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Properties

Label 11.3.656...000.1
Degree $11$
Signature $[3, 4]$
Discriminant $6.561\times 10^{21}$
Root discriminant \(96.24\)
Ramified primes $2,3,5$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $M_{11}$ (as 11T6)

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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 - 5*x^9 - 50*x^8 + 70*x^7 + 232*x^6 + 796*x^5 - 1400*x^4 - 5075*x^3 - 10950*x^2 + 2805*x + 90)
 
gp: K = bnfinit(y^11 - 2*y^10 - 5*y^9 - 50*y^8 + 70*y^7 + 232*y^6 + 796*y^5 - 1400*y^4 - 5075*y^3 - 10950*y^2 + 2805*y + 90, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^11 - 2*x^10 - 5*x^9 - 50*x^8 + 70*x^7 + 232*x^6 + 796*x^5 - 1400*x^4 - 5075*x^3 - 10950*x^2 + 2805*x + 90);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^11 - 2*x^10 - 5*x^9 - 50*x^8 + 70*x^7 + 232*x^6 + 796*x^5 - 1400*x^4 - 5075*x^3 - 10950*x^2 + 2805*x + 90)
 

\( x^{11} - 2 x^{10} - 5 x^{9} - 50 x^{8} + 70 x^{7} + 232 x^{6} + 796 x^{5} - 1400 x^{4} - 5075 x^{3} + \cdots + 90 \) 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:   \(6561000000000000000000\) \(\medspace = 2^{18}\cdot 3^{8}\cdot 5^{18}\) 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:  \(96.24\)
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:  $2^{13/6}3^{7/8}5^{39/20}\approx 270.8346732868732$
Ramified primes:   \(2\), \(3\), \(5\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{18\!\cdots\!08}a^{10}+\frac{23569802922049}{18\!\cdots\!08}a^{9}+\frac{19593327064199}{947023430976804}a^{8}-\frac{97955798617415}{473511715488402}a^{7}+\frac{302323532157431}{947023430976804}a^{6}+\frac{196073990354663}{947023430976804}a^{5}-\frac{422970158627623}{947023430976804}a^{4}-\frac{307579548304303}{947023430976804}a^{3}-\frac{800733488315849}{18\!\cdots\!08}a^{2}+\frac{40061575935705}{631348953984536}a+\frac{49459480584111}{315674476992268}$ 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:  $2$

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:   $\frac{1159417007769}{315674476992268}a^{10}-\frac{1819656396269}{315674476992268}a^{9}-\frac{371515294331}{78918619248067}a^{8}-\frac{50131292142773}{157837238496134}a^{7}+\frac{65889121906}{78918619248067}a^{6}-\frac{11818870512630}{78918619248067}a^{5}+\frac{714004225151158}{78918619248067}a^{4}+\frac{787898438993871}{78918619248067}a^{3}+\frac{97\!\cdots\!25}{315674476992268}a^{2}-\frac{62\!\cdots\!53}{315674476992268}a+\frac{446393138482111}{157837238496134}$, $\frac{26458468060787}{631348953984536}a^{10}-\frac{275759954464135}{631348953984536}a^{9}+\frac{234819293358001}{315674476992268}a^{8}+\frac{218128785801087}{157837238496134}a^{7}+\frac{30\!\cdots\!27}{315674476992268}a^{6}-\frac{11\!\cdots\!57}{315674476992268}a^{5}-\frac{15\!\cdots\!55}{315674476992268}a^{4}+\frac{16\!\cdots\!89}{315674476992268}a^{3}+\frac{29\!\cdots\!81}{631348953984536}a^{2}-\frac{73\!\cdots\!17}{631348953984536}a-\frac{990781312138379}{315674476992268}$, $\frac{360099135298729}{18\!\cdots\!08}a^{10}-\frac{55\!\cdots\!99}{18\!\cdots\!08}a^{9}+\frac{92\!\cdots\!35}{947023430976804}a^{8}-\frac{642037775782840}{236755857744201}a^{7}+\frac{37\!\cdots\!87}{947023430976804}a^{6}-\frac{16\!\cdots\!51}{947023430976804}a^{5}-\frac{15\!\cdots\!51}{947023430976804}a^{4}+\frac{13\!\cdots\!07}{947023430976804}a^{3}+\frac{55\!\cdots\!99}{18\!\cdots\!08}a^{2}-\frac{11\!\cdots\!39}{631348953984536}a-\frac{32\!\cdots\!69}{315674476992268}$, $\frac{6613342010219}{18\!\cdots\!08}a^{10}+\frac{109537184178281}{18\!\cdots\!08}a^{9}-\frac{387910435431827}{947023430976804}a^{8}+\frac{298738201686047}{473511715488402}a^{7}-\frac{35\!\cdots\!09}{947023430976804}a^{6}+\frac{16\!\cdots\!15}{947023430976804}a^{5}-\frac{21\!\cdots\!39}{947023430976804}a^{4}+\frac{46\!\cdots\!77}{947023430976804}a^{3}-\frac{37\!\cdots\!15}{18\!\cdots\!08}a^{2}+\frac{38\!\cdots\!01}{631348953984536}a-\frac{11\!\cdots\!17}{315674476992268}$, $\frac{31\!\cdots\!35}{631348953984536}a^{10}+\frac{37\!\cdots\!23}{631348953984536}a^{9}-\frac{46\!\cdots\!67}{315674476992268}a^{8}-\frac{15\!\cdots\!67}{78918619248067}a^{7}+\frac{39\!\cdots\!21}{315674476992268}a^{6}+\frac{97\!\cdots\!51}{315674476992268}a^{5}+\frac{26\!\cdots\!07}{315674476992268}a^{4}-\frac{36\!\cdots\!27}{315674476992268}a^{3}-\frac{25\!\cdots\!79}{631348953984536}a^{2}+\frac{63\!\cdots\!45}{631348953984536}a+\frac{10\!\cdots\!87}{315674476992268}$, $\frac{14\!\cdots\!87}{18\!\cdots\!08}a^{10}+\frac{32\!\cdots\!67}{18\!\cdots\!08}a^{9}+\frac{33\!\cdots\!29}{947023430976804}a^{8}-\frac{52\!\cdots\!83}{236755857744201}a^{7}-\frac{40\!\cdots\!91}{947023430976804}a^{6}-\frac{98\!\cdots\!25}{947023430976804}a^{5}+\frac{51\!\cdots\!27}{947023430976804}a^{4}+\frac{12\!\cdots\!25}{947023430976804}a^{3}+\frac{33\!\cdots\!49}{18\!\cdots\!08}a^{2}-\frac{31\!\cdots\!33}{631348953984536}a-\frac{42\!\cdots\!63}{315674476992268}$ 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:  \( 107079411.141 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{4}\cdot 107079411.141 \cdot 1}{2\cdot\sqrt{6561000000000000000000}}\cr\approx \mathstrut & 8.24138912213 \end{aligned}\] (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 - 5*x^9 - 50*x^8 + 70*x^7 + 232*x^6 + 796*x^5 - 1400*x^4 - 5075*x^3 - 10950*x^2 + 2805*x + 90)
 
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 - 5*x^9 - 50*x^8 + 70*x^7 + 232*x^6 + 796*x^5 - 1400*x^4 - 5075*x^3 - 10950*x^2 + 2805*x + 90, 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 - 5*x^9 - 50*x^8 + 70*x^7 + 232*x^6 + 796*x^5 - 1400*x^4 - 5075*x^3 - 10950*x^2 + 2805*x + 90);
 
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 - 5*x^9 - 50*x^8 + 70*x^7 + 232*x^6 + 796*x^5 - 1400*x^4 - 5075*x^3 - 10950*x^2 + 2805*x + 90);
 
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

$M_{11}$ (as 11T6):

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 7920
The 10 conjugacy class representatives for $M_{11}$
Character table for $M_{11}$

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

Degree 12 sibling: data not computed
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 R R R ${\href{/padicField/7.11.0.1}{11} }$ ${\href{/padicField/11.11.0.1}{11} }$ ${\href{/padicField/13.5.0.1}{5} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }$ ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ ${\href{/padicField/23.5.0.1}{5} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }$ ${\href{/padicField/29.5.0.1}{5} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }$ ${\href{/padicField/31.11.0.1}{11} }$ ${\href{/padicField/37.11.0.1}{11} }$ ${\href{/padicField/41.5.0.1}{5} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }$ ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ ${\href{/padicField/53.11.0.1}{11} }$ ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }$

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

# 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 $\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.4.8.7$x^{4} + 4 x^{2} + 4 x + 2$$4$$1$$8$$S_4$$[8/3, 8/3]_{3}^{2}$
2.6.10.4$x^{6} + 2 x^{5} + 4 x^{3} + 4 x + 2$$6$$1$$10$$S_4$$[8/3, 8/3]_{3}^{2}$
\(3\) Copy content Toggle raw display $\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.2.1.1$x^{2} + 6$$2$$1$$1$$C_2$$[\ ]_{2}$
3.8.7.1$x^{8} + 3$$8$$1$$7$$QD_{16}$$[\ ]_{8}^{2}$
\(5\) Copy content Toggle raw display $\Q_{5}$$x + 3$$1$$1$$0$Trivial$[\ ]$
5.5.9.4$x^{5} + 25 x^{2} + 25 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}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 10.656...000.11t6.a.a$10$ $ 2^{18} \cdot 3^{8} \cdot 5^{18}$ 11.3.6561000000000000000000.1 $M_{11}$ (as 11T6) $1$ $2$
10.314...000.330.a.a$10$ $ 2^{24} \cdot 3^{9} \cdot 5^{20}$ 11.3.6561000000000000000000.1 $M_{11}$ (as 11T6) $0$ $-2$
10.314...000.330.a.b$10$ $ 2^{24} \cdot 3^{9} \cdot 5^{20}$ 11.3.6561000000000000000000.1 $M_{11}$ (as 11T6) $0$ $-2$
11.147...000.12t272.a.a$11$ $ 2^{18} \cdot 3^{10} \cdot 5^{20}$ 11.3.6561000000000000000000.1 $M_{11}$ (as 11T6) $1$ $3$
16.306...000.144.a.a$16$ $ 2^{36} \cdot 3^{14} \cdot 5^{30}$ 11.3.6561000000000000000000.1 $M_{11}$ (as 11T6) $0$ $0$
16.306...000.144.a.b$16$ $ 2^{36} \cdot 3^{14} \cdot 5^{30}$ 11.3.6561000000000000000000.1 $M_{11}$ (as 11T6) $0$ $0$
44.216...000.55.a.a$44$ $ 2^{90} \cdot 3^{38} \cdot 5^{86}$ 11.3.6561000000000000000000.1 $M_{11}$ (as 11T6) $1$ $4$
45.199...000.110.a.a$45$ $ 2^{102} \cdot 3^{40} \cdot 5^{88}$ 11.3.6561000000000000000000.1 $M_{11}$ (as 11T6) $1$ $-3$
55.326...000.110.a.a$55$ $ 2^{120} \cdot 3^{48} \cdot 5^{108}$ 11.3.6561000000000000000000.1 $M_{11}$ (as 11T6) $1$ $-1$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.