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Properties

Label 11.3.301...296.2
Degree $11$
Signature $[3, 4]$
Discriminant $3.011\times 10^{21}$
Root discriminant \(89.66\)
Ramified primes $2,11$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $\PSL(2,11)$ (as 11T5)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^11 - 33*x^9 - 22*x^8 + 396*x^7 - 132*x^6 - 1408*x^5 + 4048*x^4 + 26048*x^3 + 16192*x^2 - 352000*x - 60672)
 
Copy content gp:K = bnfinit(y^11 - 33*y^9 - 22*y^8 + 396*y^7 - 132*y^6 - 1408*y^5 + 4048*y^4 + 26048*y^3 + 16192*y^2 - 352000*y - 60672, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^11 - 33*x^9 - 22*x^8 + 396*x^7 - 132*x^6 - 1408*x^5 + 4048*x^4 + 26048*x^3 + 16192*x^2 - 352000*x - 60672);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^11 - 33*x^9 - 22*x^8 + 396*x^7 - 132*x^6 - 1408*x^5 + 4048*x^4 + 26048*x^3 + 16192*x^2 - 352000*x - 60672)
 

\( x^{11} - 33 x^{9} - 22 x^{8} + 396 x^{7} - 132 x^{6} - 1408 x^{5} + 4048 x^{4} + 26048 x^{3} + \cdots - 60672 \) 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:   \(3011361496339065143296\) \(\medspace = 2^{16}\cdot 11^{16}\) 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:  \(89.66\)
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:  $2^{11/6}11^{84/55}\approx 138.7966460710778$
Ramified primes:   \(2\), \(11\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{5}-\frac{1}{4}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{8}a^{6}-\frac{1}{8}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{16}a^{7}-\frac{1}{16}a^{5}+\frac{1}{8}a^{4}-\frac{1}{4}a^{3}+\frac{1}{4}a^{2}$, $\frac{1}{32}a^{8}-\frac{1}{32}a^{6}+\frac{1}{16}a^{5}-\frac{1}{8}a^{4}+\frac{1}{8}a^{3}-\frac{1}{2}a$, $\frac{1}{64}a^{9}-\frac{1}{64}a^{7}+\frac{1}{32}a^{6}-\frac{1}{16}a^{5}+\frac{1}{16}a^{4}-\frac{1}{2}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{30\cdots 44}a^{10}-\frac{184998999493835}{37\cdots 68}a^{9}-\frac{25\cdots 45}{30\cdots 44}a^{8}+\frac{246793817321013}{15\cdots 72}a^{7}+\frac{14631225182049}{23\cdots 73}a^{6}+\frac{21\cdots 07}{75\cdots 36}a^{5}+\frac{491712753829421}{18\cdots 84}a^{4}-\frac{209880365091193}{46\cdots 46}a^{3}+\frac{564746769856721}{93\cdots 92}a^{2}+\frac{14\cdots 57}{46\cdots 46}a-\frac{10\cdots 37}{23\cdots 73}$ 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:  $2$

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:   $\frac{7574173599515}{30\cdots 44}a^{10}+\frac{105583407673975}{15\cdots 72}a^{9}-\frac{21\cdots 23}{30\cdots 44}a^{8}+\frac{561316889120089}{37\cdots 68}a^{7}+\frac{981929766866527}{75\cdots 36}a^{6}-\frac{40\cdots 81}{75\cdots 36}a^{5}+\frac{822885616465973}{37\cdots 68}a^{4}-\frac{512793607457743}{23\cdots 73}a^{3}+\frac{24\cdots 51}{46\cdots 46}a^{2}-\frac{30\cdots 34}{23\cdots 73}a-\frac{11\cdots 49}{23\cdots 73}$, $\frac{598675923268863}{30\cdots 44}a^{10}+\frac{43735023683927}{18\cdots 84}a^{9}-\frac{23\cdots 23}{30\cdots 44}a^{8}-\frac{40\cdots 29}{15\cdots 72}a^{7}+\frac{11\cdots 63}{75\cdots 36}a^{6}-\frac{69\cdots 17}{75\cdots 36}a^{5}-\frac{17\cdots 99}{46\cdots 46}a^{4}+\frac{60\cdots 61}{18\cdots 84}a^{3}+\frac{81\cdots 87}{93\cdots 92}a^{2}+\frac{72\cdots 41}{23\cdots 73}a+\frac{11\cdots 84}{23\cdots 73}$, $\frac{65851539929575}{15\cdots 72}a^{10}+\frac{10185011050963}{37\cdots 68}a^{9}-\frac{13\cdots 91}{15\cdots 72}a^{8}-\frac{15\cdots 17}{75\cdots 36}a^{7}+\frac{13\cdots 15}{18\cdots 84}a^{6}+\frac{240842793006739}{93\cdots 92}a^{5}-\frac{14\cdots 19}{46\cdots 46}a^{4}-\frac{23\cdots 25}{93\cdots 92}a^{3}+\frac{15\cdots 01}{93\cdots 92}a^{2}-\frac{34\cdots 79}{46\cdots 46}a-\frac{41\cdots 15}{23\cdots 73}$, $\frac{82\cdots 15}{30\cdots 44}a^{10}-\frac{36\cdots 85}{15\cdots 72}a^{9}+\frac{15\cdots 37}{30\cdots 44}a^{8}+\frac{20\cdots 25}{37\cdots 68}a^{7}-\frac{17\cdots 41}{46\cdots 46}a^{6}+\frac{63\cdots 77}{75\cdots 36}a^{5}-\frac{11\cdots 41}{37\cdots 68}a^{4}+\frac{42\cdots 81}{18\cdots 84}a^{3}-\frac{27\cdots 87}{46\cdots 46}a^{2}+\frac{99\cdots 43}{46\cdots 46}a+\frac{12\cdots 15}{23\cdots 73}$, $\frac{148213847589121}{75\cdots 36}a^{10}-\frac{147852332102697}{18\cdots 84}a^{9}-\frac{23\cdots 05}{75\cdots 36}a^{8}+\frac{860000081816817}{93\cdots 92}a^{7}+\frac{26\cdots 83}{18\cdots 84}a^{6}-\frac{81\cdots 91}{37\cdots 68}a^{5}+\frac{11\cdots 71}{18\cdots 84}a^{4}+\frac{27\cdots 27}{93\cdots 92}a^{3}+\frac{18\cdots 47}{93\cdots 92}a^{2}-\frac{25\cdots 85}{23\cdots 73}a-\frac{44\cdots 76}{23\cdots 73}$, $\frac{15\cdots 13}{30\cdots 44}a^{10}+\frac{25\cdots 39}{15\cdots 72}a^{9}-\frac{34\cdots 05}{30\cdots 44}a^{8}-\frac{37\cdots 69}{75\cdots 36}a^{7}+\frac{16\cdots 63}{37\cdots 68}a^{6}+\frac{54\cdots 29}{75\cdots 36}a^{5}-\frac{18\cdots 61}{37\cdots 68}a^{4}+\frac{91\cdots 97}{18\cdots 84}a^{3}+\frac{14\cdots 01}{93\cdots 92}a^{2}+\frac{13\cdots 35}{23\cdots 73}a+\frac{22\cdots 11}{23\cdots 73}$ 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:  \( 52170956.4414 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{4}\cdot 52170956.4414 \cdot 1}{2\cdot\sqrt{3011361496339065143296}}\cr\approx \mathstrut & 5.92688931481 \end{aligned}\] (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 - 22*x^8 + 396*x^7 - 132*x^6 - 1408*x^5 + 4048*x^4 + 26048*x^3 + 16192*x^2 - 352000*x - 60672) 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 - 22*x^8 + 396*x^7 - 132*x^6 - 1408*x^5 + 4048*x^4 + 26048*x^3 + 16192*x^2 - 352000*x - 60672, 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 - 22*x^8 + 396*x^7 - 132*x^6 - 1408*x^5 + 4048*x^4 + 26048*x^3 + 16192*x^2 - 352000*x - 60672); 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 - 22*x^8 + 396*x^7 - 132*x^6 - 1408*x^5 + 4048*x^4 + 26048*x^3 + 16192*x^2 - 352000*x - 60672); 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: 12.0.192727135765700169170944.1
Arithmetically equivalent sibling: 11.3.3011361496339065143296.1
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 R ${\href{/padicField/3.3.0.1}{3} }^{3}{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ ${\href{/padicField/5.5.0.1}{5} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }$ ${\href{/padicField/7.5.0.1}{5} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }$ R ${\href{/padicField/13.11.0.1}{11} }$ ${\href{/padicField/17.11.0.1}{11} }$ ${\href{/padicField/19.3.0.1}{3} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ ${\href{/padicField/23.5.0.1}{5} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }$ ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ ${\href{/padicField/31.3.0.1}{3} }^{3}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ ${\href{/padicField/37.11.0.1}{11} }$ ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ ${\href{/padicField/43.11.0.1}{11} }$ ${\href{/padicField/47.5.0.1}{5} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }$ ${\href{/padicField/53.5.0.1}{5} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.5.0.1}{5} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }$

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 2.1.2.3a1.4$x^{2} + 4 x + 10$$2$$1$$3$$C_2$$$[3]$$
2.1.3.2a1.1$x^{3} + 2$$3$$1$$2$$S_3$$$[\ ]_{3}^{2}$$
2.1.6.11a1.5$x^{6} + 4 x^{3} + 2$$6$$1$$11$$D_{6}$$$[3]_{3}^{2}$$
\(11\) 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)