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Properties

Label 11.3.815...000.1
Degree $11$
Signature $[3, 4]$
Discriminant $8.157\times 10^{26}$
Root discriminant \(279.58\)
Ramified primes $2,5,13$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $A_{11}$ (as 11T7)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^11 - 2*x^10 - 2250*x^7 + 900*x^6 + 700*x^5 + 1000*x^4 - 875*x^3 - 250*x^2 - 100*x + 200)
 
gp: K = bnfinit(y^11 - 2*y^10 - 2250*y^7 + 900*y^6 + 700*y^5 + 1000*y^4 - 875*y^3 - 250*y^2 - 100*y + 200, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^11 - 2*x^10 - 2250*x^7 + 900*x^6 + 700*x^5 + 1000*x^4 - 875*x^3 - 250*x^2 - 100*x + 200);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^11 - 2*x^10 - 2250*x^7 + 900*x^6 + 700*x^5 + 1000*x^4 - 875*x^3 - 250*x^2 - 100*x + 200)
 

\( x^{11} - 2x^{10} - 2250x^{7} + 900x^{6} + 700x^{5} + 1000x^{4} - 875x^{3} - 250x^{2} - 100x + 200 \) 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:  $[3, 4]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(815730721000000000000000000\) \(\medspace = 2^{18}\cdot 5^{18}\cdot 13^{8}\) 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:  \(279.58\)
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:  $2^{7/3}5^{203/100}13^{14/15}\approx 1448.7481492586514$
Ramified primes:   \(2\), \(5\), \(13\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{4}a^{5}-\frac{1}{2}a^{4}+\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{20}a^{6}-\frac{1}{10}a^{5}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{520}a^{7}+\frac{3}{130}a^{6}+\frac{47}{520}a^{5}+\frac{19}{52}a^{4}+\frac{1}{8}a^{3}+\frac{5}{26}a^{2}-\frac{1}{8}a-\frac{9}{52}$, $\frac{1}{520}a^{8}+\frac{7}{520}a^{6}-\frac{31}{260}a^{5}-\frac{27}{104}a^{4}-\frac{4}{13}a^{3}-\frac{45}{104}a^{2}+\frac{17}{52}a+\frac{1}{13}$, $\frac{1}{1040}a^{9}+\frac{1}{104}a^{6}-\frac{63}{520}a^{5}-\frac{19}{104}a^{4}-\frac{2}{13}a^{3}+\frac{25}{104}a^{2}-\frac{83}{208}a+\frac{37}{104}$, $\frac{1}{33280}a^{10}+\frac{1}{8320}a^{9}+\frac{3}{4160}a^{8}+\frac{1}{2080}a^{7}-\frac{53}{16640}a^{6}-\frac{67}{832}a^{5}-\frac{385}{1664}a^{4}-\frac{61}{416}a^{3}-\frac{1935}{6656}a^{2}+\frac{445}{1664}a+\frac{745}{1664}$ 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 $1$ (assuming GRH)

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:   $\frac{2257181}{4160}a^{10}+\frac{5717}{130}a^{9}-\frac{119472}{65}a^{8}-\frac{30965}{52}a^{7}-\frac{2539604513}{2080}a^{6}-\frac{106752457}{52}a^{5}+\frac{93685259}{208}a^{4}+\frac{60300629}{52}a^{3}+\frac{303284941}{832}a^{2}-\frac{18060415}{52}a-\frac{90541453}{208}$, $\frac{964361}{3328}a^{10}-\frac{118411}{320}a^{9}-\frac{595317}{2080}a^{8}-\frac{195547}{1040}a^{7}-\frac{5424878161}{8320}a^{6}-\frac{437037431}{2080}a^{5}+\frac{80269795}{832}a^{4}+\frac{85283919}{208}a^{3}+\frac{245188829}{3328}a^{2}-\frac{26458691}{832}a-\frac{55867875}{832}$, $\frac{24053749}{8320}a^{10}-\frac{1612655}{416}a^{9}-\frac{2514033}{1040}a^{8}-\frac{77459}{104}a^{7}-\frac{27092276649}{4160}a^{6}-\frac{361301375}{208}a^{5}+\frac{360914859}{416}a^{4}+\frac{361169379}{104}a^{3}-\frac{361453371}{1664}a^{2}-\frac{361209215}{416}a-\frac{27785551}{32}$, $\frac{27266739}{16640}a^{10}-\frac{9088933}{4160}a^{9}-\frac{3029927}{2080}a^{8}-\frac{1007107}{1040}a^{7}-\frac{30680498447}{8320}a^{6}-\frac{2045633209}{2080}a^{5}+\frac{408950093}{832}a^{4}+\frac{408987035}{208}a^{3}-\frac{409503997}{3328}a^{2}-\frac{409101497}{832}a-\frac{31465065}{64}$, $\frac{82159}{33280}a^{10}-\frac{1414377}{8320}a^{9}+\frac{343101}{4160}a^{8}+\frac{3349483}{416}a^{7}-\frac{50734683}{16640}a^{6}-\frac{10825209}{4160}a^{5}-\frac{7453951}{1664}a^{4}+\frac{1423605}{416}a^{3}+\frac{5857471}{6656}a^{2}+\frac{1108411}{1664}a-\frac{1371113}{1664}$, $\frac{44703309319}{130}a^{10}-\frac{1258468979849}{1040}a^{9}+\frac{364564521171}{260}a^{8}-\frac{85531723139}{104}a^{7}-\frac{402251297868393}{520}a^{6}+\frac{38608809920043}{26}a^{5}-\frac{107549199012557}{104}a^{4}+\frac{56347189474621}{104}a^{3}-\frac{54710301775249}{104}a^{2}+\frac{70056505617581}{208}a-\frac{7760063835683}{104}$ Copy content Toggle raw display (assuming GRH)
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:  \( 5089470030.7 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{4}\cdot 5089470030.7 \cdot 1}{2\cdot\sqrt{815730721000000000000000000}}\cr\approx \mathstrut & 1.1109093369 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^11 - 2*x^10 - 2250*x^7 + 900*x^6 + 700*x^5 + 1000*x^4 - 875*x^3 - 250*x^2 - 100*x + 200)
 
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 - 2250*x^7 + 900*x^6 + 700*x^5 + 1000*x^4 - 875*x^3 - 250*x^2 - 100*x + 200, 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 - 2250*x^7 + 900*x^6 + 700*x^5 + 1000*x^4 - 875*x^3 - 250*x^2 - 100*x + 200);
 
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 - 2250*x^7 + 900*x^6 + 700*x^5 + 1000*x^4 - 875*x^3 - 250*x^2 - 100*x + 200);
 
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

$A_{11}$ (as 11T7):

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 19958400
The 31 conjugacy class representatives for $A_{11}$
Character table for $A_{11}$

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 R ${\href{/padicField/3.11.0.1}{11} }$ R ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ ${\href{/padicField/11.11.0.1}{11} }$ R ${\href{/padicField/17.7.0.1}{7} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ ${\href{/padicField/19.11.0.1}{11} }$ ${\href{/padicField/23.7.0.1}{7} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ ${\href{/padicField/29.9.0.1}{9} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{3}$ ${\href{/padicField/43.7.0.1}{7} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ ${\href{/padicField/47.9.0.1}{9} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.11.0.1}{11} }$ ${\href{/padicField/59.11.0.1}{11} }$

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
\(2\) Copy content Toggle raw display 2.3.2.1$x^{3} + 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.8.16.30$x^{8} - 4 x^{7} + 28 x^{6} - 16 x^{5} + 24 x^{4} + 88 x^{3} + 40 x^{2} + 16 x + 52$$4$$2$$16$$C_2^3: C_4$$[2, 2, 3]^{4}$
\(5\) Copy content Toggle raw display $\Q_{5}$$x + 3$$1$$1$$0$Trivial$[\ ]$
5.5.9.3$x^{5} + 100 x^{2} + 75 x + 5$$5$$1$$9$$F_5$$[9/4]_{4}$
5.5.9.4$x^{5} + 25 x^{2} + 25 x + 5$$5$$1$$9$$F_5$$[9/4]_{4}$
\(13\) Copy content Toggle raw display 13.5.4.1$x^{5} + 13$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
13.6.4.1$x^{6} + 130 x^{3} - 1521$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$