+\frac{33}{53}a^{7}+\frac{32}{53}a^{6}+\frac{25}{53}a^{5}-\frac{190}{53}a^{4}-\frac{268}{53}a^{3}-\frac{14}{53}a^{2}+\frac{104}{53}a+\frac{47}{53}$, $\frac{49}{53}a^{10}-\frac{50}{53}a^{9}-\frac{85}{53}a^{8}+\frac{152}{53}a^{7}-\frac{10}{53}a^{6}-\frac{84}{53}a^{5}-\frac{252}{53}a^{4}-\frac{62}{53}a^{3}+\frac{382}{53}a^{2}+\frac{47}{53}a-\frac{71}{53}$, $\frac{11}{53}a^{10}+\frac{5}{53}a^{9}-\frac{18}{53}a^{8}+\frac{6}{53}a^{7}+\frac{1}{53}a^{6}+\frac{19}{53}a^{5}-\frac{49}{53}a^{4}-\frac{121}{53}a^{3}-\frac{17}{53}a^{2}-\frac{10}{53}a+\frac{76}{53}$, $\frac{10}{53}a^{10}+\frac{72}{53}a^{9}-\frac{79}{53}a^{8}-\frac{62}{53}a^{7}+\frac{184}{53}a^{6}-\frac{55}{53}a^{5}-\frac{59}{53}a^{4}-\frac{428}{53}a^{3}-\frac{160}{53}a^{2}+\frac{280}{53}a-\frac{8}{53}$ 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:  \( 72.38649544528622 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{7}\cdot(2\pi)^{2}\cdot 72.38649544528622 \cdot 1}{2\cdot\sqrt{1135258109569}}\cr\approx \mathstrut & 0.171652429738605 \end{aligned}\]

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

$S_{11}$ (as 11T8):

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 39916800
The 56 conjugacy class representatives for $S_{11}$ are not computed
Character table for $S_{11}$ 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 22 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 ${\href{/padicField/2.11.0.1}{11} }$ ${\href{/padicField/3.11.0.1}{11} }$ ${\href{/padicField/5.11.0.1}{11} }$ ${\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.3.0.1}{3} }$ ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ ${\href{/padicField/13.11.0.1}{11} }$ ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ ${\href{/padicField/19.9.0.1}{9} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ ${\href{/padicField/23.3.0.1}{3} }^{3}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ ${\href{/padicField/29.7.0.1}{7} }{,}\,{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ ${\href{/padicField/37.11.0.1}{11} }$ ${\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ ${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ ${\href{/padicField/47.11.0.1}{11} }$ ${\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{3}$ ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{3}$

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
\(71\) Copy content Toggle raw display 71.2.1.1$x^{2} + 497$$2$$1$$1$$C_2$$[\ ]_{2}$
71.3.0.1$x^{3} + 4 x + 64$$1$$3$$0$$C_3$$[\ ]^{3}$
71.6.0.1$x^{6} + x^{4} + 10 x^{3} + 13 x^{2} + 29 x + 7$$1$$6$$0$$C_6$$[\ ]^{6}$
\(73\) Copy content Toggle raw display 73.2.0.1$x^{2} + 70 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
73.2.1.2$x^{2} + 365$$2$$1$$1$$C_2$$[\ ]_{2}$
73.7.0.1$x^{7} + 10 x + 68$$1$$7$$0$$C_7$$[\ ]^{7}$
\(219034943\) Copy content Toggle raw display $\Q_{219034943}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{219034943}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $3$$1$$3$$0$$C_3$$[\ ]^{3}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$