↑

Properties

Label 11.3.243...976.1
Degree $11$
Signature $[3, 4]$
Discriminant $2.439\times 10^{23}$
Root discriminant \(133.69\)
Ramified primes $2,3,11$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $\PSL(2,11)$ (as 11T5)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / Pari/GP / SageMath

Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^11 + 33*x^9 - 110*x^8 + 363*x^7 - 1980*x^6 + 1463*x^5 - 6666*x^4 + 5016*x^3 - 6292*x^2 - 9504*x + 13392)
 
Copy content gp:K = bnfinit(y^11 + 33*y^9 - 110*y^8 + 363*y^7 - 1980*y^6 + 1463*y^5 - 6666*y^4 + 5016*y^3 - 6292*y^2 - 9504*y + 13392, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^11 + 33*x^9 - 110*x^8 + 363*x^7 - 1980*x^6 + 1463*x^5 - 6666*x^4 + 5016*x^3 - 6292*x^2 - 9504*x + 13392);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^11 + 33*x^9 - 110*x^8 + 363*x^7 - 1980*x^6 + 1463*x^5 - 6666*x^4 + 5016*x^3 - 6292*x^2 - 9504*x + 13392)
 

\( x^{11} + 33 x^{9} - 110 x^{8} + 363 x^{7} - 1980 x^{6} + 1463 x^{5} - 6666 x^{4} + 5016 x^{3} + \cdots + 13392 \) Copy content Toggle raw display

Copy content comment:Defining polynomial
 
Copy content sage:K.defining_polynomial()
 
Copy content gp:K.pol
 
Copy content magma:DefiningPolynomial(K);
 
Copy content oscar:defining_polynomial(K)
 

Invariants

Degree:  $11$
Copy content comment:Degree over Q
 
Copy content sage:K.degree()
 
Copy content gp:poldegree(K.pol)
 
Copy content magma:Degree(K);
 
Copy content oscar:degree(K)
 
Signature:  $[3, 4]$
Copy content comment:Signature
 
Copy content sage:K.signature()
 
Copy content gp:K.sign
 
Copy content magma:Signature(K);
 
Copy content oscar:signature(K)
 
Discriminant:   \(243920281203464276606976\) \(\medspace = 2^{16}\cdot 3^{4}\cdot 11^{16}\) Copy content Toggle raw display
Copy content comment:Discriminant
 
Copy content sage:K.disc()
 
Copy content gp:K.disc
 
Copy content magma:OK := Integers(K); Discriminant(OK);
 
Copy content oscar:OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(133.69\)
Copy content comment:Root discriminant
 
Copy content sage:(K.disc().abs())^(1./K.degree())
 
Copy content gp:abs(K.disc)^(1/poldegree(K.pol))
 
Copy content magma:Abs(Discriminant(OK))^(1/Degree(K));
 
Copy content oscar:OK = ring_of_integers(K); (1.0 * abs(discriminant(OK)))^(1/degree(K))
 
Galois root discriminant:  $2^{11/6}3^{1/2}11^{84/55}\approx 240.40284291526194$
Ramified primes:   \(2\), \(3\), \(11\) Copy content Toggle raw display
Copy content comment:Ramified primes
 
Copy content sage:K.disc().support()
 
Copy content gp:factor(abs(K.disc))[,1]~
 
Copy content magma:PrimeDivisors(Discriminant(OK));
 
Copy content oscar:prime_divisors(discriminant(OK))
 
Discriminant root field:  \(\Q\)
$\Aut(K/\Q)$:   $C_1$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.
This field has no CM subfields.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{6}a^{5}-\frac{1}{2}a^{4}+\frac{1}{6}a^{3}-\frac{1}{2}a^{2}-\frac{1}{3}a$, $\frac{1}{6}a^{6}-\frac{1}{3}a^{4}+\frac{1}{6}a^{2}$, $\frac{1}{6}a^{7}-\frac{1}{2}a^{3}+\frac{1}{3}a$, $\frac{1}{18}a^{8}+\frac{1}{6}a^{4}-\frac{2}{9}a^{2}$, $\frac{1}{324}a^{9}-\frac{2}{81}a^{8}-\frac{1}{12}a^{7}+\frac{1}{18}a^{6}+\frac{1}{108}a^{5}-\frac{11}{27}a^{4}+\frac{95}{324}a^{3}+\frac{43}{162}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{87342833914416}a^{10}-\frac{42280883975}{43671416957208}a^{9}+\frac{2239065379813}{87342833914416}a^{8}-\frac{191216611219}{2426189830956}a^{7}+\frac{1260652636465}{29114277971472}a^{6}+\frac{361409502427}{14557138985736}a^{5}-\frac{18014728417741}{87342833914416}a^{4}+\frac{4207656291439}{21835708478604}a^{3}-\frac{387045109289}{10917854239302}a^{2}+\frac{399898274053}{808729943652}a-\frac{201381262927}{404364971826}$ Copy content Toggle raw display

Copy content comment:Integral basis
 
Copy content sage:K.integral_basis()
 
Copy content gp:K.zk
 
Copy content magma:IntegralBasis(K);
 
Copy content oscar:basis(OK)
 

Monogenic:  No
Index:  Not computed
Inessential primes:  $2$, $3$

Class group and class number

Ideal class group:  Trivial group, which has order $1$ (assuming GRH)
Copy content comment:Class group
 
Copy content sage:K.class_group().invariants()
 
Copy content gp:K.clgp
 
Copy content magma:ClassGroup(K);
 
Copy content oscar:class_group(K)
 
Narrow class group:  Trivial group, which has order $1$ (assuming GRH)
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 

Unit group

Copy content comment:Unit group
 
Copy content sage:UK = K.unit_group()
 
Copy content magma:UK, fUK := UnitGroup(K);
 
Copy content oscar:UK, fUK = unit_group(OK)
 
Rank:  $6$
Copy content comment:Unit rank
 
Copy content sage:UK.rank()
 
Copy content gp:K.fu
 
Copy content magma:UnitRank(K);
 
Copy content oscar:rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
Copy content comment:Generator for roots of unity
 
Copy content sage:UK.torsion_generator()
 
Copy content gp:K.tu[2]
 
Copy content magma:K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Copy content oscar:torsion_units_generator(OK)
 
Fundamental units:   $\frac{417164563213}{21835708478604}a^{10}-\frac{154074380353}{5458927119651}a^{9}+\frac{4097236416229}{21835708478604}a^{8}-\frac{262380253771}{606547457739}a^{7}-\frac{12420110697269}{7278569492868}a^{6}-\frac{2380137444637}{1819642373217}a^{5}-\frac{188876829570001}{21835708478604}a^{4}+\frac{23728744732987}{5458927119651}a^{3}-\frac{213217709002013}{10917854239302}a^{2}-\frac{1630513711547}{202182485913}a+\frac{5052237754159}{202182485913}$, $\frac{6292788466213}{43671416957208}a^{10}+\frac{10681312141141}{21835708478604}a^{9}+\frac{262347310408585}{43671416957208}a^{8}-\frac{2007764967692}{606547457739}a^{7}+\frac{399384076890733}{14557138985736}a^{6}-\frac{20\cdots 45}{7278569492868}a^{5}+\frac{14\cdots 87}{43671416957208}a^{4}-\frac{24\cdots 97}{5458927119651}a^{3}+\frac{76\cdots 64}{5458927119651}a^{2}+\frac{353824708695457}{404364971826}a-\frac{282580026582310}{202182485913}$, $\frac{71\cdots 55}{43671416957208}a^{10}+\frac{12\cdots 13}{21835708478604}a^{9}+\frac{33\cdots 27}{43671416957208}a^{8}+\frac{11\cdots 41}{1213094915478}a^{7}+\frac{13\cdots 83}{14557138985736}a^{6}+\frac{56\cdots 91}{7278569492868}a^{5}+\frac{11\cdots 25}{43671416957208}a^{4}-\frac{14\cdots 21}{10917854239302}a^{3}+\frac{19\cdots 84}{5458927119651}a^{2}+\frac{10\cdots 23}{404364971826}a-\frac{12\cdots 94}{202182485913}$, $\frac{6679793733583}{43671416957208}a^{10}+\frac{11995952783023}{21835708478604}a^{9}+\frac{258921977692723}{43671416957208}a^{8}+\frac{4667404984093}{606547457739}a^{7}+\frac{913187629121059}{14557138985736}a^{6}+\frac{83017574432407}{7278569492868}a^{5}+\frac{75\cdots 29}{43671416957208}a^{4}-\frac{649368153624503}{10917854239302}a^{3}+\frac{11\cdots 18}{5458927119651}a^{2}+\frac{79278051098221}{404364971826}a-\frac{84852100573516}{202182485913}$, $\frac{226894020678473}{87342833914416}a^{10}-\frac{31\cdots 25}{43671416957208}a^{9}+\frac{26\cdots 33}{87342833914416}a^{8}-\frac{14\cdots 91}{1213094915478}a^{7}+\frac{11\cdots 09}{29114277971472}a^{6}-\frac{70\cdots 87}{14557138985736}a^{5}+\frac{11\cdots 55}{87342833914416}a^{4}-\frac{58\cdots 51}{5458927119651}a^{3}+\frac{78\cdots 89}{10917854239302}a^{2}+\frac{16\cdots 65}{808729943652}a-\frac{92\cdots 23}{404364971826}$, $\frac{69\cdots 73}{21835708478604}a^{10}-\frac{20\cdots 96}{5458927119651}a^{9}+\frac{20\cdots 03}{21835708478604}a^{8}-\frac{26\cdots 43}{606547457739}a^{7}+\frac{92\cdots 75}{7278569492868}a^{6}-\frac{10\cdots 34}{1819642373217}a^{5}+\frac{12\cdots 97}{21835708478604}a^{4}-\frac{13\cdots 77}{5458927119651}a^{3}-\frac{11\cdots 13}{10917854239302}a^{2}+\frac{16\cdots 22}{202182485913}a+\frac{32\cdots 53}{202182485913}$ Copy content Toggle raw display (assuming GRH)
Copy content comment:Fundamental units
 
Copy content sage:UK.fundamental_units()
 
Copy content gp:K.fu
 
Copy content magma:[K|fUK(g): g in Generators(UK)];
 
Copy content oscar:[K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 2824285089.59 \) (assuming GRH)
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{4}\cdot 2824285089.59 \cdot 1}{2\cdot\sqrt{243920281203464276606976}}\cr\approx \mathstrut & 35.6503700340 \end{aligned}\] (assuming GRH)

Copy content comment:Analytic class number formula
 
Copy content sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^11 + 33*x^9 - 110*x^8 + 363*x^7 - 1980*x^6 + 1463*x^5 - 6666*x^4 + 5016*x^3 - 6292*x^2 - 9504*x + 13392) 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))))
 
Copy content gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^11 + 33*x^9 - 110*x^8 + 363*x^7 - 1980*x^6 + 1463*x^5 - 6666*x^4 + 5016*x^3 - 6292*x^2 - 9504*x + 13392, 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)))]
 
Copy content magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^11 + 33*x^9 - 110*x^8 + 363*x^7 - 1980*x^6 + 1463*x^5 - 6666*x^4 + 5016*x^3 - 6292*x^2 - 9504*x + 13392); 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)));
 
Copy content oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = polynomial_ring(QQ); K, a = number_field(x^11 + 33*x^9 - 110*x^8 + 363*x^7 - 1980*x^6 + 1463*x^5 - 6666*x^4 + 5016*x^3 - 6292*x^2 - 9504*x + 13392); 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

$\PSL(2,11)$ (as 11T5):

Copy content comment:Galois group
 
Copy content sage:K.galois_group(type='pari')
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:G = GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
 
A non-solvable group of order 660
The 8 conjugacy class representatives for $\PSL(2,11)$
Character table for $\PSL(2,11)$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.
Copy content comment:Intermediate fields
 
Copy content sage:K.subfields()[1:-1]
 
Copy content gp:L = nfsubfields(K); L[2..length(L)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Degree 12 sibling: 12.0.140498081973195423325618176.3
Arithmetically equivalent sibling: 11.3.243920281203464276606976.6
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 R R ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.3.0.1}{3} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ ${\href{/padicField/7.5.0.1}{5} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }$ R ${\href{/padicField/13.5.0.1}{5} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }$ ${\href{/padicField/17.5.0.1}{5} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }$ ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ ${\href{/padicField/23.5.0.1}{5} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }$ ${\href{/padicField/29.5.0.1}{5} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }$ ${\href{/padicField/31.5.0.1}{5} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }$ ${\href{/padicField/37.3.0.1}{3} }^{3}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ ${\href{/padicField/41.5.0.1}{5} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }$ ${\href{/padicField/43.5.0.1}{5} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }$ ${\href{/padicField/47.5.0.1}{5} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }$ ${\href{/padicField/53.5.0.1}{5} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.2.0.1}{2} }$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

Copy content comment:Frobenius cycle types
 
Copy content sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
Copy content gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
Copy content magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
Copy content oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_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 2.1.2.3a1.1$x^{2} + 2$$2$$1$$3$$C_2$$$[3]$$
2.1.3.2a1.1$x^{3} + 2$$3$$1$$2$$S_3$$$[\ ]_{3}^{2}$$
2.1.6.11a1.2$x^{6} + 10$$6$$1$$11$$D_{6}$$$[3]_{3}^{2}$$
\(3\) Copy content Toggle raw display $\Q_{3}$$x + 1$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$$[\ ]$$
3.1.2.1a1.2$x^{2} + 6$$2$$1$$1$$C_2$$$[\ ]_{2}$$
3.1.2.1a1.2$x^{2} + 6$$2$$1$$1$$C_2$$$[\ ]_{2}$$
3.1.2.1a1.2$x^{2} + 6$$2$$1$$1$$C_2$$$[\ ]_{2}$$
3.1.2.1a1.2$x^{2} + 6$$2$$1$$1$$C_2$$$[\ ]_{2}$$
\(11\) Copy content Toggle raw display 11.1.11.16a2.1$x^{11} + 22 x^{6} + 11$$11$$1$$16$$C_{11}:C_5$$$[\frac{8}{5}]_{5}$$

Spectrum of ring of integers

(0)(0)(2)(3)(5)(7)(11)(13)(17)(19)(23)(29)(31)(37)(41)(43)(47)(53)(59)