Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^11 - 1265*x^9 - 759*x^8 + 511566*x^7 + 810612*x^6 - 74117109*x^5 - 171278723*x^4 + 3526980655*x^3 + 9233082947*x^2 - 12188156343*x - 5403811843)
gp: K = bnfinit(y^11 - 1265*y^9 - 759*y^8 + 511566*y^7 + 810612*y^6 - 74117109*y^5 - 171278723*y^4 + 3526980655*y^3 + 9233082947*y^2 - 12188156343*y - 5403811843, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^11 - 1265*x^9 - 759*x^8 + 511566*x^7 + 810612*x^6 - 74117109*x^5 - 171278723*x^4 + 3526980655*x^3 + 9233082947*x^2 - 12188156343*x - 5403811843);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^11 - 1265*x^9 - 759*x^8 + 511566*x^7 + 810612*x^6 - 74117109*x^5 - 171278723*x^4 + 3526980655*x^3 + 9233082947*x^2 - 12188156343*x - 5403811843)
\( x^{11} - 1265 x^{9} - 759 x^{8} + 511566 x^{7} + 810612 x^{6} - 74117109 x^{5} - 171278723 x^{4} + \cdots - 5403811843 \)
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: |
\(27869685209056005146671841116784449\)
\(\medspace = 11^{20}\cdot 23^{10}\)
| 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: |
\(11\), \(23\)
| 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$$\chi_{2783}(1024,·)$, $\chi_{2783}(1,·)$, $\chi_{2783}(2377,·)$, $\chi_{2783}(1706,·)$, $\chi_{2783}(331,·)$, $\chi_{2783}(2520,·)$, $\chi_{2783}(2003,·)$, $\chi_{2783}(2168,·)$, $\chi_{2783}(2201,·)$, $\chi_{2783}(1981,·)$, $\chi_{2783}(639,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{41}a^{9}-\frac{11}{41}a^{8}+\frac{17}{41}a^{7}+\frac{18}{41}a^{6}-\frac{9}{41}a^{5}+\frac{6}{41}a^{4}+\frac{7}{41}a^{3}+\frac{8}{41}a^{2}+\frac{20}{41}a+\frac{9}{41}$, $\frac{1}{25\!\cdots\!83}a^{10}+\frac{12\!\cdots\!11}{25\!\cdots\!83}a^{9}+\frac{96\!\cdots\!28}{25\!\cdots\!83}a^{8}+\frac{11\!\cdots\!91}{25\!\cdots\!83}a^{7}+\frac{29\!\cdots\!36}{25\!\cdots\!83}a^{6}-\frac{10\!\cdots\!05}{25\!\cdots\!83}a^{5}+\frac{11\!\cdots\!66}{25\!\cdots\!83}a^{4}-\frac{10\!\cdots\!46}{25\!\cdots\!83}a^{3}-\frac{49\!\cdots\!68}{25\!\cdots\!83}a^{2}+\frac{80\!\cdots\!22}{25\!\cdots\!83}a-\frac{78\!\cdots\!16}{28\!\cdots\!47}$
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 $2$)
| 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: |
$\frac{15\!\cdots\!45}{25\!\cdots\!83}a^{10}-\frac{14\!\cdots\!32}{62\!\cdots\!63}a^{9}-\frac{19\!\cdots\!58}{25\!\cdots\!83}a^{8}+\frac{15\!\cdots\!65}{62\!\cdots\!63}a^{7}+\frac{75\!\cdots\!80}{25\!\cdots\!83}a^{6}-\frac{17\!\cdots\!25}{25\!\cdots\!83}a^{5}-\frac{10\!\cdots\!53}{25\!\cdots\!83}a^{4}+\frac{14\!\cdots\!42}{25\!\cdots\!83}a^{3}+\frac{47\!\cdots\!68}{25\!\cdots\!83}a^{2}-\frac{42\!\cdots\!45}{25\!\cdots\!83}a-\frac{23\!\cdots\!58}{28\!\cdots\!47}$, $\frac{22\!\cdots\!24}{25\!\cdots\!83}a^{10}-\frac{10\!\cdots\!61}{25\!\cdots\!83}a^{9}-\frac{29\!\cdots\!97}{25\!\cdots\!83}a^{8}+\frac{10\!\cdots\!58}{25\!\cdots\!83}a^{7}+\frac{11\!\cdots\!99}{25\!\cdots\!83}a^{6}-\frac{26\!\cdots\!74}{25\!\cdots\!83}a^{5}-\frac{16\!\cdots\!40}{25\!\cdots\!83}a^{4}+\frac{22\!\cdots\!61}{25\!\cdots\!83}a^{3}+\frac{76\!\cdots\!49}{25\!\cdots\!83}a^{2}-\frac{68\!\cdots\!51}{25\!\cdots\!83}a-\frac{38\!\cdots\!19}{28\!\cdots\!47}$, $\frac{25\!\cdots\!43}{25\!\cdots\!83}a^{10}-\frac{26\!\cdots\!87}{25\!\cdots\!83}a^{9}-\frac{29\!\cdots\!83}{25\!\cdots\!83}a^{8}+\frac{28\!\cdots\!48}{25\!\cdots\!83}a^{7}+\frac{10\!\cdots\!40}{25\!\cdots\!83}a^{6}-\frac{83\!\cdots\!72}{25\!\cdots\!83}a^{5}-\frac{10\!\cdots\!66}{25\!\cdots\!83}a^{4}+\frac{63\!\cdots\!13}{25\!\cdots\!83}a^{3}+\frac{26\!\cdots\!80}{25\!\cdots\!83}a^{2}-\frac{29\!\cdots\!68}{25\!\cdots\!83}a-\frac{15\!\cdots\!91}{28\!\cdots\!47}$, $\frac{58\!\cdots\!14}{25\!\cdots\!83}a^{10}-\frac{51\!\cdots\!37}{25\!\cdots\!83}a^{9}-\frac{67\!\cdots\!33}{25\!\cdots\!83}a^{8}+\frac{57\!\cdots\!60}{25\!\cdots\!83}a^{7}+\frac{22\!\cdots\!78}{25\!\cdots\!83}a^{6}-\frac{17\!\cdots\!16}{25\!\cdots\!83}a^{5}-\frac{23\!\cdots\!75}{25\!\cdots\!83}a^{4}+\frac{13\!\cdots\!63}{25\!\cdots\!83}a^{3}+\frac{56\!\cdots\!32}{25\!\cdots\!83}a^{2}-\frac{63\!\cdots\!27}{25\!\cdots\!83}a-\frac{33\!\cdots\!87}{28\!\cdots\!47}$, $\frac{19\!\cdots\!04}{25\!\cdots\!83}a^{10}-\frac{43\!\cdots\!85}{25\!\cdots\!83}a^{9}-\frac{24\!\cdots\!09}{25\!\cdots\!83}a^{8}+\frac{36\!\cdots\!61}{25\!\cdots\!83}a^{7}+\frac{94\!\cdots\!95}{25\!\cdots\!83}a^{6}-\frac{20\!\cdots\!44}{25\!\cdots\!83}a^{5}-\frac{13\!\cdots\!09}{25\!\cdots\!83}a^{4}-\frac{14\!\cdots\!22}{25\!\cdots\!83}a^{3}+\frac{59\!\cdots\!98}{25\!\cdots\!83}a^{2}+\frac{12\!\cdots\!95}{25\!\cdots\!83}a-\frac{26\!\cdots\!32}{28\!\cdots\!47}$, $\frac{30\!\cdots\!12}{25\!\cdots\!83}a^{10}+\frac{26\!\cdots\!96}{25\!\cdots\!83}a^{9}-\frac{36\!\cdots\!15}{25\!\cdots\!83}a^{8}-\frac{33\!\cdots\!37}{25\!\cdots\!83}a^{7}+\frac{12\!\cdots\!95}{25\!\cdots\!83}a^{6}+\frac{13\!\cdots\!02}{25\!\cdots\!83}a^{5}-\frac{11\!\cdots\!89}{25\!\cdots\!83}a^{4}-\frac{13\!\cdots\!35}{25\!\cdots\!83}a^{3}-\frac{28\!\cdots\!95}{25\!\cdots\!83}a^{2}+\frac{84\!\cdots\!93}{25\!\cdots\!83}a-\frac{81\!\cdots\!22}{28\!\cdots\!47}$, $\frac{44\!\cdots\!19}{25\!\cdots\!83}a^{10}-\frac{10\!\cdots\!95}{25\!\cdots\!83}a^{9}-\frac{55\!\cdots\!17}{25\!\cdots\!83}a^{8}+\frac{79\!\cdots\!51}{25\!\cdots\!83}a^{7}+\frac{22\!\cdots\!86}{25\!\cdots\!83}a^{6}-\frac{73\!\cdots\!14}{25\!\cdots\!83}a^{5}-\frac{31\!\cdots\!09}{25\!\cdots\!83}a^{4}-\frac{14\!\cdots\!71}{25\!\cdots\!83}a^{3}+\frac{14\!\cdots\!04}{25\!\cdots\!83}a^{2}+\frac{12\!\cdots\!99}{25\!\cdots\!83}a-\frac{42\!\cdots\!33}{28\!\cdots\!47}$, $\frac{26\!\cdots\!24}{62\!\cdots\!63}a^{10}-\frac{11\!\cdots\!36}{25\!\cdots\!83}a^{9}-\frac{12\!\cdots\!49}{25\!\cdots\!83}a^{8}+\frac{12\!\cdots\!47}{25\!\cdots\!83}a^{7}+\frac{41\!\cdots\!82}{25\!\cdots\!83}a^{6}-\frac{35\!\cdots\!11}{25\!\cdots\!83}a^{5}-\frac{41\!\cdots\!93}{25\!\cdots\!83}a^{4}+\frac{25\!\cdots\!09}{25\!\cdots\!83}a^{3}+\frac{10\!\cdots\!92}{25\!\cdots\!83}a^{2}-\frac{12\!\cdots\!69}{25\!\cdots\!83}a-\frac{63\!\cdots\!54}{28\!\cdots\!47}$, $\frac{24\!\cdots\!80}{25\!\cdots\!83}a^{10}+\frac{30\!\cdots\!06}{25\!\cdots\!83}a^{9}-\frac{30\!\cdots\!95}{25\!\cdots\!83}a^{8}-\frac{57\!\cdots\!11}{25\!\cdots\!83}a^{7}+\frac{12\!\cdots\!47}{25\!\cdots\!83}a^{6}+\frac{35\!\cdots\!23}{25\!\cdots\!83}a^{5}-\frac{17\!\cdots\!65}{25\!\cdots\!83}a^{4}-\frac{64\!\cdots\!59}{25\!\cdots\!83}a^{3}+\frac{78\!\cdots\!80}{25\!\cdots\!83}a^{2}+\frac{32\!\cdots\!42}{25\!\cdots\!83}a+\frac{11\!\cdots\!96}{28\!\cdots\!47}$, $\frac{99\!\cdots\!16}{25\!\cdots\!83}a^{10}-\frac{10\!\cdots\!43}{25\!\cdots\!83}a^{9}-\frac{11\!\cdots\!33}{25\!\cdots\!83}a^{8}+\frac{11\!\cdots\!80}{25\!\cdots\!83}a^{7}+\frac{38\!\cdots\!56}{25\!\cdots\!83}a^{6}-\frac{33\!\cdots\!73}{25\!\cdots\!83}a^{5}-\frac{38\!\cdots\!07}{25\!\cdots\!83}a^{4}+\frac{24\!\cdots\!03}{25\!\cdots\!83}a^{3}+\frac{96\!\cdots\!90}{25\!\cdots\!83}a^{2}-\frac{11\!\cdots\!82}{25\!\cdots\!83}a-\frac{66\!\cdots\!34}{28\!\cdots\!47}$
| 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: | \( 3514735062507.8184 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{11}\cdot(2\pi)^{0}\cdot 3514735062507.8184 \cdot 11}{2\cdot\sqrt{27869685209056005146671841116784449}}\cr\approx \mathstrut & 0.237147856738074 \end{aligned}\] (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 + 511566*x^7 + 810612*x^6 - 74117109*x^5 - 171278723*x^4 + 3526980655*x^3 + 9233082947*x^2 - 12188156343*x - 5403811843)
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 + 511566*x^7 + 810612*x^6 - 74117109*x^5 - 171278723*x^4 + 3526980655*x^3 + 9233082947*x^2 - 12188156343*x - 5403811843, 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 + 511566*x^7 + 810612*x^6 - 74117109*x^5 - 171278723*x^4 + 3526980655*x^3 + 9233082947*x^2 - 12188156343*x - 5403811843);
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 + 511566*x^7 + 810612*x^6 - 74117109*x^5 - 171278723*x^4 + 3526980655*x^3 + 9233082947*x^2 - 12188156343*x - 5403811843);
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
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 | ${\href{/padicField/29.11.0.1}{11} }$ | ${\href{/padicField/31.11.0.1}{11} }$ | ${\href{/padicField/37.11.0.1}{11} }$ | ${\href{/padicField/41.1.0.1}{1} }^{11}$ | ${\href{/padicField/43.11.0.1}{11} }$ | ${\href{/padicField/47.11.0.1}{11} }$ | ${\href{/padicField/53.11.0.1}{11} }$ | ${\href{/padicField/59.11.0.1}{11} }$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(11\)
| 11.11.20.10 | $x^{11} + 110 x^{10} + 132$ | $11$ | $1$ | $20$ | $C_{11}$ | $[2]$ |
|
\(23\)
| 23.11.10.7 | $x^{11} + 138$ | $11$ | $1$ | $10$ | $C_{11}$ | $[\ ]_{11}$ |