Properties

Label 11.11.278...449.3
Degree $11$
Signature $[11, 0]$
Discriminant $2.787\times 10^{34}$
Root discriminant \(1353.24\)
Ramified primes $11,23$
Class number $11$ (GRH)
Class group [11] (GRH)
Galois group $C_{11}$ (as 11T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^11 - 1265*x^9 - 759*x^8 + 461472*x^7 + 727122*x^6 - 64557504*x^5 - 152866395*x^4 + 3259832136*x^3 + 7537075436*x^2 - 51546842380*x - 79493912081)
 
gp: K = bnfinit(y^11 - 1265*y^9 - 759*y^8 + 461472*y^7 + 727122*y^6 - 64557504*y^5 - 152866395*y^4 + 3259832136*y^3 + 7537075436*y^2 - 51546842380*y - 79493912081, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^11 - 1265*x^9 - 759*x^8 + 461472*x^7 + 727122*x^6 - 64557504*x^5 - 152866395*x^4 + 3259832136*x^3 + 7537075436*x^2 - 51546842380*x - 79493912081);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^11 - 1265*x^9 - 759*x^8 + 461472*x^7 + 727122*x^6 - 64557504*x^5 - 152866395*x^4 + 3259832136*x^3 + 7537075436*x^2 - 51546842380*x - 79493912081)
 

\( x^{11} - 1265 x^{9} - 759 x^{8} + 461472 x^{7} + 727122 x^{6} - 64557504 x^{5} - 152866395 x^{4} + 3259832136 x^{3} + 7537075436 x^{2} + \cdots - 79493912081 \) 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:  $[11, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(27869685209056005146671841116784449\) \(\medspace = 11^{20}\cdot 23^{10}\) 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:  \(1353.24\)
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:  $11^{20/11}23^{10/11}\approx 1353.24254149841$
Ramified primes:   \(11\), \(23\) 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:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $11$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2783=11^{2}\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{2783}(1,·)$, $\chi_{2783}(1475,·)$, $\chi_{2783}(100,·)$, $\chi_{2783}(903,·)$, $\chi_{2783}(1244,·)$, $\chi_{2783}(1948,·)$, $\chi_{2783}(1651,·)$, $\chi_{2783}(2773,·)$, $\chi_{2783}(2102,·)$, $\chi_{2783}(1783,·)$, $\chi_{2783}(188,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{42\!\cdots\!97}a^{10}-\frac{65\!\cdots\!33}{42\!\cdots\!97}a^{9}-\frac{17\!\cdots\!94}{42\!\cdots\!97}a^{8}+\frac{70\!\cdots\!96}{42\!\cdots\!97}a^{7}+\frac{93\!\cdots\!88}{42\!\cdots\!97}a^{6}-\frac{17\!\cdots\!20}{42\!\cdots\!97}a^{5}+\frac{82\!\cdots\!38}{42\!\cdots\!97}a^{4}+\frac{15\!\cdots\!90}{42\!\cdots\!97}a^{3}-\frac{83\!\cdots\!70}{42\!\cdots\!97}a^{2}-\frac{18\!\cdots\!71}{42\!\cdots\!97}a-\frac{48\!\cdots\!40}{42\!\cdots\!97}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  No
Index:  $1$
Inessential primes:  None

Class group and class number

$C_{11}$, which has order $11$ (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:  $10$
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:   $\frac{39\!\cdots\!23}{42\!\cdots\!97}a^{10}-\frac{40\!\cdots\!96}{42\!\cdots\!97}a^{9}-\frac{45\!\cdots\!26}{42\!\cdots\!97}a^{8}+\frac{43\!\cdots\!01}{42\!\cdots\!97}a^{7}+\frac{14\!\cdots\!97}{42\!\cdots\!97}a^{6}-\frac{11\!\cdots\!37}{42\!\cdots\!97}a^{5}-\frac{16\!\cdots\!56}{42\!\cdots\!97}a^{4}+\frac{95\!\cdots\!08}{42\!\cdots\!97}a^{3}+\frac{60\!\cdots\!56}{42\!\cdots\!97}a^{2}-\frac{22\!\cdots\!39}{42\!\cdots\!97}a-\frac{41\!\cdots\!48}{42\!\cdots\!97}$, $\frac{90\!\cdots\!98}{42\!\cdots\!97}a^{10}-\frac{64\!\cdots\!53}{42\!\cdots\!97}a^{9}-\frac{10\!\cdots\!78}{42\!\cdots\!97}a^{8}+\frac{71\!\cdots\!78}{42\!\cdots\!97}a^{7}+\frac{36\!\cdots\!09}{42\!\cdots\!97}a^{6}-\frac{19\!\cdots\!05}{42\!\cdots\!97}a^{5}-\frac{44\!\cdots\!61}{42\!\cdots\!97}a^{4}+\frac{17\!\cdots\!50}{42\!\cdots\!97}a^{3}+\frac{16\!\cdots\!86}{42\!\cdots\!97}a^{2}-\frac{50\!\cdots\!79}{42\!\cdots\!97}a-\frac{99\!\cdots\!80}{42\!\cdots\!97}$, $\frac{21\!\cdots\!53}{42\!\cdots\!97}a^{10}+\frac{20\!\cdots\!55}{42\!\cdots\!97}a^{9}-\frac{21\!\cdots\!62}{42\!\cdots\!97}a^{8}-\frac{21\!\cdots\!93}{42\!\cdots\!97}a^{7}+\frac{42\!\cdots\!38}{42\!\cdots\!97}a^{6}+\frac{48\!\cdots\!20}{42\!\cdots\!97}a^{5}-\frac{16\!\cdots\!14}{42\!\cdots\!97}a^{4}-\frac{28\!\cdots\!67}{42\!\cdots\!97}a^{3}-\frac{84\!\cdots\!12}{42\!\cdots\!97}a^{2}+\frac{43\!\cdots\!68}{42\!\cdots\!97}a+\frac{57\!\cdots\!74}{42\!\cdots\!97}$, $\frac{27\!\cdots\!21}{42\!\cdots\!97}a^{10}+\frac{61\!\cdots\!93}{42\!\cdots\!97}a^{9}+\frac{39\!\cdots\!49}{42\!\cdots\!97}a^{8}-\frac{67\!\cdots\!03}{42\!\cdots\!97}a^{7}-\frac{72\!\cdots\!33}{42\!\cdots\!97}a^{6}+\frac{16\!\cdots\!07}{42\!\cdots\!97}a^{5}+\frac{20\!\cdots\!79}{42\!\cdots\!97}a^{4}-\frac{91\!\cdots\!85}{42\!\cdots\!97}a^{3}-\frac{14\!\cdots\!97}{42\!\cdots\!97}a^{2}+\frac{90\!\cdots\!96}{42\!\cdots\!97}a+\frac{24\!\cdots\!62}{42\!\cdots\!97}$, $\frac{89\!\cdots\!81}{42\!\cdots\!97}a^{10}-\frac{65\!\cdots\!82}{42\!\cdots\!97}a^{9}-\frac{10\!\cdots\!85}{42\!\cdots\!97}a^{8}+\frac{73\!\cdots\!69}{42\!\cdots\!97}a^{7}+\frac{35\!\cdots\!49}{42\!\cdots\!97}a^{6}-\frac{20\!\cdots\!58}{42\!\cdots\!97}a^{5}-\frac{43\!\cdots\!68}{42\!\cdots\!97}a^{4}+\frac{18\!\cdots\!75}{42\!\cdots\!97}a^{3}+\frac{15\!\cdots\!32}{42\!\cdots\!97}a^{2}-\frac{50\!\cdots\!69}{42\!\cdots\!97}a-\frac{97\!\cdots\!34}{42\!\cdots\!97}$, $\frac{23\!\cdots\!18}{42\!\cdots\!97}a^{10}+\frac{32\!\cdots\!08}{42\!\cdots\!97}a^{9}-\frac{35\!\cdots\!98}{42\!\cdots\!97}a^{8}-\frac{39\!\cdots\!08}{42\!\cdots\!97}a^{7}+\frac{16\!\cdots\!28}{42\!\cdots\!97}a^{6}+\frac{13\!\cdots\!66}{42\!\cdots\!97}a^{5}-\frac{16\!\cdots\!67}{42\!\cdots\!97}a^{4}-\frac{16\!\cdots\!71}{42\!\cdots\!97}a^{3}-\frac{10\!\cdots\!39}{42\!\cdots\!97}a^{2}+\frac{53\!\cdots\!93}{42\!\cdots\!97}a+\frac{74\!\cdots\!72}{42\!\cdots\!97}$, $\frac{12\!\cdots\!38}{42\!\cdots\!97}a^{10}-\frac{93\!\cdots\!45}{42\!\cdots\!97}a^{9}-\frac{15\!\cdots\!84}{42\!\cdots\!97}a^{8}+\frac{10\!\cdots\!07}{42\!\cdots\!97}a^{7}+\frac{51\!\cdots\!37}{42\!\cdots\!97}a^{6}-\frac{28\!\cdots\!70}{42\!\cdots\!97}a^{5}-\frac{61\!\cdots\!50}{42\!\cdots\!97}a^{4}+\frac{25\!\cdots\!97}{42\!\cdots\!97}a^{3}+\frac{22\!\cdots\!11}{42\!\cdots\!97}a^{2}-\frac{71\!\cdots\!07}{42\!\cdots\!97}a-\frac{13\!\cdots\!32}{42\!\cdots\!97}$, $\frac{42\!\cdots\!06}{42\!\cdots\!97}a^{10}-\frac{67\!\cdots\!64}{42\!\cdots\!97}a^{9}-\frac{46\!\cdots\!50}{42\!\cdots\!97}a^{8}+\frac{69\!\cdots\!42}{42\!\cdots\!97}a^{7}+\frac{12\!\cdots\!05}{42\!\cdots\!97}a^{6}-\frac{15\!\cdots\!15}{42\!\cdots\!97}a^{5}-\frac{13\!\cdots\!24}{42\!\cdots\!97}a^{4}+\frac{98\!\cdots\!97}{42\!\cdots\!97}a^{3}+\frac{48\!\cdots\!93}{42\!\cdots\!97}a^{2}-\frac{19\!\cdots\!52}{42\!\cdots\!97}a-\frac{35\!\cdots\!24}{42\!\cdots\!97}$, $\frac{73\!\cdots\!51}{42\!\cdots\!97}a^{10}-\frac{52\!\cdots\!00}{42\!\cdots\!97}a^{9}-\frac{89\!\cdots\!21}{42\!\cdots\!97}a^{8}+\frac{58\!\cdots\!73}{42\!\cdots\!97}a^{7}+\frac{29\!\cdots\!45}{42\!\cdots\!97}a^{6}-\frac{16\!\cdots\!34}{42\!\cdots\!97}a^{5}-\frac{36\!\cdots\!39}{42\!\cdots\!97}a^{4}+\frac{14\!\cdots\!07}{42\!\cdots\!97}a^{3}+\frac{13\!\cdots\!02}{42\!\cdots\!97}a^{2}-\frac{41\!\cdots\!51}{42\!\cdots\!97}a-\frac{81\!\cdots\!29}{42\!\cdots\!97}$, $\frac{19\!\cdots\!14}{42\!\cdots\!97}a^{10}+\frac{15\!\cdots\!90}{42\!\cdots\!97}a^{9}-\frac{24\!\cdots\!35}{42\!\cdots\!97}a^{8}-\frac{20\!\cdots\!46}{42\!\cdots\!97}a^{7}+\frac{78\!\cdots\!25}{42\!\cdots\!97}a^{6}+\frac{75\!\cdots\!52}{42\!\cdots\!97}a^{5}-\frac{77\!\cdots\!29}{42\!\cdots\!97}a^{4}-\frac{92\!\cdots\!34}{42\!\cdots\!97}a^{3}+\frac{26\!\cdots\!23}{42\!\cdots\!97}a^{2}+\frac{19\!\cdots\!66}{42\!\cdots\!97}a+\frac{24\!\cdots\!78}{42\!\cdots\!97}$ 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:  \( 2768761764338.989 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{11}\cdot(2\pi)^{0}\cdot 2768761764338.989 \cdot 11}{2\cdot\sqrt{27869685209056005146671841116784449}}\cr\approx \mathstrut & 0.186815195613300 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^11 - 1265*x^9 - 759*x^8 + 461472*x^7 + 727122*x^6 - 64557504*x^5 - 152866395*x^4 + 3259832136*x^3 + 7537075436*x^2 - 51546842380*x - 79493912081)
 
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 - 1265*x^9 - 759*x^8 + 461472*x^7 + 727122*x^6 - 64557504*x^5 - 152866395*x^4 + 3259832136*x^3 + 7537075436*x^2 - 51546842380*x - 79493912081, 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 - 1265*x^9 - 759*x^8 + 461472*x^7 + 727122*x^6 - 64557504*x^5 - 152866395*x^4 + 3259832136*x^3 + 7537075436*x^2 - 51546842380*x - 79493912081);
 
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 - 1265*x^9 - 759*x^8 + 461472*x^7 + 727122*x^6 - 64557504*x^5 - 152866395*x^4 + 3259832136*x^3 + 7537075436*x^2 - 51546842380*x - 79493912081);
 
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

$C_{11}$ (as 11T1):

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 cyclic group of order 11
The 11 conjugacy class representatives for $C_{11}$
Character table for $C_{11}$

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 ${\href{/padicField/2.11.0.1}{11} }$ ${\href{/padicField/3.11.0.1}{11} }$ ${\href{/padicField/5.11.0.1}{11} }$ ${\href{/padicField/7.11.0.1}{11} }$ R ${\href{/padicField/13.11.0.1}{11} }$ ${\href{/padicField/17.11.0.1}{11} }$ ${\href{/padicField/19.11.0.1}{11} }$ R ${\href{/padicField/29.11.0.1}{11} }$ ${\href{/padicField/31.11.0.1}{11} }$ ${\href{/padicField/37.1.0.1}{1} }^{11}$ ${\href{/padicField/41.11.0.1}{11} }$ ${\href{/padicField/43.11.0.1}{11} }$ ${\href{/padicField/47.11.0.1}{11} }$ ${\href{/padicField/53.11.0.1}{11} }$ ${\href{/padicField/59.11.0.1}{11} }$

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
\(11\) Copy content Toggle raw display 11.11.20.11$x^{11} + 110 x^{10} + 253$$11$$1$$20$$C_{11}$$[2]$
\(23\) Copy content Toggle raw display 23.11.10.8$x^{11} + 253$$11$$1$$10$$C_{11}$$[\ ]_{11}$